U.S. patent application number 17/135061 was filed with the patent office on 2022-03-31 for encoding method and device, and apparatus.
The applicant listed for this patent is HUAWEI TECHNOLOGIES CO., LTD.. Invention is credited to Ying Chen, Rong Li, Xiaocheng Liu.
Application Number | 20220103189 17/135061 |
Document ID | / |
Family ID | 1000006208027 |
Filed Date | 2022-03-31 |
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United States Patent
Application |
20220103189 |
Kind Code |
A9 |
Liu; Xiaocheng ; et
al. |
March 31, 2022 |
ENCODING METHOD AND DEVICE, AND APPARATUS
Abstract
The present disclosure relates to encoding method and devices.
One example method includes determining N to-be-encoded bits, where
the N to-be-encoded bits include information bits and frozen bits,
obtaining a first polarization weight vector including polarization
weights of N polarized channels, where the N to-be-encoded bits
correspond to the N polarized channels, determining positions of
the information bits based on the first polarization weight vector,
and performing polar encoding on the N to-be-encoded bits to obtain
polar-encoded bits.
Inventors: |
Liu; Xiaocheng; (Hangzhou,
CN) ; Chen; Ying; (Hangzhou, CN) ; Li;
Rong; (Hangzhou, CN) |
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Applicant: |
Name |
City |
State |
Country |
Type |
HUAWEI TECHNOLOGIES CO., LTD. |
Shenzhen |
|
CN |
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Prior
Publication: |
|
Document Identifier |
Publication Date |
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US 20210194504 A1 |
June 24, 2021 |
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Family ID: |
1000006208027 |
Appl. No.: |
17/135061 |
Filed: |
December 28, 2020 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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16261220 |
Jan 29, 2019 |
10879932 |
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17135061 |
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PCT/CN2017/090359 |
Jun 27, 2017 |
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16261220 |
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Current U.S.
Class: |
1/1 |
Current CPC
Class: |
H03M 13/6362 20130101;
H04L 1/0057 20130101; H03M 13/2906 20130101; H03M 13/13 20130101;
H03M 13/616 20130101; H04L 1/0009 20130101; H03M 13/033
20130101 |
International
Class: |
H03M 13/13 20060101
H03M013/13; H04L 1/00 20060101 H04L001/00; H03M 13/03 20060101
H03M013/03; H03M 13/29 20060101 H03M013/29; H03M 13/00 20060101
H03M013/00 |
Foreign Application Data
Date |
Code |
Application Number |
Jul 29, 2016 |
CN |
201610619696.5 |
Jun 13, 2017 |
CN |
PCT/CN2017/088023 |
Claims
1. A communications method, comprising: determining, by a
communications apparatus, N bits to be encoded based on a target
code length M, wherein the N bits comprise a plurality of frozen
bits and K information bits, and wherein N is greater than or equal
to M; determining, by the communications apparatus in the N bits,
N-M positions for deleting N-M bits corresponding to the N-M
positions; determining, by the communications apparatus in the N
bits, K positions for allocating the K information bits based on a
vector, wherein the vector comprises sequence numbers of N
polarized channels corresponding to the N bits, and wherein the
sequence numbers of the N polarized channels are sorted in an
ascending order in the vector according to reliabilities of the N
polarized channels; performing, by the communications apparatus,
polar encoding on the N bits to obtain polar-encoded bits; and
performing, by the communications apparatus, rate matching on the
polar-encoded bits by deleting the N-M bits corresponding to the
N-M positions.
2. The method according to claim 1, wherein N=2.sup..left
brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where the symbol
.left brkt-bot. .right brkt-bot. denotes a flooring function.
3. (canceled)
4. The method according to claim 1, wherein determining the K
positions for allocating the K information bits comprises:
determining K positions corresponding to K most reliable polarized
channels of the N polarized channels for allocating the K
information bits.
5. The method according to claim 1, wherein reliability for each of
the N polarized channels, represented by Wi, is calculated as: W i
= j = 0 n - 1 .times. B j * ( 2 j + .PHI. ) .alpha. , ##EQU00024##
where iB.sub.n-1B.sub.n-2 . . . B.sub.0, i represents a channel
index number, B.sub.n-1B.sub.n-2 . . . B.sub.0 represents a binary
representation of i, B.sub.n-1 represents a most significant bit,
B.sub.0 represents a least significant bit, B.sub.j.di-elect
cons.{0,1}, j.di-elect cons.{0,1, . . . ,n-1}, i.di-elect
cons.{0,1, . . . ,N-1}, N=2.sup.n, n is a positive integer, and
.PHI. and .alpha. are constants.
6. The method according to claim 5, wherein .PHI.=0 and
.alpha.=1/4.
7. A communications apparatus, comprising: at least one processor;
and one or more memories coupled to the at least one processor and
storing programming instructions for execution by the at least one
processor to perform operations comprising: determining N bits to
be encoded based on a target code length M, wherein the N bits
comprise a plurality of frozen bits and K information bits, and
wherein N is greater than or equal to M; determining, in the N
bits, N-M positions for deleting N-M bits corresponding to the N-M
positions; determining, in the N bits, K positions for allocating
the K information bits based on a vector, wherein the vector
comprises sequence numbers of N polarized channels corresponding to
the N bits, and wherein the sequence numbers of the N polarized
channels are sorted in an ascending order in the vector according
to reliabilities of the N polarized channels; performing polar
encoding on the N bits to obtain polar-encoded bits; and performing
rate matching on the polar-encoded bits by deleting the N-M bits
corresponding to the N-M positions.
8. The communications apparatus according to claim 7, wherein
N=2.sup..left brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where
the symbol .left brkt-bot. .right brkt-bot. denotes a flooring
function.
9. (canceled)
10. The communications apparatus according to claim 7, wherein
determining the K positions for allocating the K information bits
comprises: determining K positions corresponding to K most reliable
polarized channels of the N polarized channels for allocating the K
information bits.
11. The communications apparatus according to claim 7, wherein
reliability for each of the N polarized channels, represented by
Wi, is calculated as: W i = j = 0 n - 1 .times. B j * ( 2 j + .PHI.
) .alpha. , ##EQU00025## where iB.sub.n-1B.sub.n-2 . . . B.sub.0, i
represents a channel index number, B.sub.n-1B.sub.n-2 . . . B.sub.0
represents a binary representation of i, B.sub.n-1 represents a
most significant bit, B.sub.0 represents a least significant bit,
B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1, . . . ,n-1},
i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n, n is a positive
integer, and .PHI. and .alpha. are constants.
12. The communications apparatus according to claim 11, wherein
.PHI.=0 and .alpha.=1/4.
13. A computer-readable storage medium storing one or more
computer-executable instructions executable by at least one
processor to perform operations comprising: determining N bits to
be encoded based on a target code length M, wherein the N bits
comprise a plurality of frozen bits and K information bits, and
wherein N is greater than or equal to M; determining, in the N
bits, N-M positions for deleting N-M bits corresponding to the N-M
positions; determining, in the N bits, K positions for allocating
the K information bits based on a vector, wherein the vector
comprises sequence numbers of N polarized channels corresponding to
the N bits, and wherein the sequence numbers of the N polarized
channels are sorted in an ascending order in the vector according
to reliabilities of the N polarized channels; performing polar
encoding on the N bits to obtain polar-encoded bits; and performing
rate matching on the polar-encoded bits by deleting the N-M bits
corresponding to the N-M positions.
14. The computer-readable storage medium according to claim 13,
wherein N=2.sup..left brkt-bot.log.sup.2.sup.(M-1).right
brkt-bot.+1, where the symbol .left brkt-bot. .right brkt-bot.
denotes a flooring function.
15. (canceled)
16. The computer-readable storage medium according to claim 13,
wherein determining the K positions for allocating the K
information bits comprises: determining K positions corresponding
to K most reliable polarized channels of the N polarized channels
for allocating the K information bits.
17. The computer-readable storage medium according to claim 13,
wherein reliability for each of the N polarized channels,
represented by Wi, is calculated as: W i = j = 0 n - 1 .times. B j
* ( 2 j + .PHI. ) .alpha. , ##EQU00026## where iB.sub.n-1B.sub.n-2
. . . B.sub.0, i represents a channel index number,
B.sub.n-1B.sub.n-2 . . . B.sub.0 represents a binary representation
of i, B.sub.n-1 represents a most significant bit, B.sub.0
represents a least significant bit, B.sub.j.di-elect cons.{0,1},
j.di-elect cons.{0,1, . . . ,n-1}, i.di-elect cons.{0,1, . . .
,N-1}, N=2.sup.n, n is a positive integer, and .PHI. and .alpha.
are constants.
18. The computer-readable storage medium according to claim 13,
wherein .PHI.=0 and .alpha.=1/4.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of U.S. patent
application Ser. No. 16/261,220, filed on Jan. 29, 2019, which is a
continuation of International Application No. PCT/CN2017/090359,
filed on Jun. 27, 2017, which claims priority to International
Application No. PCT/CN2017/088023, filed on Jun. 13, 2017 and
Chinese Patent Application No. 201610619696.5, filed on Jul. 29,
2016. All of the afore-mentioned patent applications are hereby
incorporated by reference in their entireties.
TECHNICAL FIELD
[0002] This application relates to the field of encoding and
decoding technologies, and in particular, to an encoding method and
device.
BACKGROUND
[0003] Channel encoding is a radio access technology used in a
communications system to improve data transmission reliability and
ensure communication quality. A polar (polar) code is an encoding
scheme that has low encoding and decoding complexity and that
theoretically proves that a Shannon limit can be reached. The polar
code is a linear block code whose generator matrix is G.sub.N and
whose encoding process is x.sub.1.sup.N=u.sub.1.sup.NG.sub.N, where
x.sub.1.sup.N is an encoded bit and u.sub.1.sup.N is a
to-be-encoded bit; u.sub.1.sup.N=(u.sub.1, u.sub.2, . . . ,
u.sub.N) is a binary row vector whose length is N (that is, a code
length); and G.sub.N is an N.times.N matrix, and
G N = B N .times. F 2 ( lo .times. g 2 .function. ( N ) ) . .times.
F 2 = [ 1 0 1 1 ] , ##EQU00001##
and B.sub.N is an N.times.N transposed matrix, for example, a bit
reversal (Bit Reversal) matrix. F.sub.2.sup.(log.sup.2.sup.(N)) is
defined as a Kronecker (Kronecker) product of log.sub.2 N matrices
F.sub.2. In a polar code encoding process, positions of information
bits and those of frozen bits in u.sub.1.sup.N need to be first
determined. A bit used to carry information is referred to as an
information bit, and a remaining bit that is set to a fixed value
pre-agreed on by a transmit end and a receive end is referred to as
a frozen bit. Then, encoding is performed based on
x.sub.1.sup.N=u.sub.1.sup.NG.sub.N. Specifically, polarized channel
reliability corresponding to each input bit may be estimated in
different manners, and a polarized channel with high reliability is
preferentially selected to place an information bit. In actual
application, positions of information bits needs to be indicated by
using a sequence, and positions of information bits may keep the
same between a transmit end and a receive end in an online
calculation manner or an offline storage manner.
[0004] In the prior art, there are three methods for estimating
polarized channel reliability. One method is to calculate a
Bhattacharyya (Bhattacharyya) parameter of each polarized channel,
where the parameter reflects an error probability of the polarized
channel, and then to select polarized channels corresponding to K
smallest Bhattacharyya parameters to place information bits.
However, this method is applicable to only a binary erasure
channel, and cannot accurately perform reliability estimation on
other channels, resulting in low performance. The other two methods
are a density evolution (Density evolution, DE) method and a
Gaussian approximation (Gaussian approximation, GA) method,
respectively. The DE method and the GA method have relatively high
calculation complexity and are unsuitable for online calculation.
For offline storage, the DE method and the GA method depend on a
parameter rate matching manner, a code rate, and a modulation
scheme of an actual channel during reliability calculation. A
change in any one of the foregoing parameters results in a change
in a reliability estimation result of the polarized channel and a
change in a position of a corresponding information bit. This
consequently causes excessively high storage overheads.
SUMMARY
[0005] This application provides an encoding method and device, to
reduce calculation complexity and storage complexity of polar code
encoding.
[0006] According to a first aspect, this application provides an
encoding method, including:
[0007] after input information bits are received, determining a
quantity N of to-be-encoded bits based on a target code length M of
a polar code, where N is a positive integer; afterwards obtaining a
first polarization weight vector including polarization weights of
N polarized channels, where the polarization weights are a basis
for information bit selection; then determining positions of the
information bits based on the first polarization weight vector; and
finally performing polar encoding on the N to-be-encoded bits to
obtain polar-encoded bits. Therefore, positions of the information
bits and those of frozen bits are determined by calculating
polarization weights of polar-code polarized channels, without
relating to a channel parameter and a code rate. This can reduce
calculation complexity of polar code encoding. In addition,
polarization weights corresponding to polar codes with different
code lengths and code rates have a same part, so that storage space
is greatly reduced, and storage complexity of polar code encoding
can also be reduced.
[0008] In a possible design, the target code length M of the polar
code is equal to a positive integer power of 2, M is a positive
integer, and the determining N to-be-encoded bits includes:
determining that N is equal to M, where the target code length of
the polar code is an output code length of the polar code; and the
obtaining a first polarization weight vector including polarization
weights of N polarized channels includes: calculating the
polarization weights of the N polarized channels to obtain the
first polarization weight vector.
[0009] In a possible design, the calculating the polarization
weights of the N polarized channels to obtain the first
polarization weight vector includes: calculating the polarization
weights W.sub.i of the N polarized channels according to the
following formula, to obtain the first polarization weight
vector:
W i = j = 0 n - 1 .times. B j * ( 2 j + .PHI. ) .alpha. ,
##EQU00002##
where
[0010] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n, n is a
positive integer, and .PHI. and .alpha. are parameters that are
preset based on the target code length and a code rate of the polar
code.
[0011] In a possible design, the calculating the polarization
weights of the N polarized channels to obtain the first
polarization weight vector includes: obtaining the first
polarization weight vector W.sub.0.sup.N-1 through recursion
calculation according to the following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+.beta..s-
ub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0012] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, and W.sub.0 is an
arbitrary constant.
[0013] In a possible design, the target code length M of the polar
code is equal to a positive integer power of 2, and the obtaining a
first polarization weight vector including polarization weights of
N polarized channels includes: obtaining the first polarization
weight vector from a prestored second polarization weight vector,
where the second polarization weight vector includes polarization
weights of N.sub.max polarized channels, N.sub.max is a preset
mother code length corresponding to a maximum code length supported
by a communications system, and the mother code length is a
positive integer power of 2.
[0014] After the input information bits are received, the quantity
N of to-be-encoded bits is determined based on the target code
length M of the polar code; afterwards, the first polarization
weight vector is obtained from the prestored second polarization
weight vector, where the second polarization weight vector includes
the polarization weights of the N.sub.max polarized channels, and
N.sub.max is the mother code length corresponding to the maximum
code length supported by the communications system; then, positions
of the information bits are determined based on the first
polarization weight vector; and finally, polar encoding is
performed on the N to-be-encoded bits to obtain the polar-encoded
bits. Therefore, the position of the information bits and those of
the frozen bits are determined by obtaining polarization weights of
polar-code polarized channels in an offline storage manner. For
polar codes with different code lengths and code rates, an element
set in a polarization weight vector corresponding to a short-code
length is a subset of an element set in a weight vector
corresponding to a long-code length, and polarized channels with a
same index number (sequence number) have a same polarization weight
value. Therefore, storage space can be reduced, and storage
complexity of polar code encoding can be reduced. Compared with
online calculation, when there is enough storage space, the offline
storage manner can shorten latency caused by online
calculation.
[0015] In a possible design, when the target code length M of the
polar code is less than or equal to N.sub.max, an element set in
the first polarization weight vector is a subset of an element set
in the second polarization weight vector; or when the target code
length M of the polar code is greater than or equal to N.sub.max, a
polarization weight of a polarized channel whose channel index
number is less than or equal to N.sub.max is in a one-to-one
correspondence with a polarization weight in the second
polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00003##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max,
where .PHI. and .alpha. are parameters that are preset based on the
target code length and a code rate of the polar code.
[0016] After the input information bits are received, the quantity
N of to-be-encoded bits is determined based on the target code
length M of the polar code; afterwards, the first polarization
weight vector is obtained from the prestored second polarization
weight vector, where the second polarization weight vector includes
the polarization weights of the N.sub.max polarized channels, and
N.sub.max is the maximum mother code length supported by the
communications system; then, positions of the information bits are
determined based on the first polarization weight vector; and
finally, polar encoding is performed on the N to-be-encoded bits to
obtain the polar-encoded bits. Therefore, the positions of the
information bits and those of the frozen bits are determined by
obtaining polarization weights of polar-code polarized channels in
a half-online-calculation half-offline-storage manner. For polar
codes with different code lengths and code rates, an element set in
a polar-code polarization weight vector corresponding to a code
length less than prestored N.sub.max is a subset of an element set
in a weight vector corresponding to N.sub.max, and polarized
channels with a same index number (sequence number) have a same
polarization weight value; and in a polar-code polarization weight
vector corresponding to a code length greater than prestored
N.sub.max, a polarization weight of a polarized channel whose
channel index number is less than or equal to N.sub.max is in a
one-to-one correspondence with the polarization weight in the
second polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00004##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max.
Therefore, storage space can be reduced, and storage complexity of
polar code encoding can be reduced. This manner can not only
shorten latency caused by online calculation but also reduce
storage complexity.
[0017] In a possible design, .PHI.=0, and .alpha.=1/4.
[0018] In a possible design, the target code length M of the polar
code is not equal to a positive integer power of 2, and the
determining N to-be-encoded bits includes: determining that
N=2.sup..left brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where
the symbol .left brkt-bot. .right brkt-bot. denotes rounding
down.
[0019] In a possible design, before the obtaining a first
polarization weight vector including polarization weights of N
polarized channels, the method further includes: determining
positions of NM to-be-deleted bits in the N to-be-encoded bits, and
calculating, based on a result of the determining, a set of
modification coefficients including the polarization weights
corresponding to the N to-be-encoded bits, where the modification
coefficient is used to represent an amount of modification to each
polarization weight resulting from bit deletion; and the obtaining
a first polarization weight vector including polarization weights
of N polarized channels includes: calculating the polarization
weights of the N polarized channels based on the set of
modification coefficients, to obtain the first polarization weight
vector.
[0020] In a possible design, the determining positions of NM
to-be-deleted bits in the N to-be-coded bits includes: determining
a channel index number sequence P.sub.0.sup.N-1=[0,1, . . . ,N-1]
of the N polarized channels; and performing the following
operations on each index number in the channel index number
sequence: performing bit reversal on a binary number corresponding
to each index number, and converting, to a decimal number, a binary
number obtained after bit reversal, to obtain a new channel index
number sequence P.sub.0.sup.N-1'; sorting index numbers in the new
channel index number sequence P.sub.0.sup.N-1'; and selecting, in
ascending order, NM index numbers in the channel index number
sequence P.sub.0.sup.N-1' as index numbers of the to-be-deleted
bits.
[0021] It should be noted that in this step, an operation performed
on each index number is not limited to a bit reversal operation,
for example, interleaving, scrambling, or the like may be performed
on the binary number corresponding to each index number, or no
processing is performed on each index number.
[0022] In a possible design,
[0023] the set of modification coefficients is
g = [ g ( 0 , 0 ) g ( 0 , n - 1 ) g ( 1 , 0 ) g ( 1 , n - 1 ) g ( N
- 1 , 0 ) g ( N - 1 , n - 1 ) g ( N , 0 ) g ( N , 1 ) g ( N , n - 1
) ] ; ##EQU00005##
and
[0024] the calculating, based on a result of the determining, a set
of modification coefficients including modification coefficients of
the N bits includes: calculating each item g.sub.(i,j) in the set g
of modification coefficients according to the following
formulas:
g.sub.(i,j)=(p.sub.i+p.sub.i+2.sub.j-2)/2; and
g.sub.(i+2.sub.j.sub.,j)=g.sub.(i,j), where
[0025] i.di-elect cons.{0,1, . . . ,N-1}, j.di-elect cons.{0,1, . .
. ,n-1}, n=2.sup.n, p.sub.i is an identifier indicating whether a
bit corresponding to a polarized channel whose channel index number
is i is to be deleted, and p.sub.i.di-elect cons.{0,1}.
[0026] In a possible design, the calculating the polarization
weights of the N polarized channels based on the set of
modification coefficients, to obtain the first polarization weight
vector includes: calculating the polarization weights W.sub.i of
the N polarized channels according to the following formula, to
obtain the first polarization weight vector:
W i = n - 1 j = 0 .times. B j * ( 1 - g ( i , j ) ) * ( 2 j + .PHI.
) .alpha. , ##EQU00006##
where
[0027] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n,
g.sub.(i,j) is an element in row i and column j in the set g of
modification coefficients, and .PHI. and .alpha. are parameters
that are preset based on the target code length and a code rate of
the polar code.
[0028] In a possible design, the calculating the polarization
weights of the N polarized channels based on the set of
modification coefficients, to obtain the first polarization weight
vector includes: obtaining the first polarization weight vector
W.sub.0.sup.N-1 through recursion calculation according to the
following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+g.sub.(i-
,j)*.beta..sub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0029] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, g.sub.(i,j) is an element
in row i and column j in the set g of modification coefficients,
and W.sub.0 is an arbitrary constant.
[0030] In the foregoing possible design, after the input
information bits are received, the quantity N of to-be-encoded bits
is determined based on the target code length M of the polar code;
afterwards, the positions of the NM to-be-deleted bits in the N
to-be-encoded bits are determined, and the set of modification
coefficients including the modification coefficients of the N
to-be-encoded bits is calculated based on the result of the
determining; then, the polarization weights of the N polarized
channels are calculated based on the set of modification
coefficients, to obtain the first polarization weight vector; and
finally, positions of the information bits are determined based on
the first polarization weight vector, and polar encoding is
performed on the N to-be-encoded bits to obtain the polar-encoded
bits. Therefore, for a polar code with any target code length,
positions of information bits and those of frozen bits are
determined by calculating polarization weights of polar-code
polarized channels, without relating to a channel parameter and a
code rate. This can reduce calculation and implementation
complexity of polar code encoding.
[0031] In a possible design, .PHI.=0, and .alpha.=1/4.
[0032] In a possible design, the determining positions of the
information bits based on the first polarization weight vector
includes: sorting the polarization weights in the first
polarization weight vector in descending order, and selecting bits
corresponding to the first K polarization weights as information
bits.
[0033] In a possible design, the encoding method is executed by a
base station or a terminal.
[0034] According to a second aspect, this application provides an
encoding device, including:
[0035] a first determining module, configured to determine N
to-be-encoded bits, where the N to-be-encoded bits include
information bits and frozen bits, and N is a positive integer; an
obtaining module, configured to obtain a first polarization weight
vector including polarization weights of N polarized channels,
where the polarization weights are a basis for information bit
selection, and the N to-be-encoded bits are corresponding to the N
polarized channels; a second determining module, configured to
determine positions of the information bits based on the first
polarization weight vector; and an encoding module, configured to
perform polar encoding on the N to-be-encoded bits to obtain
polar-encoded bits.
[0036] In a possible design, a target code length M of a polar code
is equal to a positive integer power of 2, M is a positive integer,
and the first determining module is specifically configured to
determine that N is equal to M, where the target code length of the
polar code is an output code length of the polar code; and the
obtaining module is specifically configured to calculate the
polarization weights of the N polarized channels to obtain the
first polarization weight vector.
[0037] In a possible design, the obtaining module is specifically
configured to calculate the polarization weights W.sub.i of the N
polarized channels according to the following formula, to obtain
the first polarization weight vector:
W i = n - 1 j = 0 .times. B j * ( 2 j + .PHI. ) .alpha. ,
##EQU00007##
where
[0038] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n, n is a
positive integer, and .PHI. and .alpha. are parameters that are
preset based on the target code length and a code rate of the polar
code.
[0039] In a possible design, the obtaining module is specifically
configured to obtain the first polarization weight vector
W.sub.0.sup.N-1 through recursion calculation according to the
following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+.beta..s-
ub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0040] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, and W.sub.0 is an
arbitrary constant.
[0041] In a possible design, a target code length M of a polar code
is equal to a positive integer power of 2, and the obtaining module
is specifically configured to obtain the first polarization weight
vector from a prestored second polarization weight vector, where
the second polarization weight vector includes polarization weights
of N.sub.max polarized channels, N.sub.max is a preset mother code
length corresponding to a maximum code length supported by a
communications system, and the mother code length is a positive
integer power of 2.
[0042] In a possible design, when the target code length M of the
polar code is less than or equal to N.sub.max, an element set in
the first polarization weight vector is a subset of an element set
in the second polarization weight vector; or when the target code
length M of the polar code is greater than or equal to N.sub.max, a
polarization weight of a polarized channel whose channel index
number is less than or equal to N.sub.max is in a one-to-one
correspondence with a polarization weight in the second
polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00008##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max,
where .PHI. and .alpha. are parameters that are preset based on the
target code length and a code rate of the polar code.
[0043] In a possible design, .PHI.=0, and .alpha.=1/4.
[0044] In a possible design, a target code length M of a polar code
is not equal to a positive integer power of 2, and the first
determining module is specifically configured to determine that
N=2.sup..left brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where
the symbol .left brkt-bot. .right brkt-bot. denotes rounding
down.
[0045] In a possible design, the encoding device further includes:
a third determining module, configured to: before the obtaining
module obtains the first polarization weight vector including the
polarization weights of the N polarized channels, determine
positions of NM to-be-deleted bits in the N to-be-encoded bits; and
a calculation module, configured to calculate, based on a result of
the determining performed by the third determining module, a set of
modification coefficients including the polarization weights
corresponding to the N to-be-encoded bits, where the modification
coefficient is used to represent an amount of modification to each
polarization weight resulting from bit deletion, where the
obtaining module is specifically configured to calculate the
polarization weights of the N polarized channels based on the set
of modification coefficients, to obtain the first polarization
weight vector.
[0046] In a possible design, the third determining module is
specifically configured to: determine a channel index number
sequence P.sub.0.sup.N-1=[0,1, . . . ,N-1] of the N polarized
channels; and perform the following operations on each index number
in the channel index number sequence: performing bit reversal on a
binary number corresponding to each index number, and converting,
to a decimal number, a binary number obtained after bit reversal,
to obtain a new channel index number sequence P.sub.0.sup.N-1';
sorting index numbers in the new channel index number sequence
P.sub.0.sup.N-1'; and selecting, in ascending order, NM index
numbers in the channel index number sequence P.sub.0.sup.N-1' as
index numbers of the to-be-deleted bits.
[0047] In a possible design, the set of modification coefficients
is
g = [ g ( 0 , 0 ) g ( 0 , n - 1 ) g ( 1 , 0 ) g ( 1 , n - 1 ) g ( N
- 1 , 0 ) g ( N - 1 , n - 1 ) g ( N , 0 ) g ( N , 1 ) g ( N , n - 1
) ] ; ##EQU00009##
and the calculation module is specifically configured to calculate
each item g.sub.(i,j) in the set g of modification coefficients
according to the following formulas:
g.sub.(i,j)=(p.sub.i+p.sub.i+2.sub.j-2)/2; and
g.sub.(i+2.sub.j.sub.,j)=g.sub.(i,j), where
[0048] i.di-elect cons.{0,1, . . . ,N-1}, j.di-elect cons.{0,1, . .
. ,n-1}, n=2.sup.n, p.sub.i is an identifier indicating whether a
bit corresponding to a polarized channel whose channel index number
is i is to be deleted, and p.sub.i.di-elect cons.{0,1}.
[0049] In a possible design, the obtaining module is specifically
configured to calculate the polarization weights W.sub.i of the N
polarized channels according to the following formula, to obtain
the first polarization weight vector:
W i = n - 1 j = 0 .times. B j * ( 1 - g ( i , j ) ) * ( 2 j + .PHI.
) .alpha. , ##EQU00010##
where
[0050] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n,
g.sub.(i,j) is an element in row i and column j in the set g of
modification coefficients, and .PHI. and .alpha. are parameters
that are preset based on the target code length and a code rate of
the polar code.
[0051] In a possible design, the obtaining module is specifically
configured to obtain the first polarization weight vector
W.sub.0.sup.N-1 through recursion calculation according to the
following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+g.sub.(i-
,j)*.beta..sub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0052] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, g.sub.(i,j) is an element
in row i and column j in the set g of modification coefficients,
and W.sub.0 is an arbitrary constant.
[0053] In a possible design, .PHI.=0, and .alpha.=1/4.
[0054] In a possible design, the second determining module is
specifically configured to: sort the polarization weights in the
first polarization weight vector in descending order, and select
bits corresponding to the first K polarization weights as
information bits.
[0055] In a possible design, the encoding device is a base station
or a terminal.
[0056] For beneficial effects of the encoding device provided in
the second aspect and all the possible designs of the second
aspect, refer to beneficial effects brought by the first aspect and
all the possible designs of the first aspect, and details are not
repeated herein.
[0057] According to a third aspect, this application provides a
communications apparatus, including:
[0058] a memory, configured to store an instruction for performing
an operation; and
[0059] a processor, coupled with the memory, and according to the
instruction, configured to: determine N to-be-encoded bits, where
the N to-be-encoded bits include information bits and frozen bits,
and N is a positive integer; obtain a first polarization weight
vector including polarization weights of N polarized channels,
where the polarization weights are a basis for information bit
selection, and the N to-be-encoded bits are corresponding to the N
polarized channels; determine positions of the information bits
based on the first polarization weight vector; and perform polar
encoding on the N to-be-encoded bits to obtain polar-encoded
bits.
[0060] In a possible design, the communications apparatus is a base
station or a terminal.
[0061] According to a fourth aspect, this application provides a
computer storage medium, where the computer storage medium stores a
computer-executable instruction, and when the computer-executable
instruction is invoked by a computer, the computer executes the
method according to all the foregoing embodiments.
BRIEF DESCRIPTION OF DRAWINGS
[0062] FIG. 1 is a schematic diagram of an existing polar code
encoding process;
[0063] FIG. 2 is a block diagram of an encoding structure in
Embodiment 1 of an encoding method according to this
application;
[0064] FIG. 3 is a flowchart of Embodiment 1 of an encoding method
according to this application;
[0065] FIG. 4 is a flowchart of Embodiment 2 of an encoding method
according to this application;
[0066] FIG. 5 is a schematic diagram of polarization weight vectors
respectively corresponding to polar codes with code lengths N being
4, 8, and 16 according to Embodiment 2 of this application;
[0067] FIG. 6 is a schematic diagram of polarization weight vectors
respectively corresponding to polar codes with code lengths N being
4, 8, and 16 according to Embodiment 3 of this application;
[0068] FIG. 7 is a flowchart of Embodiment 4 of an encoding method
according to this application;
[0069] FIG. 8 is a schematic structural diagram of Embodiment 1 of
an encoding device according to this application;
[0070] FIG. 9 is a schematic structural diagram of Embodiment 2 of
an encoding device according to this application; and
[0071] FIG. 10 is a schematic structural diagram of an embodiment
of a communications apparatus according to this application.
DESCRIPTION OF EMBODIMENTS
[0072] FIG. 1 is a schematic diagram of an existing polar code
encoding process. Polar code encoding is performed by a transmit
end. As shown in FIG. 1, a polar code encoding process includes
three processing processes: construction, encoding, and rate
matching. First, in the construction process, after input data
(namely, input information bits, for example, N.sub.1 information
bits) is received, a quantity N of to-be-encoded bits needs to be
determined based on a target code length M obtained after encoding,
where the to-be-encoded bits include information bits (whose
quantity is N.sub.1) and frozen bits (whose quantity is N.sub.2),
and N=N.sub.1+N.sub.2; and positions of the information bits and
those of the frozen bits need to be determined, where one bit is
corresponding to one polarized channel. Then, in the encoding
process, polar encoding is performed on the to-be-encoded bits to
obtain polar-encoded bits. Finally, in the rate matching process,
because a code length obtained after polar code encoding is a
positive integer power of 2, and during actual implementation, the
code length needs to be flexibly configured, rate matching needs to
be performed to obtain a code with any code length. A specific
implementation is deleting some NM bits from a code with the code
length of a positive integer power of 2, to obtain a code with a
target code length. N, M, N.sub.1, and N.sub.2 are all positive
integers. An encoding method provided in this application mainly
relates to the polar code encoding process shown in FIG. 1, and
mainly relates to the construction process. In an existing
construction process, in the prior art, channel reliability
estimation is performed mostly based on actual parameters and code
rates of all polarized channels, so as to determine positions of
information bits, and remaining positions are positions of frozen
bits. In actual application, the positions of the information bits
need to be indicated by using a sequence, and positions of
information bits may keep the same in an online calculation manner
or an offline storage manner. During offline storage, a quite large
table is required to store the sequence, and due to different
channel parameters, code rates, and code lengths, millions of
sequences need to be stored. Therefore, in an existing manner of
determining positions of information bits, both online calculation
and offline storage are relatively complex. In view of the
foregoing problem, this application provides an encoding method.
Positions of information bits and those of frozen bits are
determined by calculating polarization weights of polar-code
polarized channels, without relating to a channel parameter and a
code rate. This can reduce calculation complexity and storage
complexity of polar code encoding. The following describes in
detail the encoding method and device provided in this application
with reference to the accompanying drawings.
[0073] This application is mainly applied to various wireless
communications systems, and acts on digital signal processing units
in the systems, to improve reliability of wireless
communication.
[0074] Network elements in this application are mainly a base
station and a terminal. The base station implements communication
with the terminal.
[0075] FIG. 2 is a block diagram of an encoding structure in
Embodiment 1 of an encoding method according to this application.
As shown in FIG. 2, both information bits (whose quantity is
N.sub.1) and a target code length M of a polar code are inputs, and
then polar code construction is performed. N to-be-encoded bits are
first determined based on the target code length M of the polar
code. A quantity of frozen bits is N.sub.2, and N=N.sub.1+N.sub.2.
Positions of the information bits and those of the frozen bits are
then determined. Then, encoding is performed by using an encoding
matrix G.sub.N, to obtain polar-encoded bits.
[0076] FIG. 3 is a flowchart of Embodiment 1 of an encoding method
according to this application. As shown in FIG. 3, the method in
this embodiment may include the following steps.
[0077] S101: Determine N to-be-encoded bits, where the N
to-be-encoded bits include information bits and frozen bits.
[0078] Specifically, a code length obtained after polar code
encoding is a positive integer power of 2, and during actual
implementation, the code length needs to be flexibly configured, in
other words, a polar code with any code length needs to be
obtained. Therefore, after input information bits are received, the
quantity N of to-be-encoded bits needs to be determined based on a
target code length M of a polar code. The N to-be-encoded bits
include the information bits and the frozen bits. The target code
length of the polar code is an output code length of the polar
code. Specifically, there are two cases. One case is that the
target code length M of the polar code is equal to a positive
integer power of 2, to be specific, there is no need to perform
rate matching. In this case, the determined quantity N of
to-be-encoded bits is equal to M. The other case is that the target
code length M of the polar code is not equal to a positive integer
power of 2, to be specific, rate matching needs to be performed. In
this case, the determined quantity of to-be-encoded bits is
N=2.sup..left brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where
the symbol .left brkt-bot. .right brkt-bot. denotes rounding
down.
[0079] S102: Obtain a first polarization weight vector including
polarization weights of N polarized channels, where the
polarization weights are a basis for information bit selection, and
the N to-be-encoded bits are corresponding to the N polarized
channels.
[0080] Specifically, one bit is corresponding to one polarized
channel, and a polarization weight of the polarized channel is a
basis for information bit selection. The polarization weight is
different from channel reliability in the prior art. For a case in
which the target code length M of the polar code is equal to a
positive integer power of 2 and a case in which the target code
length M of the polar code is not equal to a positive integer power
of 2, manners of obtaining the polarization weights are different.
The following describes in detail how to obtain the first
polarization weight vector when the quantity N of to-be-encoded
bits is equal to the target code length M of the polar code.
Specifically, the polarization weights W.sub.i of the N polarized
channels in the first polarization weight vector are calculated
according to the following formula:
W i = n - 1 j = 0 .times. B j * ( 2 j + .PHI. ) .alpha. ,
##EQU00011##
where
[0081] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n, and
.PHI. and .alpha. are parameters that are preset based on the
target code length and a code rate of the polar code. For polar
codes with different code lengths and code rates, the parameters
.PHI. and .alpha. may be adjusted, to obtain an information bit
indication sequence of the polar code for better performance.
[0082] Optionally, in
W i = j = 0 n - 1 .times. B j * ( 2 j + .PHI. ) .alpha. ,
##EQU00012##
.PHI.=0, .alpha.=1/4, and therefore the calculation formula is
W i = j = 0 n - 1 .times. B j * 2 j 4 . ##EQU00013##
The polarization weight of the polarized channel is calculated
according to the formula, to determine the information bit, to be
specific, to obtain the information bit indication sequence for
better performance. For example, if N=8, n=log.sub.2(8)=3. For i=3
(i011), a process of calculating a polarization weight W.sub.3 is
as follows:
W.sub.3=1.times.2{circumflex over (
)}(0.times.(1/4))+1.times.2{circumflex over (
)}(1.times.(1/4))+0.times.2{circumflex over (
)}(2.times.(1/4))=2.8192.
[0083] It should be noted that corresponding polarization weights
in all the embodiments of this application are corresponding to a
polar-code encoding matrix F.sub.N excluding a bit reversal
operation B.sub.N. If the encoding matrix includes the bit reversal
operation B.sub.N, a corresponding bit reversal operation needs to
be performed on the calculated polarization weights.
[0084] In another implementation, the first polarization weight
vector W.sub.0.sup.N-1 is obtained through recursion calculation
according to the following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1]; and
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+.beta..s-
ub.i; where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0085] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, and W.sub.0 is an
arbitrary constant.
[0086] It can be learned from the foregoing formulas that a
relationship between polarization weights of polar codes in two
adjacent orders may be simply described as W.sup.N=[W.sup.N/2,
W.sup.N/2+.beta..sub.i], where i=log.sub.2(N). When i=1, the value
is substituted into the foregoing recursion calculation formulas,
to obtain initial formulas for recursion calculation:
.beta..sub.1=(1+.PHI.).sup..alpha.;
W.sub.1=W.sub.0+.beta..sub.1; and
W.sub.0.sup.1=[W.sub.0,W.sub.1].
[0087] For example, when W.sub.0=0, a calculation formula
W i = j = 0 n - 1 .times. B j * ( 2 j + .PHI. ) .alpha.
##EQU00014##
may be obtained. When W.sub.0=0, .PHI.=0, and .alpha.=1/4, a
calculation formula
W i = j = 0 n - 1 .times. B j * 2 j 4 ##EQU00015##
may be obtained.
[0088] The following uses N=8 as an example, and a recursion
calculation process is as follows:
[0089] First, a polarization weight of a polarized channel is
initialized as W.sub.0=a, where a is a constant value, and usually
may be set to 0. Then, based on a code length N (N=2.sup.n) of the
polar code, i is set to 1 to n, and the polarization weight of the
polarized channel increases order by order to the target code
length N, and the calculation process is shown as follows:
[0090] (1) Set an initial value: W.sub.0=a.
[0091] (2) i=1, .beta..sub.1=(1+.PHI.).sup..alpha.,
W.sub.1=W.sub.0.beta..sub.1, and W.sub.0.sup.1=[W.sub.0,
W.sub.1]=[W.sub.0, W.sub.0+.beta..sub.1].
[0092] (3) i=2, .beta..sub.2=(2+.PHI.).sup..alpha., and [W.sub.0,
W.sub.1, W.sub.2, W.sub.3]=[W.sub.0, W.sub.1, W.sub.0+.beta..sub.1,
W.sub.1+.beta..sub.2].
[0093] (4) i=3, .beta..sub.3=(4+.PHI.).sup..alpha., and [W.sub.0,
W.sub.1, W.sub.2, W.sub.3, W.sub.4, W.sub.5, W.sub.6,
W.sub.7]=[W.sub.0, W.sub.1, W.sub.2, W.sub.3, W.sub.0+.beta..sub.3,
W.sub.1+.beta..sub.3, W.sub.2+.beta..sub.3,
W.sub.3+.beta..sub.3].
[0094] Finally, W.sub.0.sup.7=[W.sub.0, W.sub.1, W.sub.2, W.sub.3,
W.sub.4, W.sub.5, W.sub.6, W.sub.7] is obtained.
[0095] S103: Determine positions of the information bits based on
the first polarization weight vector.
[0096] Specifically, the polarization weights in the first
polarization weight vector may be sorted in descending order, and
bits corresponding to the first K polarization weights are selected
as information bits. To be specific, positions of the information
bits are selected, and remaining positions are set for the frozen
bits, so as to obtain the complete to-be-encoded bits with a length
of N.
[0097] S104: Perform polar encoding on the N to-be-encoded bits to
obtain polar-encoded bits.
[0098] Specifically, the polar-code encoding matrix F.sub.N is used
to complete an encoding process of the to-be-encoded bits.
[0099] According to the encoding method provided in this
embodiment, after the input information bits are received, the
quantity N of to-be-encoded bits is determined based on the target
code length M of the polar code; afterwards, the first polarization
weight vector including the polarization weights of the N polarized
channels is obtained; then, positions of the information bits are
determined based on the first polarization weight vector; and
finally, polar encoding is performed on the N to-be-encoded bits to
obtain the polar-encoded bits. Therefore, the positions of the
information bits and those of the frozen bits are determined by
calculating polarization weights of polar-code polarized channels,
without relating to a channel parameter and a code rate. This can
reduce calculation complexity of polar code encoding. In addition,
polarization weights corresponding to polar codes with different
code lengths and code rates have a same part, so that storage space
is greatly reduced, and storage complexity of polar code encoding
can also be reduced.
[0100] Still further, it can be learned, according to the
calculation formula of the polarization weights in Embodiment 1
shown in FIG. 3, that a difference between polarization weights of
polar codes in two adjacent orders is a weight increment
.beta..sub.i. In a specific implementation of S102 in Embodiment 1,
the first polarization weight vector is obtained through online
calculation. In another implementation, the first polarization
weight vector may be obtained in an offline storage manner.
Compared with online calculation, when there is enough storage
space, the offline storage manner can shorten latency caused by
online calculation. The following describes in detail the offline
storage manner with reference to FIG. 4. It should be noted that
the offline storage manner is applicable to a scenario in which a
target code length M of a polar code is equal to a positive integer
power of 2.
[0101] FIG. 4 is a flowchart of Embodiment 2 of an encoding method
according to this application. As shown in FIG. 4, the method in
this embodiment may include the following steps.
[0102] S201: Determine N to-be-encoded bits, where the N
to-be-encoded bits include information bits and frozen bits.
[0103] S202: Obtain a first polarization weight vector from a
prestored second polarization weight vector, where an element set
in the first polarization weight vector is a subset of an element
set in the second polarization weight vector, the second
polarization weight vector includes polarization weights of
N.sub.max polarized channels, N.sub.max is a preset mother code
length corresponding to a maximum code length supported by a
communications system, and the mother code length is a positive
integer power of 2.
[0104] Specifically, the mother code length N.sub.max corresponding
to the maximum code length supported by the communications system
may be preset based on an actual communication scenario, where the
mother code length is a positive integer power of 2; and the second
polarization weight vector is calculated in advance by using the
calculation formulas shown in Embodiment 1, and is stored offline.
During usage, the first polarization weight vector is directly
obtained from the prestored second polarization weight vector. It
can be learned from the calculation formulas shown in Embodiment 1
that when a code length is a positive integer power of 2, an
element set in a polarization weight vector corresponding to a
short-code length is a subset of an element set in a weight vector
corresponding to a long-code length, and polarized channels with a
same index number (sequence number) have a same polarization weight
value.
[0105] An example of polar codes with code lengths N being 4, 8,
and 16 is used. FIG. 5 is a schematic diagram of polarization
weight vectors respectively corresponding to polar codes with code
lengths N being 4, 8, and 16 according to Embodiment 2 of this
application. As shown in FIG. 5, an element set in the polarization
weight vector corresponding to the code length N=8 is a subset of
an element set in the polarization weight vector corresponding to
the code length N=16, and an element set in the polarization weight
vector corresponding to the code length N=4 is a subset of an
element set in the polarization weight vector corresponding to the
code length N=8. In addition, polarization weights of polarized
channels are the same in a same position. Therefore, a polarization
weight vector, corresponding to a maximum code length N.sub.max, of
polarized channels is prestored, so that a polarized channel weight
vector corresponding to any code length N less than or equal to
N.sub.max can be obtained.
[0106] S203: Determine positions of the information bits based on
the first polarization weight vector.
[0107] Specifically, the polarization weights in the first
polarization weight vector may be sorted in descending order, and
bits corresponding to the first K polarization weights are selected
as information bits. To be specific, positions of the information
bits are selected, and remaining positions are set for the frozen
bits, so as to obtain the complete to-be-encoded bits with a length
of N.
[0108] S204: Perform polar encoding on the N to-be-encoded bits to
obtain polar-encoded bits.
[0109] Specifically, a polar-code encoding matrix F.sub.N is used
to complete an encoding process of the to-be-encoded bits.
[0110] According to the encoding method provided in this
embodiment, after input information bits are received, the quantity
N of to-be-encoded bits is determined based on the target code
length M of the polar code; afterwards, the first polarization
weight vector is obtained from the prestored second polarization
weight vector, where the second polarization weight vector includes
the polarization weights of the N.sub.max polarized channels, and
N.sub.max is the mother code length corresponding to the maximum
code length supported by the communications system; then, positions
of the information bits are determined based on the first
polarization weight vector; and finally, polar encoding is
performed on the N to-be-encoded bits to obtain the polar-encoded
bits. Therefore, the positions of the information bits and those of
the frozen bits are determined by obtaining polarization weights of
polar-code polarized channels in an offline storage manner. For
polar codes with different code lengths and code rates, an element
set in a polarization weight vector corresponding to a short-code
length is a subset of an element set in a weight vector
corresponding to a long-code length, and polarized channels with a
same index number (sequence number) have a same polarization weight
value. Therefore, storage space can be reduced, and storage
complexity of polar code encoding can be reduced. Compared with
online calculation, when there is enough storage space, the offline
storage manner can shorten latency caused by online
calculation.
[0111] Optionally, in a specific implementation different from that
in S202 in Embodiment 2, the first polarization weight vector is
obtained in the offline storage manner. In another implementation,
the first polarization weight vector may be obtained in a
half-online-calculation half-offline-storage manner. The following
describes in detail the half-online-calculation
half-offline-storage manner in Embodiment 3 of this application. It
should be noted that this manner is applicable to a scenario in
which a target code length M of a polar code is equal to a positive
integer power of 2.
[0112] The obtaining a first polarization weight vector from a
prestored second polarization weight vector in S202 may be
specifically: When the target code length M of the polar code is
less than or equal to N.sub.max, the element set in the first
polarization weight vector is the subset of the element set in the
second polarization weight vector; or when the target code length M
of the polar code is greater than or equal to N.sub.max, a
polarization weight of a polarized channel whose channel index
number is less than or equal to N.sub.max is in a one-to-one
correspondence with a polarization weight in the second
polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00016##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max,
where .PHI. and .alpha. are parameters that are preset based on the
target code length and a code rate of the polar code. For example,
FIG. 6 is a schematic diagram of polarization weight vectors
respectively corresponding to polar codes with code lengths N being
4, 8, and 16 according to Embodiment 3 of this application. In this
embodiment, N.sub.max=8. As shown in FIG. 6, polarization weights
stored in the second polarization weight vector are W0 to W7.
Polarization weights corresponding to code lengths less than
N.sub.max=8 are extracted from the second polarization weight
vector, and polarization weights corresponding to index numbers 8
to 15 in a polarization weight vector corresponding to a code
length 16 are obtained by adding a weight increment
(8+.PHI.).sup..alpha. to the polarization weights corresponding to
index numbers 0 to 7.
[0113] Optionally, in the foregoing embodiment, .PHI.=0, and
.alpha.=1/4.
[0114] According to the encoding method provided in this
embodiment, after input information bits are received, the quantity
N of to-be-encoded bits is determined based on the target code
length M of the polar code; afterwards, the first polarization
weight vector is obtained from the prestored second polarization
weight vector, where the second polarization weight vector includes
the polarization weights of the N.sub.max polarized channels, and
N.sub.max is the maximum mother code length supported by the
communications system; then, positions of the information bits are
determined based on the first polarization weight vector; and
finally, polar encoding is performed on the N to-be-encoded bits to
obtain the polar-encoded bits. Therefore, the positions of the
information bits and those of the frozen bits are determined by
obtaining polarization weights of polar-code polarized channels in
the half-online-calculation half-offline-storage manner. For polar
codes with different code lengths and code rates, an element set in
a polar-code polarization weight vector corresponding to a code
length less than prestored N.sub.max is a subset of an element set
in a weight vector corresponding to N.sub.max, and polarized
channels with a same index number (sequence number) have a same
polarization weight value; and in a polar-code polarization weight
vector corresponding to a code length greater than prestored
N.sub.max, a polarization weight of a polarized channel whose
channel index number is less than or equal to N.sub.max is in a
one-to-one correspondence with the polarization weight in the
second polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00017##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max.
Therefore, storage space can be reduced, and storage complexity of
polar code encoding can be reduced. This manner can not only
shorten latency caused by online calculation but also reduce
storage complexity.
[0115] FIG. 7 is a flowchart of Embodiment 4 of an encoding method
according to this application. In this embodiment, the following
describes in detail how to obtain a first polarization weight
vector when a target code length M of a polar code is not equal to
a positive integer power of 2, to be specific, when rate matching
needs to be performed. As shown in FIG. 7, the method in this
embodiment may include the following steps.
[0116] S301: Determine N to-be-encoded bits, where the N
to-be-encoded bits include information bits and frozen bits.
[0117] Specifically, a code length obtained after polar code
encoding is a positive integer power of 2, and during actual
implementation, the code length needs to be flexibly configured, in
other words, a polar code with any code length needs to be
obtained. Therefore, after input information bits are received, the
quantity N of to-be-encoded bits needs to be determined based on a
target code length M of a polar code. The N to-be-encoded bits
include the information bits and the frozen bits. When the target
code length M of the polar code is not equal to a positive integer
power of 2, to be specific, rate matching needs to be performed,
the determined quantity of to-be-encoded bits is N=2.sup..left
brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where the symbol
.left brkt-bot. .right brkt-bot. denotes rounding down.
[0118] S302: Determine positions of NM to-be-deleted bits in the N
to-be-encoded bits, and calculate, based on a result of the
determining, a set of modification coefficients including
polarization weights corresponding to the N to-be-encoded bits,
where the modification coefficient is used to represent an amount
of modification to each polarization weight resulting from bit
deletion.
[0119] Specifically, the determining positions of NM to-be-deleted
bits in the N to-be-encoded bits includes the following steps:
[0120] (1) Determine a channel index number sequence
P.sub.0.sup.N-1=[0,1, . . . ,N-1] of N polarized channels.
[0121] (2) Perform the following operations on each index number in
the channel index number sequence:
[0122] (3) Perform bit reversal on a binary number corresponding to
each index number, and convert, to a decimal number, a binary
number obtained after bit reversal, to obtain a new channel index
number sequence P.sub.0.sup.N-1'.
[0123] It should be noted that in this step, an operation performed
on each index number is not limited to a bit reversal operation,
for example, interleaving, scrambling, or the like may be performed
on the binary number corresponding to each index number, or no
processing is performed on each index number.
[0124] (4) Sort index numbers in the new channel index number
sequence P.sub.0.sup.N-1'.
[0125] (5) Select, in ascending order, NM index numbers in the
channel index number sequence P.sub.0.sup.N-1' as index numbers of
the to-be-deleted bits.
[0126] For example, in the original channel index number sequence
P.sub.0.sup.N-1=[0,1, . . . ,N-1], a binary number corresponding to
an index number in the original channel index number sequence is
A.sub.iB.sub.n-1B.sub.n-2 . . . B.sub.0, and a binary number
obtained after bit reversal is A'.sub.iB.sub.0B.sub.1 . . .
B.sub.n-1, where B.sub.j.di-elect cons.{0,1} and j.di-elect
cons.{0,1, . . . ,n-1}. After the bit reversal operation is
performed on all elements in P.sub.0.sup.N-1, P.sub.0.sup.N-1' is
obtained. After all the index numbers in P.sub.0.sup.N-1' are
re-sorted, bits corresponding to NM index numbers are selected from
the channel index number sequence P.sub.0.sup.N-1' in ascending
order as to-be-deleted bits.
[0127] Due to introduction of bit deletion, coefficients of the
polarization weights need to be modified, and the corresponding set
g of modification coefficients is in a one-to-one correspondence
with each computing node in a butterfly decoding tree, and
therefore is an N.times.n vector. The set of modification
coefficients is
g = [ g ( 0 , 0 ) g ( 0 , n - 1 ) g ( 1 , 0 ) g ( 1 , n - 1 ) g ( N
- 1 , 0 ) g ( N - 1 , n - 1 ) g ( N , 0 ) g ( N , 1 ) g ( N , n - 1
) ] ; ##EQU00018##
and
[0128] the calculating, based on a result of the determining, a set
of modification coefficients including modification coefficients of
the N bits may be specifically:calculating each item g.sub.(i,j) in
the set g of modification coefficients according to the following
formulas:
g.sub.(i,j)=(p.sub.i+p.sub.i+2.sub.j-2)/2; and
g.sub.(i+2.sub.j.sub.,j)=g.sub.(i,j), where
[0129] i.di-elect cons.{0,1, . . . ,N-1}, j.di-elect cons.{0,1, . .
. ,n-1}, N=2.sup.n, p.sub.i is an identifier indicating whether a
bit corresponding to a polarized channel whose channel index number
is i is to be deleted, and p.sub.i.di-elect cons.{0,1}.
[0130] Specifically, when j=n-1, initial g is calculated by using
the following formulas:
g.sub.(i,n-1)=(p.sub.i+p.sub.i+2.sub.n-1)/2; and
g.sub.(i+2.sub.n-1.sub.,n-1)=g.sub.(i,n-1); and
[0131] when j=[n-2, n-3, . . . , 1, 0], remaining coefficients of g
are calculated by using the following formulas:
g.sub.(i,j)=(g.sub.(i,j+1)+g.sub.(i+2.sub.j.sub.,j+1))/2; and
g.sub.(i+2.sub.j.sub.,j)=g.sub.(i,j).
[0132] The following provides an example for description. For
example, N=2.sup.n:
when j=2,
g.sub.(0,2)=(P.sub.0+P.sub.4)/2, and g.sub.(4,2)=g.sub.(0,2);
g.sub.(1,2)=(P.sub.1+P.sub.5)/2, and g.sub.(5,2)=g.sub.(1,2);
g.sub.(2,2)=(P.sub.2+P.sub.6)/2, and g.sub.(6,2)=g.sub.(2,2);
and
g.sub.(3,2)=(P.sub.3+P.sub.7)/2, and g.sub.(7,2)=g.sub.(3,2);
when j.di-elect cons.{0,1}
g.sub.(0,1)=(g.sub.(0,2)+g.sub.(2,2))/2, and
g.sub.(2,1)=g.sub.(0,1);
g.sub.(1,1)=(g.sub.(1,2)+g.sub.(3,2))/2, and
g.sub.(3,1)=g.sub.(1,1); and
g.sub.(0,0)=(g.sub.(0,1)+g.sub.(1,1))/2, and
g.sub.(1,0)=g.sub.(0,0);
[0133] Remaining g.sub.(i,j) is calculated according to the same
rule.
[0134] S303: Calculate polarization weights of N polarized channels
based on the set of modification coefficients, to obtain a first
polarization weight vector.
[0135] Specifically, in an implementation, the polarization weights
W.sub.i of the N polarized channels are calculated according to the
following formula, to obtain the first polarization weight
vector:
W i = j = 0 n - 1 .times. B j * ( 1 - g ( i , j ) ) * ( 2 j + .PHI.
) .alpha. , ##EQU00019##
where
[0136] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n,
g.sub.(i,j) is an element in row i and column j in the set g of
modification coefficients, and .PHI. and .alpha. are parameters
that are preset based on the target code length and a code rate of
the polar code.
[0137] In another implementation, the first polarization weight
vector W.sub.0.sup.N-1 is obtained through recursion calculation
according to the following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+g.sub.(i-
,j)*.beta..sub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0138] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, g.sub.(i,j) is an element
in row i and column j in the set g of modification coefficients,
and W.sub.0 is an arbitrary constant.
[0139] Optionally, in the foregoing formula, .PHI.=0, and
.alpha.=1/4.
[0140] S304: Determine positions of the information bits based on
the first polarization weight vector.
[0141] Specifically, the polarization weights in the first
polarization weight vector may be sorted in descending order, and
bits corresponding to the first K polarization weights are selected
as information bits. To be specific, positions of the information
bits are selected, and remaining positions are set for the frozen
bits, so as to obtain the complete to-be-encoded bits with a length
of N.
[0142] S305: Perform polar encoding on the N to-be-encoded bits to
obtain polar-encoded bits.
[0143] Specifically, a polar-code encoding matrix F.sub.N is used
to complete an encoding process of the to-be-encoded bits.
[0144] According to the encoding method provided in this
embodiment, after the input information bits are received, the
quantity N of to-be-encoded bits is determined based on the target
code length M of the polar code; afterwards, the positions of the
N-M to-be-deleted bits in the N to-be-encoded bits are determined,
and the set of modification coefficients including the modification
coefficients of the N to-be-encoded bits is calculated based on the
result of the determining; then, the polarization weights of the N
polarized channels are calculated based on the set of modification
coefficients, to obtain the first polarization weight vector; and
finally, positions of the information bits are determined based on
the first polarization weight vector, and polar encoding is
performed on the N to-be-encoded bits to obtain the polar-encoded
bits. Therefore, for a polar code with any target code length,
positions of information bits and those of frozen bits are
determined by calculating polarization weights of polar-code
polarized channels, without relating to a channel parameter and a
code rate. This can reduce calculation and implementation
complexity of polar code encoding.
[0145] FIG. 8 is a schematic structural diagram of Embodiment 1 of
an encoding device according to this application. As shown in FIG.
8, the encoding device 100 in this embodiment may include a first
determining module 11, an obtaining module 12, a second determining
module 13, and an encoding module 14. The first determining module
11 is configured to determine N to-be-encoded bits. The N
to-be-encoded bits include information bits and frozen bits. The
obtaining module 12 is configured to obtain a first polarization
weight vector including polarization weights of N polarized
channels. The polarization weights are a basis for information bit
selection, and the N to-be-encoded bits are corresponding to the N
polarized channels. The second determining module 13 is configured
to determine positions of the information bits based on the first
polarization weight vector. The encoding module 14 is configured to
perform polar encoding on the N to-be-encoded bits to obtain
polar-encoded bits.
[0146] Further, a target code length M of a polar code is equal to
a positive integer power of 2, and the first determining module 11
is specifically configured to determine that N is equal to M. The
target code length of the polar code is an output code length of
the polar code. The obtaining module 12 is specifically configured
to calculate the polarization weights of the N polarized channels
to obtain the first polarization weight vector.
[0147] Optionally, the obtaining module 12 is specifically
configured to calculate the polarization weights W.sub.i of the N
polarized channels according to the following formula, to obtain
the first polarization weight vector:
W i = j = 0 n - 1 .times. B j * ( 2 j + .PHI. ) .alpha. ,
##EQU00020##
where
[0148] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code.
[0149] Optionally, the obtaining module 12 is specifically
configured to obtain the first polarization weight vector
W.sub.0.sup.N-1 through recursion calculation according to the
following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+.beta..s-
ub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0150] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, and W.sub.0 is an
arbitrary constant.
[0151] The encoding device in this embodiment may be configured to
execute the technical solution in the method embodiment shown in
FIG. 3. Their implementation principles are similar, and details
are not repeated herein.
[0152] It should be noted that the encoding device in this
application may be a base station or a terminal.
[0153] After receiving input information bits, the encoding device
provided in this embodiment determines the quantity N of
to-be-encoded bits based on the target code length M of the polar
code; afterwards obtains the first polarization weight vector
including the polarization weights of the N polarized channels,
where the polarization weights are a basis for information bit
selection; then determines positions of the information bits based
on the first polarization weight vector; and finally performs polar
encoding on the N to-be-encoded bits to obtain the polar-encoded
bits. Therefore, the position of the information bits and those of
the frozen bits are determined by calculating polarization weights
of polar-code polarized channels, without relating to a channel
parameter and a code rate. This can reduce calculation complexity
of polar code encoding. In addition, polarization weights
corresponding to polar codes with different code lengths and code
rates have a same part, so that storage space is greatly reduced,
and storage complexity of polar code encoding can also be
reduced.
[0154] Optionally, a target code length M of a polar code is equal
to a positive integer power of 2, and the obtaining module 12 is
specifically configured to obtain the first polarization weight
vector from a prestored second polarization weight vector. The
second polarization weight vector includes polarization weights of
N.sub.max polarized channels, N.sub.max is a preset mother code
length corresponding to a maximum code length supported by a
communications system, and the mother code length is a positive
integer power of 2.
[0155] In this implementation, the polarization weights are
obtained in an offline storage manner. Compared with online
calculation, when there is enough storage space, the offline
storage manner can shorten latency caused by online
calculation.
[0156] Optionally, when the target code length M of the polar code
is less than or equal to N.sub.max, an element set in the first
polarization weight vector is a subset of an element set in the
second polarization weight vector; or when the target code length M
of the polar code is greater than or equal to N.sub.max, a
polarization weight of a polarized channel whose channel index
number is less than or equal to N.sub.max is in a one-to-one
correspondence with a polarization weight in the second
polarization weight vector, and a polarization weight of a
polarized channel whose channel index number is greater than
N.sub.max is equal to a weight increment
.beta. = j = log 2 .function. ( N max ) log 2 .function. ( M - 1 )
.times. ( 2 j + .PHI. ) .alpha. ##EQU00021##
plus the corresponding polarization weight of the polarized channel
whose channel index number is less than or equal to N.sub.max,
where .PHI. and .alpha. are parameters that are preset based on the
target code length and a code rate of the polar code.
[0157] In this implementation, a half-online-calculation
half-offline-storage manner is used, so that not only latency
caused by online calculation can be shortened, but also storage
complexity can be reduced.
[0158] In the foregoing implementation, further, .PHI.=0, and
.alpha.=1/4.
[0159] Further, a target code length M of a polar code is not equal
to a positive integer power of 2, and the first determining module
11 is specifically configured to determine that N=2.sup..left
brkt-bot.log.sup.2.sup.(M-1).right brkt-bot.+1, where the symbol
.left brkt-bot. .right brkt-bot. denotes rounding down.
[0160] FIG. 9 is a schematic structural diagram of Embodiment 2 of
an encoding device according to this application. As shown in FIG.
9, based on a structure of the encoding device shown in FIG. 8, the
encoding device 200 in this embodiment may further include a third
determining module 15 and a calculation module 16. The third
determining module 15 is configured to: before the obtaining module
12 obtains the first polarization weight vector including the
polarization weights of the N polarized channels, determine
positions of NM to-be-deleted bits in the N to-be-encoded bits. The
calculation module 16 is configured to calculate, based on a result
of the determining performed by the third determining module 15, a
set of modification coefficients including the polarization weights
corresponding to the N to-be-encoded bits. The modification
coefficient is used to represent an amount of modification to each
polarization weight resulting from bit deletion. The obtaining
module 12 is specifically configured to calculate the polarization
weights of the N polarized channels based on the set of
modification coefficients, to obtain the first polarization weight
vector.
[0161] Further, the third determining module 15 is specifically
configured to: determine a channel index number sequence
P.sub.0.sup.N-1=[0,1, . . . ,N-1] of the N polarized channels; and
perform the following operations on each index number in the
channel index number sequence: performing bit reversal on a binary
number corresponding to each index number, and converting, to a
decimal number, a binary number obtained after bit reversal, to
obtain a new channel index number sequence P.sub.0.sup.N-1';
sorting index numbers in the new channel index number sequence
P.sub.0.sup.N-1'; and selecting, in ascending order, N-M index
numbers in the channel index number sequence P.sub.0.sup.N-1' as
index numbers of the to-be-deleted bits.
[0162] Optionally, the set of modification coefficients is
g = [ g ( 0 , 0 ) g ( 0 , n - 1 ) g ( 1 , 0 ) g ( 1 , n - 1 ) g ( N
- 1 , 0 ) g ( N - 1 , n - 1 ) g ( N , 0 ) g ( N , 1 ) g ( N , n - 1
) ] ; ##EQU00022##
and
[0163] the calculation module 16 is specifically configured to
calculate each item g.sub.(i,j) in the set g of modification
coefficients according to the following formulas:
g.sub.(i,j)=(p.sub.i+p.sub.i+2.sub.j-2)/2; and
g.sub.(i+2.sub.j.sub.,j)=g.sub.(i,j), where
[0164] i.di-elect cons.{0,1, . . . ,N-1}, j.di-elect cons.{0,1, . .
. ,n-1}, N=2.sup.n, p.sub.i is an identifier indicating whether a
bit corresponding to a polarized channel whose channel index number
is i is to be deleted, and p.sub.i.di-elect cons.{0,1}.
[0165] Optionally, the obtaining module 12 is specifically
configured to calculate the polarization weights W.sub.i of the N
polarized channels according to the following formula, to obtain
the first polarization weight vector:
W i = j = 0 n - 1 .times. B j * ( 1 - g ( i , j ) ) * ( 2 j + .PHI.
) .alpha. , ##EQU00023##
where
[0166] iB.sub.n-1B.sub.n-2 . . . B.sub.0, i is a channel index
number, B.sub.n-1B.sub.n-2 . . . B.sub.0 is a binary representation
of i, B.sub.n-1 is a most significant bit, B.sub.0 is a least
significant bit, B.sub.j.di-elect cons.{0,1}, j.di-elect cons.{0,1,
. . . ,n-1}, i.di-elect cons.{0,1, . . . ,N-1}, N=2.sup.n,
g.sub.(i,j) is an element in row i and column j in the set g of
modification coefficients, and .PHI. and .alpha. are parameters
that are preset based on the target code length and a code rate of
the polar code.
[0167] Optionally, the obtaining module 12 is specifically
configured to obtain the first polarization weight vector
W.sub.0.sup.N-1 through recursion calculation according to the
following formulas:
.beta..sub.i=(2.sup.i-1+.PHI.).sup..alpha.;
W.sub.2.sub.i-1.sup.2.sup.i.sup.-1=W.sub.0.sup.2.sup.i-1.sup.-1+g.sub.(i-
,j)*.beta..sub.i; and
W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0.sup.2.sup.i-1.sup.-1,W.sub.2.sub.i-1-
.sup.2.sup.i.sup.-1], where
W.sub.0.sup.1=[W.sub.0,W.sub.1],W.sub.0.sup.2.sup.i.sup.-1=[W.sub.0,W.su-
b.1, . . . ,W.sub.2.sub.i-1], where
[0168] i.di-elect cons.{1, . . . ,N-1}, .beta..sub.i is an
intermediate variable related to a channel index number i, .PHI.
and .alpha. are parameters that are preset based on the target code
length and a code rate of the polar code, g.sub.(i,j) is an element
in row i and column j in the set g of modification coefficients,
and W.sub.0 is an arbitrary constant.
[0169] In the foregoing implementation, .PHI.=0, and
.alpha.=1/4.
[0170] In the foregoing implementation, the second determining
module 13 is specifically configured to: sort the polarization
weights in the first polarization weight vector in descending
order, and select bits corresponding to the first K polarization
weights as information bits.
[0171] The encoding device in this embodiment may be configured to
execute the technical solution in the method embodiment shown in
FIG. 7. Their implementation principles are similar, and details
are not repeated herein.
[0172] After receiving input information bits, the encoding device
provided in this embodiment determines the quantity N of
to-be-encoded bits based on the target code length M of the polar
code; afterwards determines the positions of the NM to-be-deleted
bits in the N to-be-encoded bits, and calculates, based on the
result of the determining, the set of modification coefficients
including modification coefficients of the N to-be-encoded bits;
then calculates the polarization weights of the N polarized
channels based on the set of modification coefficients, to obtain
the first polarization weight vector; and finally determines
positions of the information bits based on the first polarization
weight vector, and performs polar encoding on the N to-be-encoded
bits to obtain the polar-encoded bits. Therefore, for a polar code
with any target code length, positions of information bits and
those of frozen bits are determined by calculating polarization
weights of polar-code polarized channels, without relating to a
channel parameter and a code rate. This can reduce calculation and
implementation complexity of polar code encoding.
[0173] A person of ordinary skill in the art may understand that
some or all of the steps in the method embodiments may be
implemented by a program instructing relevant hardware. The program
may be stored in a computer-readable storage medium. When the
program runs, the steps in the method embodiments are performed.
The storage medium includes various media that can store program
code, such as a ROM, a RAM, a magnetic disk, or an optical
disc.
[0174] As shown in the foregoing method and the corresponding
device, when the polarization weights in the first polarization
weight vector are sorted in descending order, according to a method
for calculating a polarization weight according to a formula or a
polynomial corresponding to a binary representation manner of a
polarized channel sequence number, if a polarization weight of one
polarized channel is greater than a polarization weight of another
polarized channel, a position of the "one polarized channel" in the
first polarization weight vector is necessarily before that of the
"another polarized channel". Certainly, when the polarization
weights in the first polarization weight vector are sorted in
ascending order, if a polarization weight of one polarized channel
is greater than a polarization weight of another polarized channel,
a position of the "one polarized channel" in the first polarization
weight vector is necessarily after that of the "another polarized
channel". In addition, a position of each polarized channel in the
first polarization weight vector may be considered as a
polarization weight multiplied by a factor.
[0175] According to the foregoing method and the corresponding
device for generating the first polarization weight vector
including the polarization weights of the N polarized channels, the
following several examples of the first polarization weight vector
may be provided. The polarization weights may be sorted in
descending order, or the polarization weights may be sorted in
ascending order. Without loss of generality, sequence numbers in
all the examples are ranked from 1, in other words, a minimum
sequence number value is 1. This is equivalent to that the sequence
numbers are ranked from 0. If the sequence numbers are ranked from
0, 1 only needs to be subtracted from each sequence number in a
corresponding sequence.
[0176] The polarization weights in the first polarization weight
vector may be sorted in descending order. Herein, a minimum
sequence number value is 1. Correspondingly, if a minimum sequence
number value is 0, 1 may be subtracted from each of all sequence
numbers.
[0177] (1) N=16, and the first polarization weight vector is:
[0178] 16,15,14,12,8,13,11,10,7,6,4,9,5,3,2,1.
[0179] (2) N=32, and the first polarization weight vector is:
[0180]
32,31,30,28,24,16,29,27,26,23,22,15,20,14,12,25,8,21,19,13,18,11,10-
,7,6,4,17,9,5,3,2,1.
[0181] (3) N=64, and the first polarization weight vector is:
[0182]
64,63,62,60,56,48,61,32,59,58,55,54,47,52,46,31,44,30,57,40,28,53,2-
4,51,45,50,16,43,29,42,39,27,38,26,23,36,22,49,15,20,14,41,12,37,25,8,35,2-
1,34,19,13,18,11,10,7,6,33,4,17,9,5,3,2,1.
[0183] (4) N=128, and the first polarization weight vector is:
[0184]
128,127,126,124,120,112,125,96,123,122,119,64,118,111,116,110,95,10-
8,94,121,104,92,
63,117,62,88,115,109,60,114,80,107,56,93,106,103,91,48,102,90,61,87,100,3-
2,59,86,113,79,58,84,5
5,78,105,54,76,47,101,52,89,46,72,99,31,44,85,98,30,57,83,40,77,28,82,53,-
75,24,51,74,45,71,50,16,
43,70,97,29,42,68,39,27,81,38,26,23,36,73,22,49,15,69,20,14,41,67,12,66,3-
7,25,8,35,21,34,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.
[0185] (5) N=256, and the first polarization weight vector is:
[0186]
256,255,254,252,248,240,253,224,251,250,247,192,246,239,244,238,128-
,223,236,222,24
9,232,220,191,245,190,216,243,237,188,242,127,208,235,184,126,221,234,231-
,124,219,176,230,21
8,189,120,215,228,160,187,214,241,112,207,186,212,183,125,206,233,182,96,-
123,204,175,229,180,
122,217,174,119,200,227,64,159,118,172,213,226,111,158,185,116,211,168,11-
0,205,156,210,181,95,108,203,152,179,94,121,202,173,104,199,178,92,63,144,-
117,171,198,225,62,157,88,170,115,196,1
67,109,60,155,114,209,166,80,107,154,56,151,164,93,106,201,103,150,177,91-
,48,143,102,197,148,
90,61,142,87,169,100,195,32,59,86,140,113,194,165,79,58,153,84,55,136,163-
,78,105,54,149,162,76,47,101,52,147,89,46,141,72,99,146,31,44,85,139,98,19-
3,30,57,138,83,40,135,77,28,82,53,134,161,
75,24,51,132,74,45,71,50,145,16,43,70,97,29,42,137,68,39,27,81,38,133,26,-
23,36,131,73,22,49,130,
15,69,20,14,41,67,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,1-
7,9,5,3,2,1.
[0187] (6) N=512, and the first polarization weight vector is:
[0188]
512,511,510,508,504,496,509,480,507,506,503,448,502,495,500,494,384-
,479,492,478,50
5,488,476,447,501,256,446,472,499,493,444,498,383,464,491,440,382,477,490-
,487,380,475,432,48
6,474,255,445,376,471,484,254,416,443,470,497,368,463,252,442,468,439,381-
,462,489,248,438,35
2,379,460,431,485,436,378,473,240,430,375,456,483,253,320,415,374,428,469-
,482,367,224,414,251,441,372,467,424,366,461,250,412,466,247,437,351,364,4-
59,192,246,408,435,350,377,458,239,42
9,360,455,244,434,348,238,319,400,373,427,454,481,128,223,318,413,236,344-
,426,371,452,423,36
5,222,249,316,411,370,465,232,422,336,363,220,410,191,245,312,407,420,349-
,362,457,190,359,21
6,406,243,433,347,237,304,399,358,188,453,242,404,346,127,208,317,398,235-
,343,425,356,451,18
4,126,221,288,234,315,342,396,369,450,231,421,335,124,219,314,409,176,340-
,230,311,392,419,33
4,361,218,189,120,215,310,405,228,418,332,303,160,357,187,214,241,308,403-
,345,112,207,302,39
7,328,186,355,212,402,183,125,206,287,233,300,341,395,354,449,182,96,286,-
123,204,313,394,175,
339,229,296,391,180,333,122,217,284,174,338,119,200,309,390,227,417,331,6-
4,159,118,172,213,2
80,226,307,388,330,111,301,158,327,185,116,211,306,401,168,110,205,272,29-
9,326,156,353,210,1
81,95,285,108,203,298,393,324,295,152,179,94,121,202,283,173,337,104,199,-
294,389,178,92,282,6
3,144,117,171,198,279,225,292,387,329,62,157,88,170,278,115,196,305,386,1-
67,109,271,60,325,15
5,114,209,276,166,80,270,107,297,154,323,56,151,164,93,106,201,268,322,10-
3,293,150,177,91,281,48,143,102,197,264,291,148,90,61,142,87,169,277,100,1-
95,290,385,32,59,86,140,113,194,275,165,
79,269,58,153,84,274,55,136,163,78,105,267,321,54,149,162,76,266,47,101,2-
63,52,147,89,46,141,7
2,262,99,289,146,31,44,85,139,98,193,260,30,57,138,83,273,40,135,77,28,82-
,53,134,161,75,265,24,
51,132,74,45,71,261,50,145,16,43,70,97,259,29,42,137,68,258,39,27,81,38,1-
33,26,23,36,131,73,22,
49,130,15,69,20,14,41,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10-
,7,6,33,4,17,9,5,3,2,1.
[0189] (7) N=1024, and the first polarization weight vector is:
[0190]
1024,1023,1022,1020,1016,1008,1021,992,1019,1018,1015,960,1014,1007-
,1012,1006,89
6,991,1004,990,1017,1000,988,959,1013,768,958,984,1011,1005,956,1010,895,-
976,1003,952,894,9
89,1002,999,512,892,987,944,998,986,767,957,888,983,996,766,928,955,982,1-
009,880,975,764,954,980,951,893,974,1001,760,950,864,511,891,972,943,997,9-
48,510,890,985,752,942,887,968,995,50
8,765,832,927,886,940,981,994,879,736,926,763,953,504,884,979,936,878,973-
,762,924,978,759,94
9,863,496,876,971,704,758,920,947,862,509,889,970,751,941,872,967,756,946-
,480,860,507,750,83
1,912,885,939,966,993,506,640,735,830,925,748,856,938,503,883,964,935,877-
,734,448,761,828,92
3,502,882,977,744,934,848,495,875,732,922,703,500,757,824,919,932,861,494-
,874,969,702,871,72
8,918,755,945,384,479,859,492,749,816,911,870,700,965,754,916,478,858,505-
,639,720,829,910,74
7,855,937,488,868,963,696,476,638,733,800,447,746,827,854,908,501,881,962-
,743,933,847,256,44
6,636,731,826,921,472,688,852,499,742,823,904,931,846,493,873,730,444,701-
,498,632,727,822,91
7,740,930,383,464,844,491,815,672,869,699,726,440,753,820,915,382,477,857-
,490,624,719,814,90
9,840,698,487,867,724,914,695,380,475,637,718,799,432,745,812,853,907,486-
,866,961,694,474,60
8,255,798,445,635,716,825,376,906,471,687,851,484,741,808,903,692,845,254-
,416,634,729,796,44
3,470,686,850,497,631,712,821,902,739,929,368,463,843,252,442,576,671,468-
,630,684,725,792,43
9,738,819,900,381,462,842,489,623,813,670,839,697,248,438,628,723,818,913-
,352,680,379,460,62
2,717,784,431,811,838,668,485,865,722,436,693,378,473,607,240,797,430,620-
,715,810,375,905,45
6,836,483,807,664,691,606,253,320,415,633,714,795,374,428,469,685,849,482-
,616,711,806,901,69
0,367,224,414,604,251,794,441,575,656,372,467,629,683,710,791,424,737,804-
,899,366,461,841,25
0,412,574,669,466,600,682,247,790,437,627,708,817,898,351,679,364,459,621-
,783,192,572,837,66
7,246,408,626,721,788,435,350,678,377,458,592,239,782,429,619,809,360,666-
,455,835,244,434,56
8,663,348,676,605,238,319,400,618,713,780,373,427,454,834,481,615,805,662-
,689,128,223,318,41
3,603,236,793,344,426,560,655,371,452,614,709,776,423,803,660,365,222,602-
,249,316,411,573,65
4,370,465,599,681,232,789,422,612,707,802,897,336,363,220,410,544,191,571-
,598,652,245,312,40
7,625,706,787,420,349,677,362,457,591,781,190,570,359,665,216,406,596,243-
,786,433,567,648,34
7,675,590,237,304,399,617,779,358,188,453,833,242,404,566,661,346,674,127-
,208,317,398,588,23
5,778,343,425,559,356,451,613,775,184,564,659,126,221,288,601,234,315,342-
,396,558,653,369,45
0,584,231,774,421,611,801,658,335,124,219,314,409,543,176,340,556,597,651-
,230,311,392,610,70
5,772,419,334,361,218,542,189,569,120,650,215,310,405,595,228,785,418,552-
,647,332,589,303,16
0,540,357,187,214,594,241,308,403,565,646,345,673,112,207,302,397,587,777-
,328,186,355,212,40
2,536,183,563,644,125,206,287,586,233,300,341,395,557,354,449,583,773,182-
,562,657,96,286,123,
204,313,394,528,175,339,555,582,229,296,391,609,771,180,333,122,217,284,5-
41,174,338,554,119,
649,200,309,390,580,227,770,417,551,331,64,159,539,118,172,213,280,593,22-
6,307,388,550,645,3
30,111,301,158,538,327,185,116,211,306,401,535,168,548,643,110,205,272,58-
5,299,326,156,353,2
10,534,181,561,642,95,285,108,203,298,393,527,324,581,295,152,532,179,94,-
121,202,283,526,173,
337,553,104,199,294,389,579,769,178,92,282,63,144,524,117,171,198,279,578-
,225,292,387,549,32
9,62,157,537,88,170,278,115,196,305,386,520,167,547,109,271,60,325,155,11-
4,209,276,533,166,54
6,641,80,270,107,297,154,323,56,151,531,164,93,106,201,268,525,322,103,29-
3,150,530,177,91,281,48,143,523,102,197,264,577,291,148,90,61,142,522,87,1-
69,277,100,195,290,385,519,32,59,86,140,
113,194,275,518,165,545,79,269,58,153,84,274,55,136,516,163,78,105,267,32-
1,54,149,529,162,76,
266,47,101,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193-
,260,517,30,57,138,83,273,40,135,515,77,28,82,53,134,514,161,75,265,24,51,-
132,74,45,71,261,50,145,16,43,70,97,259,29,
42,137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,4-
1,67,257,12,66,37,25,8,35,21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,-
1.
[0191] (8) N=2048, and the first polarization weight vector is:
[0192]
2048,2047,2046,2044,2040,2032,2045,2016,2043,2042,2039,1984,2038,20-
31,2036,2030,
1920,2015,2028,2014,2041,2024,2012,1983,2037,1792,1982,2008,2035,2029,198-
0,2034,1919,2000,
2027,1976,1918,2013,2026,2023,1536,1916,2011,1968,2022,2010,1791,1981,191-
2,2007,2020,1790,
1952,1979,2006,2033,1904,1999,1788,1978,2004,1975,1917,1998,2025,1024,178-
4,1974,1888,1535,
1915,1996,1967,2021,1972,1534,1914,2009,1776,1966,1911,1992,2019,1532,178-
9,1856,1951,1910,
1964,2005,2018,1903,1760,1950,1787,1977,1528,1908,2003,1960,1902,1997,178-
6,1948,1023,2002,
1783,1973,1887,1520,1900,1995,1728,1022,1782,1944,1971,1886,1533,1913,199-
4,1775,1965,1896,
1991,1020,1780,1970,1504,1884,1531,1774,1855,1936,1909,1963,1990,2017,101-
6,1530,1664,1759,
1854,1949,1772,1880,1962,1527,1907,1988,1959,1901,1758,1472,1785,1852,194-
7,1526,1906,1008,
2001,1768,1958,1872,1519,1899,1756,1946,1727,1021,1524,1781,1848,1943,195-
6,1885,1518,1898,
1993,1726,992,1895,1752,1019,1942,1779,1969,1408,1503,1883,1516,1773,1840-
,1935,1894,1724,1
989,1018,1778,1940,1502,1015,1882,1529,1663,1744,1853,1934,1771,1879,1961-
,1512,1892,1987,9
60,1720,1014,1500,1662,1757,1824,1471,1770,1851,1878,1932,1525,1905,1007,-
1986,1767,1957,18
71,1012,1280,1470,1660,1755,1850,1945,1496,1712,1876,1006,1523,1766,1847,-
1928,1955,1870,15
17,1897,1754,1468,1725,896,991,1522,1656,1751,1846,1004,1941,1764,1954,14-
07,1488,1868,1515,
1839,1696,990,1893,1723,1750,1017,1464,1777,1844,1939,1406,1501,1000,1881-
,1514,1648,1743,
1838,1933,1864,1722,1511,988,1891,1748,959,1938,1719,1013,1404,1499,1661,-
1742,1823,1456,17
69,1836,1877,1931,1510,1890,768,1985,958,1718,984,1498,1632,1011,1279,182-
2,1469,1659,1740,
1849,1400,1930,1495,1711,1875,1005,1508,1765,1832,1927,956,1716,1869,1010-
,1278,1440,1658,1
753,1820,1467,1494,1710,895,1874,976,1521,1655,1736,1845,1003,1926,1763,1-
953,1392,1487,186
7,952,1276,1466,1600,1695,894,989,1492,1654,1708,1749,1816,1002,1463,1762-
,1843,1924,1405,1
486,999,1866,1513,1647,1837,1694,512,1863,1721,892,1272,987,1462,1652,174-
7,1842,944,1937,1
376,1704,998,1403,1484,1646,1741,1808,1455,1835,1862,1692,1509,986,1889,7-
67,1746,957,1460,
1717,888,983,1402,1497,1631,996,1264,1821,1454,1644,1739,1834,1399,1929,1-
480,1860,766,1507,
928,1831,1688,955,982,1715,1630,1009,1277,1344,1439,1657,1738,1819,1398,1-
452,1493,1709,88
0,1873,975,1506,1640,1735,1830,764,1925,954,1714,1391,980,1248,1438,1628,-
951,1275,1818,146
5,1599,1680,893,1396,974,1491,1653,1707,1734,1815,1001,1448,1761,1828,192-
3,1390,1485,760,1
865,950,1274,1436,1598,1693,864,511,1490,1624,1706,891,1271,1814,972,1461-
,1651,1732,1841,9
43,1922,1375,1703,997,1388,1483,1645,1807,948,1216,510,1596,1861,1691,890-
,1270,985,1432,16
50,752,1745,1812,942,1459,1374,1702,887,968,1401,1482,1616,995,1263,1806,-
1453,1643,1833,13
84,1690,1479,508,1859,765,832,1268,927,1458,1592,1687,886,940,981,1372,17-
00,1629,994,1262,1
343,1424,1642,1737,1804,1397,1451,1478,879,1858,736,1505,1639,926,1829,76-
3,1686,953,504,17
13,884,1152,979,1247,1342,1437,1627,936,1260,1817,1368,1450,1584,1679,878-
,1395,973,1476,16
38,1733,1800,762,1447,924,1827,1684,1389,978,1246,759,1626,949,1273,1340,-
1435,1597,1678,86
3,1394,496,1489,1623,1705,876,1256,1813,971,1446,1636,1731,1826,704,1921,-
1360,758,1387,920,
1244,1434,1568,947,1215,862,509,1595,1622,1676,889,1269,1336,970,1431,164-
9,751,1730,1811,9
41,1444,1373,1701,872,967,1386,1481,1615,756,1805,946,1214,480,1594,1383,-
1689,860,1240,507,
1430,1620,750,831,1267,1810,912,1457,1591,1672,885,939,966,1371,1699,1614-
,993,1261,1328,14
23,1641,1803,1382,1212,1477,506,640,1857,735,830,1266,925,1428,1590,748,1-
685,856,938,503,13
70,1698,883,1151,964,1232,1341,1422,1612,935,1259,1802,1367,1449,1583,877-
,1380,734,1475,16
37,448,1799,761,828,1208,923,502,1588,1683,882,1150,977,1245,1312,744,162-
5,934,1258,1339,13
66,1420,1582,1677,848,1393,495,1474,1608,875,1255,1798,732,1445,1635,922,-
1825,703,1682,135
9,500,757,824,1148,919,1243,1338,1433,1567,932,1200,861,1364,494,1580,162-
1,1675,874,1254,13
35,969,1416,1634,1729,1796,702,1443,1358,871,728,1385,918,1242,755,1566,9-
45,1213,384,479,15
93,1144,1674,859,1239,1334,492,1429,1619,749,816,1252,1809,911,1442,1576,-
1671,870,700,965,1
356,1613,754,1327,916,1184,478,1564,1381,1211,858,1238,505,639,1618,720,8-
29,1265,1332,910,1
427,1589,747,1670,855,937,488,1369,1697,868,1136,963,1231,1326,1421,1611,-
696,1801,1352,121
0,476,638,1379,733,800,1236,447,1426,1560,746,827,1207,854,908,501,1587,1-
668,881,1149,962,1
230,1311,743,1610,933,1257,1324,1365,1419,1581,847,1378,256,1473,1607,446-
,636,1797,731,826,
1206,921,472,1586,688,1681,852,1120,499,1310,742,823,1147,904,1228,1337,1-
418,1552,931,1199,
846,1363,493,1579,1606,873,1253,1320,730,1415,1633,444,1795,701,1204,1357-
,498,632,727,822,1
146,917,1241,1308,740,1565,930,1198,383,1362,464,1578,1143,1673,844,1224,-
1333,491,1414,160
4,815,1251,1794,672,1441,1575,869,699,726,1355,440,753,820,1088,915,1183,-
382,477,1563,1142,
1196,857,1237,1304,490,624,1617,719,814,1250,1331,909,1412,1574,1669,840,-
698,487,1354,867,1
135,724,1325,914,1182,695,1562,1351,1209,380,475,637,1140,718,799,1235,13-
30,432,1425,1559,7
45,812,1192,853,907,486,1572,1667,866,1134,961,1229,1296,1609,694,1323,13-
50,1180,474,608,13
77,255,798,1234,445,635,1558,716,825,1205,376,906,471,1585,687,1666,851,1-
119,484,1309,741,8
08,1132,903,1227,1322,1417,1551,692,845,1348,254,1605,416,634,1319,729,79-
6,1176,443,470,155
6,686,1203,850,1118,497,631,712,821,1145,902,1226,1307,739,1550,929,1197,-
368,1361,463,1577,
1128,843,1223,1318,252,1413,1603,442,576,1793,671,1202,468,630,684,725,79-
2,1116,439,1306,73
8,819,1087,900,1168,381,462,1548,1141,1195,842,1222,1303,489,623,1602,813-
,1249,1316,670,141
1,1573,839,697,248,1353,438,628,723,818,1086,913,1181,352,680,1561,1112,1-
194,379,1302,460,6
22,1139,717,784,1220,1329,431,1410,1544,811,1191,838,668,485,1571,865,113-
3,722,1295,436,693,
1084,1349,1179,378,473,607,1138,240,797,1233,1300,430,620,1557,715,810,11-
90,375,905,456,15
70,1665,836,1104,483,1294,807,1131,664,1321,1178,691,606,1347,253,320,415-
,633,1080,714,795,
1175,374,428,469,1555,685,1188,849,1117,482,616,711,806,1130,901,1225,129-
2,1549,690,367,134
6,224,1127,414,604,1317,251,794,1174,441,575,1554,656,1201,372,467,629,68-
3,710,791,1115,424,
1305,737,804,1072,899,1167,366,461,1547,1126,841,1221,1288,250,1601,412,5-
74,1315,669,1172,4
66,600,682,247,790,1114,437,627,708,817,1085,898,1166,351,679,1546,1111,1-
193,364,1301,459,62
1,1124,783,1219,1314,192,1409,1543,572,837,667,246,408,626,721,788,1056,4-
35,350,678,1083,11
10,1164,377,458,592,1137,239,782,1218,1299,429,619,1542,809,1189,360,666,-
455,1569,835,1103,
244,1293,434,568,663,1082,1177,348,676,605,1108,238,319,1298,400,618,1079-
,713,780,1160,373,
427,454,1540,1187,834,1102,481,615,805,1129,662,1291,689,128,1345,223,318-
,413,603,1078,236,
793,1173,344,426,560,1553,655,1186,371,452,614,709,776,1100,423,1290,803,-
1071,660,365,222,1
125,602,1287,249,316,411,573,1076,654,1171,370,465,599,681,232,789,1113,4-
22,612,707,802,107
0,897,1165,336,1545,1096,363,1286,220,1123,410,544,1313,191,1170,571,598,-
652,245,312,407,62
5,706,787,1055,420,349,677,1068,1109,1163,362,457,591,1122,781,1217,1284,-
190,1541,570,359,6
65,216,406,596,243,786,1054,433,567,648,1081,1162,347,675,590,1107,237,30-
4,1297,399,617,106
4,779,1159,358,188,453,1539,833,1101,242,404,566,661,1052,346,674,127,110-
6,208,317,398,588,1
077,235,778,1158,343,425,559,1538,1185,356,451,613,775,1099,184,1289,564,-
659,126,221,288,60
1,1048,234,315,342,396,558,1075,653,1156,369,450,584,231,774,1098,421,611-
,801,1069,658,335,1
095,124,1285,219,314,409,543,1074,176,1169,340,556,597,651,230,311,392,61-
0,705,772,1040,419,
334,1067,1094,361,218,1121,542,1283,189,569,120,650,215,310,405,595,228,7-
85,1053,418,552,64
7,1066,1161,332,589,1092,303,1282,160,1063,540,357,187,214,594,241,308,40-
3,565,646,1051,345,
673,112,1105,207,302,397,587,1062,777,1157,328,186,1537,355,212,402,536,1-
83,1050,563,644,12
5,206,287,586,1047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182-
,562,657,96,286,123,
1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039,1-
80,333,1093,122,21
7,284,541,1044,174,338,554,119,649,200,309,390,580,227,770,1038,417,551,1-
065,331,1091,64,128
1,159,539,118,172,213,280,593,226,307,388,550,645,1036,330,111,1090,301,1-
58,1061,538,327,185,
116,211,306,401,535,168,1049,548,643,110,205,272,585,1032,299,326,156,105-
9,353,210,534,181,
561,642,95,285,108,1045,203,298,393,527,1058,1153,324,581,295,152,532,179-
,94,121,202,283,526,
1043,173,337,553,104,199,294,389,579,769,1037,178,92,282,63,1042,144,524,-
117,171,198,279,57
8,225,292,387,549,1035,329,1089,62,157,537,88,170,278,115,196,305,386,520-
,167,1034,547,109,2
71,1031,60,325,155,114,209,276,533,166,546,641,80,270,107,1030,297,154,10-
57,323,56,151,531,1
64,93,106,201,268,525,1028,322,103,293,150,530,177,91,281,48,1041,143,523-
,102,197,264,577,29
1,148,90,61,142,522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,-
194,275,518,165,545,
79,269,1029,58,153,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,1-
62,76,266,47,1026,10
1,263,52,147,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,-
30,57,138,83,273,40,1
35,515,77,28,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,-
16,43,70,97,259,29,42,
137,68,258,39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,6-
7,257,12,66,37,25,8,35,
21,34,129,19,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.
[0193] (9) N=4096, and the first polarization weight vector is:
4096,4095,4094,4092,4088,4080,4093,4064,4091,4090,4087,4032,4086,4079,408-
4,4078,
3968,4063,4076,4062,4089,4072,4060,4031,4085,3840,4030,4056,4083,4-
077,4028,4082,3967,4048,
4075,4024,3966,4061,4074,4071,3584,3964,4059,4016,4070,4058,3839,4029,396-
0,4055,4068,3838,
4000,4027,4054,4081,3952,4047,3836,4026,4052,4023,3965,4046,4073,3072,383-
2,4022,3936,3583,
3963,4044,4015,4069,4020,3582,3962,4057,3824,4014,3959,4040,4067,3580,383-
7,3904,3999,3958,
4012,4053,4066,3951,3808,3998,3835,4025,3576,3956,4051,4008,3950,4045,383-
4,3996,3071,4050,
3831,4021,2048,3935,3568,3948,4043,3776,3070,3830,3992,4019,3934,3581,396-
1,4042,3823,4013,
3944,4039,3068,3828,4018,3552,3932,3579,3822,3903,3984,3957,4011,4038,406-
5,3064,3578,3712,
3807,3902,3997,3820,3928,4010,3575,3955,4036,4007,3949,3806,3520,3833,390-
0,3995,3574,3954,
3056,4049,3816,4006,2047,3920,3567,3947,3804,3994,3775,3069,3572,3829,389-
6,3991,2046,4004,
3933,3566,3946,4041,3774,3040,3943,3800,3067,3990,3827,4017,3456,3551,204-
4,3931,3564,3821,
3888,3983,3942,3772,4037,3066,3826,3988,3550,3063,3930,3577,3711,3792,390-
1,3982,2040,3819,
3927,4009,3560,3940,4035,3008,3768,3062,3548,3710,3805,3872,3519,3818,389-
9,3926,3980,3573,
3953,3055,4034,3815,4005,2032,3919,3060,3328,3518,3708,3803,3898,3993,354-
4,3760,3924,3054,
3571,3814,3895,3976,2045,4003,3918,3565,3945,3802,3516,3773,2944,3039,357-
0,3704,3799,3894,
3052,3989,3812,2016,4002,3455,3536,2043,3916,3563,3887,3744,3038,3941,377-
1,3798,3065,3512,
3825,3892,3987,3454,3549,2042,3048,3929,3562,3696,3791,3886,3981,2039,391-
2,3770,3559,3036,
3939,3796,3007,3986,3767,3061,3452,3547,3709,1984,3790,3871,3504,2038,381-
7,3884,3925,3979,
3558,3938,2816,4033,3006,3766,3032,2031,3546,3680,3059,3327,3870,3517,370-
7,3788,3897,3448,
3978,3543,2036,3759,3923,3053,3556,3813,3880,3975,3004,2030,3764,3917,305-
8,3326,3488,3706,
3801,3868,3515,3542,3758,2943,3922,3024,3569,3703,3784,3893,3051,3974,192-
0,3811,2015,4001,
3440,3535,2028,3915,3000,3324,3514,3648,3743,2942,3037,3540,3702,3756,379-
7,3864,3050,3511,
3810,3891,2014,3972,3453,3534,2041,3047,3914,3561,3695,3885,3742,2024,256-
0,3911,3769,2940,
3320,3035,3510,3700,3795,3890,2992,3985,3424,3752,2012,3046,3451,3532,369-
4,1983,3789,3856,
3503,2037,3883,3910,3740,3557,3034,3937,2815,3794,3005,3508,3765,2936,303-
1,3450,1792,3545,
3679,1982,3044,3312,3869,3502,3692,2008,3787,3882,3447,3977,3528,2035,390-
8,2814,3555,2976,
3879,3736,3003,3030,2029,3763,3678,3057,3325,3392,3487,3705,1980,3786,386-
7,3446,3500,3541,
2034,3757,2928,3921,3023,3554,3688,3783,3878,2812,3973,3002,1919,2000,376-
2,3439,3028,3296,
2027,3486,3676,2999,3323,3866,3513,3647,3728,2941,3444,3022,3539,3701,197-
6,3755,3782,3863,
3049,3496,1918,3809,3876,2013,3971,3438,3533,2026,2808,3913,2998,3322,348-
4,3646,3741,2912,
2023,2559,3538,3672,3754,2939,3319,3862,3020,3509,3699,3780,3889,1536,299-
1,3970,3423,1916,
3751,2011,3045,3436,3531,3693,1968,3855,2996,3264,2022,2558,3644,3909,373-
9,2938,3318,3033,
3480,3698,2800,3793,3860,2990,3507,3422,3750,2935,2010,3016,3449,1791,353-
0,3664,1981,3043,
3311,3854,1912,3501,3691,2007,3881,3432,3738,3527,2020,2556,3907,2813,288-
0,3316,2975,3506,
3640,3735,2934,2988,3029,3420,1790,3748,3677,1952,3042,3310,3391,3472,369-
0,1979,2006,3785,
3852,3445,3499,3526,2033,2927,3906,2784,3553,3687,2974,3877,2811,3734,300-
1,1904,2552,1999,
3761,2932,3200,3027,3295,3390,1788,3485,3675,1978,2984,3308,3865,3416,349-
8,3632,2004,3727,
2926,3443,3021,3524,3686,1975,3781,3848,2810,3495,2972,1917,3875,1998,373-
2,3437,3026,3294,
2025,2807,3674,2997,3321,3388,3483,3645,1024,3726,2911,3442,1784,2544,353-
7,3671,1974,3753,
2924,3304,3861,3019,3494,3684,1888,3779,3874,1535,2752,3969,3408,1915,199-
6,2806,3435,2968,
3292,1967,3482,3616,2995,3263,2910,2021,2557,3643,3670,3724,2937,3317,338-
4,3018,3479,3697,
1972,2799,3778,3859,1534,2989,3492,3421,1914,3749,2920,2009,3015,3434,177-
6,3529,3663,1966,
2804,3853,2994,1911,3262,1992,2528,3642,3431,3737,2908,3288,2019,2555,347-
8,3668,2798,2879,
3315,3858,2960,3505,3639,3720,2933,1532,2987,3014,3419,1789,1856,3747,366-
2,1951,3041,3309,
3376,1910,3471,3689,1964,2005,3851,3430,3260,3525,2018,2554,2688,3905,278-
3,2878,3314,2973,
3476,3638,2796,3733,2904,2986,1903,2551,3418,1760,3746,2931,3199,1950,301-
2,3280,3389,1787,
3470,3660,1977,1528,2983,3307,3850,3415,1908,3497,3631,2003,2925,3428,278-
2,3523,3685,1960,
2496,3847,2809,2876,3256,2971,1902,2550,3636,1997,3731,2930,3198,3025,329-
3,3360,1786,2792,
3673,1948,2982,3306,3387,3414,3468,3630,1023,2002,3725,2896,3441,1783,254-
3,3522,3656,1973,
2923,3303,3846,2780,3493,3683,2970,1887,3873,1520,2751,3730,3407,1900,254-
8,1995,2805,2872,
3196,2967,3291,3386,1728,3481,3615,1022,2980,3248,2909,3412,1782,2542,362-
8,3669,1944,3723,
2922,3302,3383,3017,3464,3682,1971,1886,3777,3844,1533,2750,3491,3406,191-
3,2919,1994,2776,
3433,1775,2966,3290,1965,2803,3614,2993,1896,3261,2432,1991,2527,3641,102-
0,3192,3722,2907,
3287,3382,1780,2540,3477,3667,1970,2797,2864,3300,3857,1504,2959,3490,362-
4,1884,3719,2918,
1531,2748,3013,3404,1774,1855,3661,1936,2802,3375,2964,1909,3232,1963,199-
0,2526,3612,3429,
3259,2906,3286,2017,2553,2687,3666,2768,2877,3313,3380,2958,3475,3637,101-
6,2795,3718,2903,
1530,2985,1664,2536,3417,1759,1854,3745,2916,3184,1949,3011,3279,3374,177-
2,1880,3469,3659,
1962,1527,2744,3849,3400,1907,3258,1988,2524,2686,3427,2781,2848,3284,195-
9,2495,3474,3608,
2794,2875,3255,2902,2956,1901,2549,3635,1758,3716,2929,3197,1472,3010,327-
8,3359,1785,1852,
2791,3658,1947,1526,2981,3305,3372,3413,1906,3467,3629,1008,2001,2895,342-
6,1768,2304,3521,
3655,1958,2494,2684,3845,2779,2874,3254,2969,1872,2520,3634,1519,2736,372-
9,2900,3168,1899,
2547,3358,1756,2790,2871,3195,1946,2952,3276,3385,1727,3466,3600,1021,152-
4,2979,3247,2894,
3411,1781,1848,2541,3627,3654,1943,2921,3301,3368,2778,3463,3681,1956,249-
2,1885,3843,1518,
2749,3252,3405,1898,2546,2680,1993,2775,2870,3194,2965,3289,3356,1726,278-
8,3613,992,2978,1
895,3246,2431,3410,1752,2512,3626,1019,3191,1942,3721,2892,3272,3381,1779-
,2539,3462,3652,1
969,1408,2863,3299,3842,1503,2720,3489,3623,1883,2917,1516,2747,2774,3403-
,1773,1840,2488,1
935,2801,2868,3136,2963,1894,3231,2430,1724,1989,2525,3611,1018,3190,3244-
,2905,3285,3352,1
778,2538,2672,3665,1940,2767,2862,3298,3379,1502,2957,3460,3622,1015,1882-
,3717,2888,1529,2
746,1663,2535,3402,1744,1853,2915,3183,1934,2772,3373,1771,2962,1879,3230-
,1961,1512,2743,3
610,3399,1892,3257,2428,1987,2523,2685,960,3188,2766,2847,3283,3378,1720,-
2480,3473,3607,10
14,2793,2860,3240,2901,1500,2955,1662,2534,3620,1757,1824,3715,2914,3182,-
1471,3009,3277,33
44,1770,1851,1878,3657,1932,1525,2742,3371,3398,1905,3228,1007,1986,2522,-
2656,3425,1767,23
03,2846,3282,1957,2493,2683,3606,2764,2873,3253,2424,2954,1871,2519,3633,-
1012,1280,2735,37
14,2899,3167,1470,1660,2532,3357,1755,1850,2789,2856,3180,1945,1496,2951,-
3275,3370,1712,18
76,3465,3599,1006,1523,2740,2893,3396,1766,1847,2302,3653,1928,2464,2682,-
3367,2777,2844,32
24,1955,2491,1870,2518,3604,1517,2734,3251,2898,3166,1897,2545,2679,1754,-
2760,2869,3193,14
68,2950,3274,3355,1725,896,2787,3598,991,1522,2977,1656,3245,2416,3409,17-
51,1846,2511,3625,
1004,3176,1941,2891,3271,3366,1764,2300,3461,3651,1954,2490,1407,2624,384-
1,1488,2719,3250,
1868,2516,2678,1515,2732,2773,2840,3164,1839,2487,3354,1696,2786,2867,313-
5,990,2948,1893,
3216,2429,1723,1750,2510,3596,1017,3189,1464,3243,2890,3270,3351,1777,184-
4,2537,2671,3650,
1939,1406,2861,3297,3364,1501,2718,3459,3621,1000,1881,2887,1514,2745,164-
8,2296,3401,1743,
1838,2486,2676,1933,2771,2866,3134,2961,1864,3229,2400,1722,1511,2728,360-
9,988,3160,1891,3
242,2427,3350,1748,2508,2670,959,3187,1938,2765,2832,3268,3377,1719,2479,-
3458,3592,1013,14
04,2859,3239,2886,1499,2716,1661,2533,3619,1742,1823,2913,3181,1456,2770,-
3343,1769,1836,24
84,1877,1931,1510,2741,3132,3397,1890,3227,2426,768,1985,2521,2655,958,31-
86,2288,2845,3281,
3348,1718,2478,2668,3605,984,2763,2858,3238,2423,1498,2953,1632,2504,3618-
,1011,1279,1822,
3713,2884,3152,1469,1659,2531,3342,1740,1849,1400,2855,3179,1930,1495,271-
2,3369,1711,1875,
3226,1005,1508,2739,2654,3395,1765,1832,2301,2368,1927,2463,2681,956,3128-
,2762,2843,3223,2
422,1716,2476,1869,2517,3603,1010,1278,2733,3236,2897,3165,1440,1658,2530-
,2664,1753,1820,2
759,2854,3178,1467,1494,2949,3273,3340,1710,895,1874,3597,976,1521,2738,1-
655,2415,3394,173
6,1845,2272,1003,3175,1926,2462,2652,3365,1763,2299,2842,3222,1953,2489,1-
392,2623,3602,148
7,2704,3249,2420,1867,2515,2677,952,1276,2731,2758,2839,3163,1466,1600,24-
72,3353,1695,894,
2785,2852,3120,989,1492,2947,1654,3215,2414,1708,1749,1816,2509,3595,1002-
,3174,1463,2889,3
269,3336,1762,1843,2298,3649,1924,2460,1405,2622,3363,1486,2717,3220,999,-
1866,2514,2648,15
13,2730,1647,2295,2838,3162,1837,2485,2675,1694,2756,2865,3133,512,2946,1-
863,3214,2399,172
1,892,1272,2727,3594,987,3159,1462,1652,3241,2412,3349,1747,1842,2507,266-
9,944,3172,1937,1
376,2831,3267,3362,1704,2240,3457,3591,998,1403,2620,2885,1484,2715,1646,-
2294,1741,1808,24
56,2674,1455,2769,2836,3104,1835,2483,1862,2398,1692,1509,2726,3131,986,3-
158,1889,3212,242
5,767,1746,2506,2640,957,3185,1460,2287,2830,3266,3347,1717,888,2477,2667-
,3590,983,1402,28
57,3237,2408,1497,2714,1631,2503,3617,996,1264,1821,2883,3151,1454,1644,2-
292,3341,1739,183
4,2482,1399,2616,1929,1480,2711,3130,1860,3225,2396,766,1507,2724,2653,92-
8,3156,1831,2286,
2367,3346,1688,2448,2666,955,3127,982,2761,2828,3208,2421,1715,2475,1630,-
2502,3588,1009,12
77,1344,3235,2882,3150,1439,1657,2529,2663,1738,1819,1398,2853,3177,1452,-
1493,2710,3339,17
09,880,1873,975,1506,2737,1640,2176,3393,1735,1830,2271,2366,764,1925,246-
1,2651,954,3126,2
284,2841,3221,2392,1714,2474,1391,2608,3601,980,1248,2703,3234,2419,1438,-
1628,2500,2662,95
1,1275,1818,2757,2824,3148,1465,1599,2471,3338,1680,893,1396,2851,3119,97-
4,1491,2708,1653,
2413,1707,1734,1815,2270,1001,3173,1448,2650,3335,1761,1828,2297,2364,192-
3,2459,1390,2621,
3124,1485,2702,3219,2418,760,1865,2513,2647,950,1274,2729,2280,2837,3161,-
1436,1598,2470,26
60,1693,864,2755,2850,3118,511,1490,2945,1624,3213,2384,1706,891,1271,181-
4,3593,972,3144,1
461,1651,2411,3334,1732,1841,2268,943,3171,1922,2458,1375,2592,3361,1703,-
2239,3218,997,138
8,2619,2646,1483,2700,1645,2293,2360,1807,2455,2673,948,1216,2754,2835,31-
03,510,1596,2468,
1861,2397,1691,890,1270,2725,3116,985,3157,1432,1650,3211,2410,752,1745,1-
812,2505,2639,942,
3170,1459,1374,2829,3265,3332,1702,887,2238,3589,968,1401,2618,2407,1482,-
2713,1616,2264,9
95,1263,1806,2454,2644,1453,1643,2291,2834,3102,1833,2481,1384,2615,1690,-
1479,2696,3129,50
8,1859,3210,2395,765,832,1268,2723,2638,927,3155,1458,1592,2285,2352,3345-
,1687,886,2447,26
65,940,3112,981,1372,2827,3207,2406,1700,2236,1629,2501,3587,994,1262,134-
3,2881,3149,1424,
1642,2290,1737,1804,2452,1397,2614,1451,1478,2709,3100,879,1858,2394,736,-
1505,2722,1639,21
75,926,3154,1829,2256,2365,763,1686,2446,2636,953,3125,504,2283,2826,3206-
,2391,1713,884,24
73,1152,2607,3586,979,1247,1342,3233,2404,1437,1627,2499,2661,936,1260,18-
17,1368,2823,3147,
1450,1584,2232,3337,1679,878,1395,2612,973,1476,2707,1638,2174,1733,1800,-
2269,2336,762,14
47,2649,924,3096,1827,2282,2363,2390,1684,2444,1389,2606,3123,978,1246,27-
01,3204,2417,759,
1626,2498,2632,949,1273,1340,2279,2822,3146,1435,1597,2469,2659,1678,863,-
1394,2849,3117,49
6,1489,2706,1623,2383,1705,876,1256,1813,971,3143,1446,1636,2172,3333,173-
1,1826,2267,2362,
704,1921,2457,1360,2591,3122,2224,3217,2388,758,1387,2604,2645,920,1244,2-
699,2278,2359,143
4,1568,2440,2658,947,1215,862,2753,2820,3088,509,1595,2467,1622,2382,1676-
,889,1269,1336,31
15,970,3142,1431,1649,2409,751,1730,1811,2266,941,3169,1444,1373,2590,333-
1,1701,872,2237,9
67,1386,2617,2168,1481,2698,1615,2263,2358,756,1805,2453,2643,946,1214,22-
76,2833,3101,480,
1594,2466,1383,2600,1689,860,1240,2695,3114,507,1430,1620,3209,2380,750,8-
31,1267,1810,2637,
912,3140,1457,1591,2351,3330,1672,885,2208,939,3111,966,1371,2588,2405,16-
99,2235,1614,226
2,993,1261,1328,2642,1423,1641,2289,2356,1803,2451,1382,2613,1212,1477,26-
94,3099,506,640,1
857,2393,735,830,1266,2721,2160,925,3153,1428,1590,2255,2350,748,1685,856-
,2445,2635,938,31
10,503,1370,2825,3205,2376,1698,883,2234,1151,3585,964,1232,1341,2403,142-
2,1612,2260,935,1
259,1802,2450,1367,2584,1449,1583,2231,3098,877,1380,2611,734,1475,2692,1-
637,2173,448,1799,
2254,2335,761,828,1208,2634,923,3095,502,1588,2281,2348,2389,1683,882,244-
3,1150,2605,3108,
977,1245,1312,3203,2402,744,1625,2497,2631,934,1258,1339,1366,2821,3145,1-
420,1582,2230,167
7,848,1393,2610,495,1474,2705,1608,2144,875,1255,1798,2334,732,1445,1635,-
2171,922,3094,182
5,2252,2361,703,1682,2442,1359,2576,3121,500,2223,3202,2387,757,824,1148,-
2603,2630,919,124
3,1338,2277,2344,1433,1567,2439,2657,932,1200,861,1364,2819,3087,494,1580-
,2228,1621,2381,1
675,874,1254,1335,969,3141,1416,1634,2170,1729,1796,2265,2332,702,1443,13-
58,2589,3092,871,
2222,2386,728,1385,2602,2167,918,1242,2697,2248,2357,755,1566,2438,2628,9-
45,1213,384,2275,
2818,3086,479,1593,2465,1144,2599,1674,859,1239,1334,3113,492,1429,1619,2-
379,749,816,1252,
1809,911,3139,1442,1576,2112,3329,1671,870,2207,700,965,1356,2587,2166,22-
20,1613,2261,2328,
754,1327,2641,916,1184,2274,2355,478,1564,2436,1381,2598,1211,858,1238,26-
93,3084,505,639,1
618,2378,720,829,1265,1332,2159,910,3138,1427,1589,2349,747,1670,855,2206-
,937,3109,488,136
9,2586,2375,1697,868,2233,1136,963,1231,1326,2164,1421,1611,2259,2354,696-
,1801,2449,1352,2
583,1210,2216,3097,476,638,1379,2596,733,800,1236,2691,2158,447,1426,1560-
,2253,2320,746,82
7,1207,854,2633,908,3080,501,1587,2347,2374,1668,881,2204,1149,3107,962,1-
230,1311,2401,743,
1610,2258,933,1257,1324,1365,2582,1419,1581,2229,847,1378,2609,256,1473,2-
690,1607,2143,446,
636,1797,2333,731,826,1206,2156,921,3093,472,1586,2251,2346,688,1681,852,-
2441,1120,2575,31
06,499,1310,3201,2372,742,823,1147,2629,904,1228,1337,2343,1418,1552,2200-
,931,1199,846,136
3,2580,493,1579,2227,1606,2142,873,1253,1320,730,1415,1633,2169,444,1795,-
2250,2331,701,120
4,1357,2574,3091,498,632,2221,2385,727,822,1146,2601,2152,917,1241,1308,2-
247,2342,740,1565,
2437,2627,930,1198,383,1362,2817,3085,464,1578,2226,1143,1673,844,1224,13-
33,491,1414,1604,
2140,815,1251,1794,2330,672,1441,1575,2111,3090,869,2192,699,726,1355,257-
2,2165,440,2219,2
246,2327,753,820,1088,2626,915,1183,382,2273,2340,477,1563,2435,1142,2597-
,1196,857,1237,13
04,3083,490,624,1617,2377,719,814,1250,1331,909,3137,1412,1574,2110,1669,-
840,2205,698,487,1
354,2585,2136,867,2218,1135,2326,724,1325,2163,914,1182,2244,2353,695,156-
2,2434,1351,2568,
1209,380,2215,3082,475,637,1140,2595,718,799,1235,1330,2157,432,1425,1559-
,2319,745,812,119
2,853,907,3079,486,1572,2108,2373,1667,866,2203,1134,961,1229,1296,2162,1-
609,2257,2324,694,
1323,1350,2581,1180,2214,474,608,1377,2594,255,798,1234,2689,2128,445,635-
,1558,2318,716,82
5,1205,376,2155,906,3078,471,1585,2345,687,1666,851,2202,1119,3105,484,13-
09,2371,741,808,11
32,903,1227,1322,2104,1417,1551,2199,692,845,1348,2579,254,2212,1605,2141-
,416,634,1319,729,
796,1176,2154,443,470,1556,2249,2316,686,1203,850,1118,2573,3076,497,631,-
2370,712,821,1145,
2151,902,1226,1307,2341,739,1550,2198,929,1197,368,1361,2578,463,1577,222-
5,1128,843,1223,1
318,252,1413,1603,2139,442,576,1793,2329,671,1202,2096,3089,468,630,2191,-
684,725,792,1116,2
571,2150,439,1306,2245,2312,738,819,1087,2625,900,1168,381,2339,462,1548,-
2196,1141,1195,84
2,1222,1303,489,623,1602,2138,813,1249,1316,670,1411,1573,2109,839,2190,6-
97,248,1353,2570,2
135,438,628,2217,2325,723,818,1086,2148,913,1181,352,2243,2338,680,1561,2-
433,1112,2567,119
4,379,1302,3081,460,622,1139,717,784,1220,1329,431,1410,1544,2080,811,119-
1,838,668,485,1571,
2107,2134,865,2188,1133,722,1295,2161,436,2242,2323,693,1084,1349,2566,11-
79,378,2213,473,6
07,1138,2593,240,797,1233,1300,2127,430,620,1557,2317,715,810,1190,375,90-
5,3077,456,1570,21
06,1665,836,2201,1104,483,1294,2132,807,1131,2322,664,1321,2103,1178,2184-
,691,606,1347,256
4,253,320,2211,2126,415,633,1080,714,795,1175,374,2153,428,469,1555,2315,-
685,1188,849,1117,
3075,482,616,2369,711,806,1130,901,1225,1292,2102,1549,2197,690,367,1346,-
2577,224,2210,112
7,414,604,1317,251,794,1174,2124,441,575,1554,2314,656,1201,372,2095,3074-
,467,629,683,710,7
91,1115,2149,424,1305,2311,737,804,1072,899,1167,366,2100,461,1547,2195,1-
126,841,1221,1288,
250,1601,2137,412,574,1315,669,1172,2094,466,600,2189,682,247,790,1114,25-
69,2120,437,627,23
10,708,817,1085,2147,898,1166,351,2337,679,1546,2194,1111,1193,364,1301,4-
59,621,1124,783,12
19,1314,192,1409,1543,2079,572,837,667,246,2092,2133,408,626,2187,721,788-
,1056,2146,435,350,
2241,2308,678,1083,1110,2565,1164,377,458,592,1137,239,782,1218,1299,429,-
619,1542,2078,809,
1189,360,666,455,1569,2105,835,2186,1103,244,1293,2131,434,568,2321,663,1-
082,2088,1177,348,
2183,676,605,1108,2563,238,319,1298,2125,400,618,1079,713,780,1160,373,42-
7,454,1540,2076,1
187,834,1102,481,615,2130,805,1129,662,1291,2101,2182,689,128,1345,2562,2-
23,318,2209,413,60
3,1078,236,793,1173,344,2123,426,560,1553,2313,655,1186,371,3073,452,614,-
709,776,1100,423,1
290,2072,803,1071,660,365,2099,222,2180,1125,602,1287,249,316,2122,411,57-
3,1076,654,1171,37
0,2093,465,599,681,232,789,1113,2119,422,612,2309,707,802,1070,897,1165,3-
36,2098,1545,2193,
1096,363,1286,220,1123,410,544,1313,191,1170,2064,571,598,652,245,312,209-
1,2118,407,625,706,
787,1055,2145,420,349,2307,677,1068,1109,1163,362,457,591,1122,781,1217,1-
284,190,1541,2077,
570,359,665,216,2090,406,596,2185,243,786,1054,2116,433,567,2306,648,1081-
,2087,1162,347,67
5,590,1107,237,304,1297,399,617,1064,779,1159,358,188,453,1539,2075,833,1-
101,242,2129,404,5
66,661,1052,2086,346,2181,674,127,1106,2561,208,317,398,588,1077,235,778,-
1158,343,425,559,1
538,2074,1185,356,451,613,775,1099,184,1289,2071,564,659,126,2084,221,288-
,2179,601,1048,234,
315,342,2121,396,558,1075,653,1156,369,450,584,231,774,1098,421,611,2070,-
801,1069,658,335,2
097,2178,1095,124,1285,219,314,409,543,1074,176,1169,340,2063,556,597,651-
,230,311,2117,392,
610,705,772,1040,419,334,2068,1067,1094,361,218,1121,542,1283,189,2062,56-
9,120,650,215,310,
2089,405,595,228,785,1053,2115,418,552,2305,647,1066,1161,332,589,1092,30-
3,1282,160,1063,54
0,357,187,214,2060,594,241,308,2114,403,565,646,1051,2085,345,673,112,110-
5,207,302,397,587,1
062,777,1157,328,186,1537,2073,355,212,402,536,183,1050,2056,563,644,125,-
2083,206,287,586,1
047,233,300,341,395,557,1060,1155,354,449,583,773,1097,182,2069,562,657,9-
6,2082,286,2177,12
3,1046,204,313,394,528,1073,175,1154,339,555,582,229,296,391,609,771,1039-
,180,333,2067,1093,
122,217,284,541,1044,174,338,2061,554,119,649,200,309,390,580,227,770,103-
8,417,551,2066,106
5,331,1091,64,1281,159,539,118,172,213,280,2059,593,226,307,2113,388,550,-
645,1036,330,111,10
90,301,158,1061,538,327,185,2058,116,211,306,401,535,168,1049,2055,548,64-
3,110,205,272,585,1
032,299,326,156,1059,353,210,534,181,2054,561,642,95,2081,285,108,1045,20-
3,298,393,527,1058,
1153,324,581,295,152,532,179,94,2052,121,202,283,526,1043,173,337,553,104-
,199,294,389,579,76
9,1037,178,2065,92,282,63,1042,144,524,117,171,198,279,578,225,292,387,54-
9,1035,329,1089,62,
157,537,88,170,278,2057,115,196,305,386,520,167,1034,547,109,271,1031,60,-
325,155,114,209,276,
533,166,2053,546,641,80,270,107,1030,297,154,1057,323,56,151,531,164,93,2-
051,106,201,268,52
5,1028,322,103,293,150,530,177,2050,91,281,48,1041,143,523,102,197,264,57-
7,291,148,90,61,142,
522,87,169,277,100,195,290,385,519,1033,32,59,86,140,113,194,275,518,165,-
545,79,269,1029,58,1
53,84,274,55,136,516,163,78,105,267,1027,321,54,149,529,162,2049,76,266,4-
7,1026,101,263,52,14
7,89,46,141,521,72,262,99,289,146,31,44,85,139,98,193,260,517,30,57,138,8-
3,273,40,135,515,77,2
8,82,53,134,514,161,75,265,1025,24,51,132,74,45,71,261,50,145,16,43,70,97-
,259,29,42,137,68,258,
39,27,81,38,133,513,26,23,36,131,73,22,49,130,15,69,20,14,41,67,257,12,66-
,37,25,8,35,21,34,129,1 9,13,18,11,65,10,7,6,33,4,17,9,5,3,2,1.
[0195] The polarization weights in the first polarization weight
vector may alternatively be sorted in ascending order. Herein, a
minimum sequence number value is 1. Similarly, if a minimum
sequence number value is 0, only subtracting 1 from all sequence
numbers is required.
[0196] (1) N=16, and the first polarization weight vector is:
[0197] 1,2,3,5,9,4,6,7,10,11,13,8,12,14,15,16.
[0198] (2) N=32, and the first polarization weight vector is:
[0199]
1,2,3,5,9,17,4,6,7,10,11,18,13,19,21,8,25,12,14,20,15,22,23,26,27,2-
9,16,24,28,30,31,32.
[0200] (3) N=64, and the first polarization weight vector is:
[0201]
1,2,3,5,9,17,4,33,6,7,10,11,18,13,19,34,21,35,8,25,37,12,41,14,20,1-
5,49,22,36,23,26,38,2
7,39,42,29,43,16,50,45,51,24,53,28,40,57,30,44,31,46,52,47,54,55,58,59,32-
,61,48,56,60,62,63,64.
[0202] (4) N=128, and the first polarization weight vector is:
[0203]
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,34,21,35,8,25,37,66,12,67,4-
1,14,20,69,15,49,22,7
3,36,23,26,38,81,27,39,68,42,29,97,70,43,16,50,71,45,74,51,24,75,53,82,28-
,77,40,83,57,30,98,85,44,
31,99,72,46,89,52,101,47,76,54,105,78,55,84,58,79,113,86,59,32,100,87,61,-
90,102,48,91,103,106,9
3,56,107,80,114,60,109,115,88,62,117,63,92,104,121,94,108,95,110,116,111,-
118,64,119,122,123,96, 125,112,120,124,126,127,128.
[0204] (5) N=256, and the first polarization weight vector is:
[0205]
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,-
67,41,14,20,69,15,13
0,49,22,73,131,36,23,26,133,38,81,27,39,68,137,42,29,97,70,43,16,145,50,7-
1,45,74,132,51,24,75,16
1,134,53,82,28,77,135,40,83,138,57,30,193,98,139,85,44,31,146,99,72,141,4-
6,89,147,52,101,47,76,
162,149,54,105,78,163,136,55,84,153,58,79,165,194,113,140,86,59,32,195,10-
0,169,87,142,61,90,14
8,197,102,143,48,91,177,150,103,201,106,93,164,151,56,154,107,80,166,209,-
114,155,60,109,167,1
96,115,170,88,157,62,225,198,171,117,144,63,92,178,199,104,173,202,121,94-
,179,152,203,108,95,
181,210,156,205,110,168,211,116,185,158,111,226,213,172,118,159,64,227,20-
0,119,174,217,122,18
0,229,175,204,123,96,182,233,206,125,183,212,186,207,112,241,214,187,160,-
228,215,120,189,218,
230,176,219,124,231,234,221,126,184,235,208,127,242,188,237,243,216,190,2-
45,191,220,232,249,
222,236,223,128,238,244,239,246,192,247,250,251,224,253,240,248,252,254,2-
55,256.
[0206] (6) N=512, and the first polarization weight vector is:
[0207]
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,-
257,67,41,14,20,69,1
5,130,49,22,73,131,36,23,26,133,38,81,27,39,258,68,137,42,29,259,97,70,43-
,16,145,50,261,71,45,7
4,132,51,24,265,75,161,134,53,82,28,77,135,40,273,83,138,57,30,260,193,98-
,139,85,44,31,146,289,
99,262,72,141,46,89,147,52,263,101,47,266,76,162,149,54,321,267,105,78,16-
3,136,55,274,84,153,5
8,269,79,165,275,194,113,140,86,59,32,385,290,195,100,277,169,87,142,61,9-
0,148,291,264,197,10
2,143,48,281,91,177,150,293,103,322,268,201,106,93,164,151,56,323,154,297-
,107,270,80,166,276,
209,114,155,325,60,271,109,167,386,305,196,115,278,170,88,157,62,329,387,-
292,225,279,198,171,
117,144,63,282,92,178,389,294,199,104,337,173,283,202,121,94,179,152,295,-
324,393,298,203,108,
285,95,181,210,353,156,326,299,272,205,110,168,401,306,211,116,185,327,15-
8,301,111,330,388,3
07,226,280,213,172,118,159,64,331,417,227,390,309,200,119,338,174,284,217-
,122,333,180,391,29
6,229,339,175,394,313,204,123,286,96,182,449,354,395,341,300,233,287,206,-
125,183,402,212,355,
186,328,397,302,207,112,345,403,308,241,214,187,357,160,303,332,418,228,4-
05,310,215,120,189,
218,361,334,419,392,311,230,340,176,409,314,219,124,335,421,231,450,369,3-
96,342,315,234,288,
221,126,184,451,356,425,343,235,398,317,208,127,346,404,242,453,188,358,3-
99,304,237,347,433,
243,406,216,359,190,457,362,349,420,407,312,245,191,410,220,363,336,422,2-
32,465,370,411,316,
249,222,365,423,452,371,426,344,236,413,318,223,128,481,454,427,373,400,3-
19,238,348,434,244,
455,360,429,239,458,377,350,435,408,246,192,459,364,351,437,247,466,412,2-
50,461,366,424,467,
372,441,251,414,224,367,482,469,428,374,415,320,253,483,456,375,430,240,4-
73,378,436,485,431,
460,379,352,438,248,489,462,381,439,468,442,252,463,368,497,470,443,416,2-
54,484,471,376,445,
255,474,486,432,475,380,487,490,477,382,440,491,464,383,498,444,493,499,4-
72,446,256,501,447,
476,488,505,478,492,479,384,494,500,495,502,448,503,506,507,480,509,496,5-
04,508,510,511,512.
[0208] (7) N=1024, and the first polarization weight vector is:
[0209]
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,-
257,67,41,14,20,69,1
5,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,7-
0,43,16,145,50,261,71,
45,74,132,51,24,265,75,161,514,134,53,82,28,77,515,135,40,273,83,138,57,3-
0,517,260,193,98,139,
85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,47,266,76,162,529-
,149,54,321,267,105,7
8,163,516,136,55,274,84,153,58,269,79,545,165,518,275,194,113,140,86,59,3-
2,519,385,290,195,10
0,277,169,87,522,142,61,90,148,291,577,264,197,102,523,143,48,281,91,177,-
530,150,293,103,322,
525,268,201,106,93,164,531,151,56,323,154,297,107,270,80,641,546,166,533,-
276,209,114,155,325,
60,271,109,547,167,520,386,305,196,115,278,170,88,537,157,62,329,549,387,-
292,225,578,279,198,
171,117,524,144,63,282,92,178,769,579,389,294,199,104,553,337,173,526,283-
,202,121,94,179,532,
152,295,581,324,527,393,298,203,108,285,95,642,561,181,534,210,353,156,32-
6,299,585,272,205,1
10,643,548,168,535,401,306,211,116,185,327,538,158,301,111,330,645,550,38-
8,307,226,593,280,2
13,172,118,539,159,64,331,551,417,770,227,580,390,309,200,649,119,554,338-
,174,541,284,217,12
2,333,180,771,609,391,296,229,582,555,339,175,528,394,313,204,123,286,96,-
657,562,182,773,583,
449,354,557,395,341,300,233,586,287,206,125,644,563,183,536,402,212,355,1-
86,328,777,587,397,
302,207,112,673,345,646,565,403,308,241,594,214,187,357,540,160,303,589,3-
32,647,552,418,785,
228,595,405,310,215,650,120,569,189,542,218,361,334,419,772,705,610,392,3-
11,230,651,597,556,
340,176,543,409,314,219,124,335,658,801,611,421,774,231,584,450,369,653,5-
58,396,342,315,234,
601,288,221,126,659,564,184,775,613,451,356,559,425,343,778,235,588,398,3-
17,208,127,674,346,
661,566,404,242,833,453,188,358,779,617,399,304,237,590,675,347,648,567,4-
33,786,243,596,406,
216,665,359,570,190,781,591,457,362,677,349,420,787,706,625,407,312,245,6-
52,598,571,191,544,
410,220,363,336,897,802,707,612,422,789,232,681,599,465,370,654,573,411,3-
16,249,602,222,365,
660,803,423,776,709,614,452,371,655,560,426,344,793,236,603,413,318,223,1-
28,689,662,805,615,
481,834,454,427,373,780,713,618,400,319,238,605,676,348,663,568,434,244,8-
35,455,666,360,809,
619,429,782,239,592,458,377,678,350,435,788,721,626,408,246,667,837,572,1-
92,783,621,459,364,
679,351,898,817,708,627,437,790,247,682,600,466,669,574,412,250,841,461,3-
66,899,804,737,424,
791,710,683,629,467,372,656,575,441,794,251,604,414,224,367,690,901,806,7-
11,616,482,849,685,
469,428,374,795,714,633,415,320,253,606,691,664,807,483,836,456,905,375,8-
10,715,620,430,797,
240,607,473,378,693,436,722,865,485,668,838,811,431,784,717,622,460,379,6-
80,352,913,818,723,
628,438,248,697,839,670,813,623,489,842,462,381,900,819,738,439,792,725,6-
84,630,468,671,576,
442,252,843,463,368,929,739,902,821,712,631,497,850,686,470,443,796,729,6-
34,416,254,845,692,
903,808,741,484,851,687,471,906,376,825,716,635,445,798,255,608,474,694,9-
61,866,486,907,853,
812,745,432,799,718,637,475,380,695,914,724,867,487,698,840,909,814,719,6-
24,490,857,477,382,
915,820,753,440,726,699,869,672,815,491,844,464,383,930,740,917,822,727,6-
32,498,701,444,730,
873,493,846,931,904,823,742,499,852,688,472,921,826,731,636,446,256,847,9-
33,743,962,881,501,
908,854,827,746,447,800,733,638,476,696,963,868,488,937,855,747,910,829,7-
20,639,505,858,478,
916,754,965,700,870,911,816,749,492,859,479,384,945,755,918,728,871,702,9-
69,874,494,861,932,
919,824,757,500,703,922,732,875,495,848,934,744,977,882,502,923,828,761,4-
48,734,877,935,964,
883,503,938,856,748,925,830,735,640,506,993,966,939,885,912,831,750,507,8-
60,480,946,756,967,
872,941,751,970,889,509,862,947,920,758,704,971,876,496,863,949,759,978,9-
24,762,973,878,936,
979,884,504,953,763,926,736,879,994,981,940,886,927,832,765,508,995,968,8-
87,942,752,985,890,
510,948,997,943,972,891,511,864,950,760,1001,974,893,951,980,954,764,975,-
880,1009,982,955,92
8,766,996,983,888,957,767,986,998,944,987,892,512,999,1002,989,894,952,10-
03,976,895,1010,956,
1005,1011,984,958,768,1013,959,988,1000,1017,990,1004,991,896,1006,1012,1-
007,1014,960,1015,
1018,1019,992,1021,1008,1016,1020,1022,1023,1024.
[0210] (8) N=2048, and the first polarization weight vector is:
[0211]
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,-
257,67,41,14,20,69,1
5,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,7-
0,43,16,145,50,261,71,
45,74,132,51,24,1025,265,75,161,514,134,53,82,28,77,515,135,40,273,83,138-
,57,30,517,260,193,98,
139,85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,1026,47,266,7-
6,162,529,149,54,321,
1027,267,105,78,163,516,136,55,274,84,153,58,1029,269,79,545,165,518,275,-
194,113,140,86,59,3
2,1033,519,385,290,195,100,277,169,87,522,142,61,90,148,291,577,264,197,1-
02,523,143,1041,48,2
81,91,177,530,150,293,103,322,1028,525,268,201,106,93,164,531,151,56,323,-
1057,154,297,1030,1
07,270,80,641,546,166,533,276,209,114,155,325,60,1031,271,109,547,1034,16-
7,520,386,305,196,1
15,278,170,88,537,157,62,1089,329,1035,549,387,292,225,578,279,198,171,11-
7,524,144,1042,63,2
82,92,178,1037,769,579,389,294,199,104,553,337,173,1043,526,283,202,121,9-
4,179,532,152,295,5
81,324,1153,1058,527,393,298,203,1045,108,285,95,642,561,181,534,210,353,-
1059,156,326,299,10
32,585,272,205,110,643,548,1049,168,535,401,306,211,116,185,327,538,1061,-
158,301,1090,111,33
0,1036,645,550,388,307,226,593,280,213,172,118,539,159,1281,64,1091,331,1-
065,551,417,1038,77
0,227,580,390,309,200,649,119,554,338,174,1044,541,284,217,122,1093,333,1-
80,1039,771,609,391,
296,229,582,555,339,1154,175,1073,528,394,313,204,1046,123,286,96,657,562-
,182,1097,773,583,
449,354,1155,1060,557,395,341,300,233,1047,586,287,206,125,644,563,1050,1-
83,536,402,212,355,
1537,186,328,1157,777,1062,587,397,302,207,1105,112,673,345,1051,646,565,-
403,308,241,594,21
4,187,357,540,1063,160,1282,303,1092,589,332,1161,1066,647,552,418,1053,7-
85,228,595,405,310,
215,650,120,569,189,1283,542,1121,218,361,1094,1067,334,419,1040,772,705,-
610,392,311,230,65
1,597,556,340,1169,176,1074,543,409,314,219,1285,124,1095,335,658,1069,80-
1,611,421,1098,774,
231,584,450,369,1156,653,1075,558,396,342,315,234,1048,601,288,221,126,65-
9,564,1289,184,109
9,775,613,451,356,1185,1538,559,425,343,1158,778,235,1077,588,398,317,208-
,1106,127,674,346,1
052,661,566,404,242,1101,833,1539,453,188,358,1159,779,1064,617,399,1297,-
304,237,1107,590,6
75,347,1162,1081,648,567,433,1054,786,243,596,406,216,665,359,570,1541,19-
0,1284,1217,781,11
22,591,457,362,1163,1109,1068,677,349,420,1055,787,706,625,407,312,245,65-
2,598,571,1170,191,
1313,544,410,1123,220,1286,363,1096,1545,336,1165,897,1070,802,707,612,42-
2,1113,789,232,681,
599,465,370,1171,654,1076,573,411,316,249,1287,602,1125,222,365,660,1071,-
803,1290,423,1100,
776,709,614,452,371,1186,655,1553,560,426,344,1173,793,236,1078,603,413,3-
18,223,1345,128,68
9,1291,662,1129,805,615,481,1102,834,1187,1540,454,427,373,1160,780,713,1-
079,618,400,1298,3
19,238,1108,605,676,348,1177,1082,663,568,434,1293,244,1103,835,1569,455,-
666,360,1189,809,1
542,619,429,1299,1218,782,239,1137,592,458,377,1164,1110,1083,678,350,435-
,1056,788,721,626,
408,246,667,837,572,1543,1409,192,1314,1219,783,1124,621,459,1301,364,119-
3,1111,1546,679,35
1,1166,898,1085,817,708,627,437,1114,790,247,682,600,466,1172,669,1315,57-
4,412,1601,250,128
8,1221,841,1126,1547,461,366,1167,899,1072,804,737,1305,424,1115,791,710,-
683,629,467,372,12
01,656,1554,575,441,1174,794,251,1317,604,414,1127,224,1346,367,690,1549,-
1292,1225,901,1130,
806,711,616,482,1117,849,1188,685,1555,469,428,374,1175,795,714,1080,633,-
415,320,253,1347,6
06,691,1178,1321,664,1131,807,1294,483,1104,836,1665,1570,456,905,375,119-
0,810,715,1557,620,
430,1300,1233,797,240,1138,607,473,378,1179,1349,1084,693,436,1295,722,11-
33,865,1571,485,6
68,838,1191,811,1544,1410,431,1329,1220,784,717,1139,622,460,1302,379,119-
4,1112,1561,680,35
2,1181,913,1086,818,723,628,438,1353,248,697,839,1573,1411,670,1316,1249,-
813,1602,623,489,1
303,1222,842,1195,1141,1548,462,381,1168,900,1087,819,738,1306,439,1116,7-
92,725,684,630,468,
1202,671,1793,576,442,1603,1413,252,1318,1223,843,1128,1577,463,1361,368,-
1197,929,1550,739,
1307,1226,902,1145,821,712,631,497,1118,850,1203,686,1556,470,443,1176,79-
6,729,1319,634,41
6,1605,254,1348,845,692,1551,1417,1322,1227,903,1132,808,741,1309,484,111-
9,851,1666,687,158
5,471,906,376,1205,825,716,1558,635,445,1234,798,255,1377,608,474,1180,13-
50,1323,694,1609,1
296,1229,961,1134,866,1667,1572,486,907,853,1192,812,745,1559,1425,432,13-
30,1235,799,718,11
40,637,475,380,1209,1351,1562,695,1182,914,1325,724,1135,867,1354,487,698-
,840,1669,1574,141
2,909,1331,1250,814,719,1617,624,490,1304,1237,857,1196,1142,1563,477,382-
,1183,915,1088,820,
753,440,1355,726,699,869,1575,1441,672,1794,1251,815,1604,1414,491,1333,1-
224,844,1673,1143,
1578,464,1362,383,1198,930,1565,740,1308,1241,917,1146,822,727,632,498,13-
57,1204,701,1795,
444,1633,1415,730,1320,1253,873,1606,1579,493,1363,846,1199,931,1552,1418-
,1337,1228,904,11
47,823,742,1310,499,1120,852,1681,688,1586,472,921,1206,826,731,1797,636,-
446,1607,1473,256,
1378,847,1581,1419,1365,1324,1257,933,1610,743,1311,1230,962,1149,881,166-
8,1587,501,908,85
4,1207,827,746,1560,1426,447,1236,800,733,1379,638,476,1210,1352,1801,696-
,1611,1421,1326,12
31,963,1136,868,1697,1369,488,937,855,1670,747,1589,1427,910,1332,1265,82-
9,720,1618,639,505,
1238,858,1211,1381,1564,478,1184,916,1327,754,1613,1356,965,700,870,1671,-
1576,1442,911,180
9,1252,816,749,1619,1429,492,1334,1239,859,1674,1144,1593,479,384,1213,94-
5,1566,755,1242,91
8,1385,728,871,1358,1443,702,1796,1729,1634,1416,969,1335,1254,874,1675,1-
621,1580,494,1364,
861,1200,932,1567,1433,1338,1243,919,1148,824,757,500,1359,1682,703,1825,-
922,1635,1445,732,
1798,1255,875,1608,1474,495,1393,848,1677,1582,1420,1366,1339,1258,934,16-
25,744,1312,1245,
977,1150,882,1683,1588,502,923,1208,828,761,1799,448,1637,1475,734,1380,8-
77,1583,1449,1367,
1802,1259,935,1612,1422,1341,1232,964,1151,883,1698,1370,503,938,856,1685-
,748,1590,1428,92
5,1266,830,735,1857,640,506,1477,1212,1382,1803,1641,1423,1328,1261,993,1-
614,1699,1371,966,
939,885,1672,1591,1457,912,1810,1267,831,750,1620,1430,507,1240,860,1689,-
1383,1594,480,121
4,946,1805,756,1615,1481,1386,967,872,1701,1373,1444,941,1811,1730,751,16-
49,1431,970,1336,1
269,889,1676,1622,1595,509,862,1215,947,1568,1434,1244,920,1387,758,1360,-
1921,704,1826,173
1,1636,1446,971,1813,1256,876,1705,1623,1489,496,1394,863,1678,1597,1435,-
1340,1273,949,162
6,759,1246,978,1389,1684,1827,924,1447,762,1800,1733,1638,1476,973,1395,8-
78,1679,1584,1450,
1368,1817,1260,936,1627,1437,1342,1247,979,1152,884,1713,504,953,1686,763-
,1829,926,1639,15
05,736,1858,879,1478,1451,1397,1804,1737,1642,1424,1343,1262,994,1629,170-
0,1372,981,940,88
6,1687,1592,1458,927,1268,832,765,1859,508,1479,1690,1384,1833,1643,1453,-
1806,1263,995,161
6,1482,1401,968,887,1702,1374,1459,942,1812,1745,752,1650,1432,985,1270,8-
90,1691,1861,1596,
510,1216,948,1807,1645,1483,1388,997,1703,1375,1922,943,1841,1732,1651,14-
61,972,1814,1271,
891,1706,1624,1490,511,864,1693,1598,1436,1274,950,1865,760,1485,1390,192-
3,1828,1761,1448,
1001,1815,1734,1707,1653,1491,974,1396,893,1680,1599,1465,1818,1275,951,1-
628,1438,1248,980,
1391,1714,954,1925,764,1830,1735,1640,1506,975,1873,880,1709,1493,1452,13-
98,1819,1738,165
7,1439,1344,1277,1009,1630,1715,982,955,1688,1831,928,1507,766,1860,1480,-
1929,1399,1834,17
39,1644,1454,1821,1264,996,1631,1497,1402,983,888,1717,1460,957,1746,767,-
1889,986,1509,169
2,1862,1835,1455,1808,1741,1646,1484,1403,998,1704,1376,1937,944,1842,174-
7,1652,1462,987,1
272,892,1721,1863,512,1694,1837,1647,1513,1866,999,1486,1405,1924,1843,17-
62,1463,1002,1816,
1749,1708,1654,1492,989,894,1695,1600,1466,1276,952,1867,1487,1392,1953,1-
763,1926,1003,18
45,1736,1655,1521,976,1874,895,1710,1494,1467,1820,1753,1658,1440,1278,10-
10,1869,1716,956,
1927,1832,1765,1508,1005,1875,1711,1495,1930,1400,1849,1740,1659,1469,182-
2,1279,1011,1632,
1498,984,1718,958,1985,768,1890,1510,1931,1877,1836,1769,1456,1823,1742,1-
661,1499,1404,101
3,1719,1938,959,1748,1891,988,1511,1722,1864,1933,1838,1743,1648,1514,188-
1,1000,1501,1406,
1939,1844,1777,1464,1017,1750,1723,1893,990,1696,1839,1515,1868,1488,1407-
,1954,1764,1941,1
004,1846,1751,1656,1522,991,896,1725,1468,1754,1897,1517,1870,1955,1928,1-
847,1766,1523,100
6,1876,1712,1496,1945,1850,1755,1660,1470,1280,1012,1871,1957,1767,1986,1-
007,1905,1525,193
2,1878,1851,1770,1471,1824,1757,1662,1500,1014,1720,960,1987,1892,1512,19-
61,1879,1771,1934,
1853,1744,1663,1529,1882,1015,1502,1940,1778,1018,1989,1724,1894,1935,184-
0,1773,1516,1883,
1503,1408,1969,1779,1942,1019,1752,1895,992,1726,1993,1898,1518,1885,1956-
,1943,1848,1781,
1524,1021,1727,1946,1756,1899,1519,1872,1958,1768,2001,1008,1906,1526,194-
7,1852,1785,1472,
1758,1901,1959,1988,1907,1527,1962,1880,1772,1949,1854,1759,1664,1530,101-
6,2017,1990,1963,
1909,1936,1855,1774,1531,1884,1504,1970,1780,1020,1991,1896,1965,1775,199-
4,1913,1533,1886,
1971,1944,1782,1022,1728,1995,1900,1520,1887,1973,1783,2002,1023,1948,178-
6,1997,1902,1960,
2003,1908,1528,1977,1787,1950,1760,1903,2018,2005,1964,1910,1951,1856,178-
9,1532,2019,1992,
1911,1966,1776,2009,1914,1534,1972,2021,1967,1996,1915,1535,1888,1974,178-
4,1024,2025,1998,
1917,1975,2004,1978,1788,1999,1904,2033,2006,1979,1952,1790,2020,2007,191-
2,1981,1791,2010,
2022,1968,2011,1916,1536,2023,2026,2013,1918,1976,2027,2000,1919,2034,198-
0,2029,2035,2008,
1982,1792,2037,1983,2012,2024,2041,2014,2028,2015,1920,2030,2036,2031,203-
8,1984,2039,2042, 2043,2016,2045,2032,2040,2044,2046,2047,2048.
[0212] (9) N=4096, and the first polarization weight vector is:
1,2,3,5,9,17,4,33,6,7,10,65,11,18,13,19,129,34,21,35,8,25,37,66,12,257,67-
,41,14,20,69,1
5,130,49,22,73,131,36,23,26,513,133,38,81,27,39,258,68,137,42,29,259,97,7-
0,43,16,145,50,261,71,
45,74,132,51,24,1025,265,75,161,514,134,53,82,28,77,515,135,40,273,83,138-
,57,30,517,260,193,98,
139,85,44,31,146,289,99,262,72,521,141,46,89,147,52,263,101,1026,47,266,7-
6,2049,162,529,149,5
4,321,1027,267,105,78,163,516,136,55,274,84,153,58,1029,269,79,545,165,51-
8,275,194,113,140,86,
59,32,1033,519,385,290,195,100,277,169,87,522,142,61,90,148,291,577,264,1-
97,102,523,143,1041,
48,281,91,2050,177,530,150,293,103,322,1028,525,268,201,106,2051,93,164,5-
31,151,56,323,1057,
154,297,1030,107,270,80,641,546,2053,166,533,276,209,114,155,325,60,1031,-
271,109,547,1034,16
7,520,386,305,196,115,2057,278,170,88,537,157,62,1089,329,1035,549,387,29-
2,225,578,279,198,1
71,117,524,144,1042,63,282,92,2065,178,1037,769,579,389,294,199,104,553,3-
37,173,1043,526,283,
202,121,2052,94,179,532,152,295,581,324,1153,1058,527,393,298,203,1045,10-
8,285,2081,95,642,
561,2054,181,534,210,353,1059,156,326,299,1032,585,272,205,110,643,548,20-
55,1049,168,535,40
1,306,211,116,2058,185,327,538,1061,158,301,1090,111,330,1036,645,550,388-
,2113,307,226,593,2
059,280,213,172,118,539,159,1281,64,1091,331,1065,2066,551,417,1038,770,2-
27,580,390,309,200,
649,119,554,2061,338,174,1044,541,284,217,122,1093,2067,333,180,1039,771,-
609,391,296,229,58
2,555,339,1154,175,1073,528,394,313,204,1046,123,2177,286,2082,96,657,562-
,2069,182,1097,773,
583,449,354,1155,1060,557,395,341,300,233,1047,586,287,206,2083,125,644,5-
63,2056,1050,183,5
36,402,212,355,2073,1537,186,328,1157,777,1062,587,397,302,207,1105,112,6-
73,345,2085,1051,6
46,565,403,2114,308,241,594,2060,214,187,357,540,1063,160,1282,303,1092,5-
89,332,1161,1066,6
47,2305,552,418,2115,1053,785,228,595,405,2089,310,215,650,120,569,2062,1-
89,1283,542,1121,2
18,361,1094,1067,2068,334,419,1040,772,705,610,392,2117,311,230,651,597,5-
56,2063,340,1169,1
76,1074,543,409,314,219,1285,124,1095,2178,2097,335,658,1069,801,2070,611-
,421,1098,774,231,
584,450,369,1156,653,1075,558,396,2121,342,315,234,1048,601,2179,288,221,-
2084,126,659,564,2
071,1289,184,1099,775,613,451,356,1185,2074,1538,559,425,343,1158,778,235-
,1077,588,398,317,
208,2561,1106,127,674,2181,346,2086,1052,661,566,404,2129,242,1101,833,20-
75,1539,453,188,35
8,1159,779,1064,617,399,1297,304,237,1107,590,675,347,1162,2087,1081,648,-
2306,567,433,2116,
1054,786,243,2185,596,406,2090,216,665,359,570,2077,1541,190,1284,1217,78-
1,1122,591,457,362,
1163,1109,1068,677,2307,349,420,2145,1055,787,706,625,407,2118,2091,312,2-
45,652,598,571,20
64,1170,191,1313,544,410,1123,220,1286,363,1096,2193,1545,2098,336,1165,8-
97,1070,802,707,23
09,612,422,2119,1113,789,232,681,599,465,2093,370,1171,654,1076,573,411,2-
122,316,249,1287,6
02,1125,2180,222,2099,365,660,1071,803,2072,1290,423,1100,776,709,614,452-
,3073,371,1186,655,
2313,1553,560,426,2123,344,1173,793,236,1078,603,413,2209,318,223,2562,13-
45,128,689,2182,2
101,1291,662,1129,805,2130,615,481,1102,834,1187,2076,1540,454,427,373,11-
60,780,713,1079,61
8,400,2125,1298,319,238,2563,1108,605,676,2183,348,1177,2088,1082,663,232-
1,568,434,2131,129
3,244,1103,2186,835,2105,1569,455,666,360,1189,809,2078,1542,619,429,1299-
,1218,782,239,1137,
592,458,377,1164,2565,1110,1083,678,2308,2241,350,435,2146,1056,788,721,2-
187,626,408,2133,
2092,246,667,837,572,2079,1543,1409,192,1314,1219,783,1124,621,459,1301,3-
64,1193,1111,2194,
1546,679,2337,351,1166,898,2147,1085,817,708,2310,627,437,2120,2569,1114,-
790,247,682,2189,6
00,466,2094,1172,669,1315,574,412,2137,1601,250,1288,1221,841,1126,2195,1-
547,461,2100,366,1
167,899,1072,804,737,2311,1305,424,2149,1115,791,710,683,629,467,3074,209-
5,372,1201,656,231
4,1554,575,441,2124,1174,794,251,1317,604,414,1127,2210,224,2577,1346,367-
,690,2197,1549,210
2,1292,1225,901,1130,806,711,2369,616,482,3075,1117,849,1188,685,2315,155-
5,469,428,2153,374,
1175,795,714,1080,633,415,2126,2211,320,253,2564,1347,606,691,2184,1178,2-
103,1321,664,2322,
1131,807,2132,1294,483,1104,2201,836,1665,2106,1570,456,3077,905,375,1190-
,810,715,2317,155
7,620,430,2127,1300,1233,797,240,2593,1138,607,473,2213,378,1179,2566,134-
9,1084,693,2323,22
42,436,2161,1295,722,1133,2188,865,2134,2107,1571,485,668,838,1191,811,20-
80,1544,1410,431,1
329,1220,784,717,1139,622,460,3081,1302,379,1194,2567,1112,2433,1561,680,-
2338,2243,352,118
1,913,2148,1086,818,723,2325,2217,628,438,2135,2570,1353,248,697,2190,839-
,2109,1573,1411,67
0,1316,1249,813,2138,1602,623,489,1303,1222,842,1195,1141,2196,1548,462,2-
339,381,1168,900,2
625,1087,819,738,2312,2245,1306,439,2150,2571,1116,792,725,684,2191,630,4-
68,3089,2096,1202,
671,2329,1793,576,442,2139,1603,1413,252,1318,1223,843,1128,2225,1577,463-
,2578,1361,368,11
97,929,2198,1550,739,2341,1307,1226,902,2151,1145,821,712,2370,631,497,30-
76,2573,1118,850,1
203,686,2316,2249,1556,470,443,2154,1176,796,729,1319,634,416,2141,1605,2-
212,254,2579,1348,
845,692,2199,1551,1417,2104,1322,1227,903,1132,808,741,2371,1309,484,3105-
,1119,2202,851,16
66,687,2345,1585,471,3078,906,2155,376,1205,825,716,2318,1558,635,445,212-
8,2689,1234,798,25
5,2594,1377,608,474,2214,1180,2581,1350,1323,694,2324,2257,1609,2162,1296-
,1229,961,1134,22
03,866,1667,2373,2108,1572,486,3079,907,853,1192,812,745,2319,1559,1425,4-
32,2157,1330,1235,
799,718,2595,1140,637,475,3082,2215,380,1209,2568,1351,2434,1562,695,2353-
,2244,1182,914,21
63,1325,724,2326,1135,2218,867,2136,2585,1354,487,698,2205,840,1669,2110,-
1574,1412,3137,90
9,1331,1250,814,719,2377,1617,624,490,3083,1304,1237,857,1196,2597,1142,2-
435,1563,477,2340,
2273,382,1183,915,2626,1088,820,753,2327,2246,2219,440,2165,2572,1355,726-
,699,2192,869,309
0,2111,1575,1441,672,2330,1794,1251,815,2140,1604,1414,491,1333,1224,844,-
1673,1143,2226,15
78,464,3085,2817,1362,383,1198,930,2627,2437,1565,740,2342,2247,1308,1241-
,917,2152,2601,11
46,822,727,2385,2221,632,498,3091,2574,1357,1204,701,2331,2250,1795,444,2-
169,1633,1415,730,
1320,1253,873,2142,1606,2227,1579,493,2580,1363,846,1199,931,2200,1552,14-
18,2343,1337,1228,
904,2629,1147,823,742,2372,3201,1310,499,3106,2575,1120,2441,852,1681,688-
,2346,2251,1586,4
72,3093,921,2156,1206,826,731,2333,1797,636,446,2143,1607,2690,1473,256,2-
609,1378,847,2229,
1581,1419,2582,1365,1324,1257,933,2258,1610,743,2401,1311,1230,962,3107,1-
149,2204,881,1668,
2374,2347,1587,501,3080,908,2633,854,1207,827,746,2320,2253,1560,1426,447-
,2158,2691,1236,8
00,733,2596,1379,638,476,3097,2216,1210,2583,1352,2449,1801,696,2354,2259-
,1611,1421,2164,1
326,1231,963,1136,2233,868,1697,2375,2586,1369,488,3109,937,2206,855,1670-
,747,2349,1589,14
27,3138,910,2159,1332,1265,829,720,2378,1618,639,505,3084,2693,1238,858,1-
211,2598,1381,243
6,1564,478,2355,2274,1184,916,2641,1327,754,2328,2261,1613,2220,2166,2587-
,1356,965,700,220
7,870,1671,3329,2112,1576,1442,3139,911,1809,1252,816,749,2379,1619,1429,-
492,3113,1334,123
9,859,1674,2599,1144,2465,1593,479,3086,2818,2275,384,1213,945,2628,2438,-
1566,755,2357,224
8,2697,1242,918,2167,2602,1385,728,2386,2222,871,3092,2589,1358,1443,702,-
2332,2265,1796,17
29,2170,1634,1416,3141,969,1335,1254,874,1675,2381,1621,2228,1580,494,308-
7,2819,1364,861,1
200,932,2657,2439,1567,1433,2344,2277,1338,1243,919,2630,2603,1148,824,75-
7,2387,3202,2223,
500,3121,2576,1359,2442,1682,703,2361,2252,1825,3094,922,2171,1635,1445,7-
32,2334,1798,1255,
875,2144,1608,2705,1474,495,2610,1393,848,1677,2230,1582,1420,3145,2821,1-
366,1339,1258,93
4,2631,2497,1625,744,2402,3203,1312,1245,977,3108,2605,1150,2443,882,1683-
,2389,2348,2281,1
588,502,3095,923,2634,1208,828,761,2335,2254,1799,448,2173,1637,2692,1475-
,734,2611,1380,87
7,3098,2231,1583,1449,2584,1367,2450,1802,1259,935,2260,1612,1422,2403,13-
41,1232,964,3585,
1151,2234,883,1698,2376,3205,2825,1370,503,3110,938,2635,2445,856,1685,74-
8,2350,2255,1590,
1428,3153,925,2160,2721,1266,830,735,2393,1857,640,506,3099,2694,1477,121-
2,2613,1382,2451,
1803,2356,2289,1641,1423,2642,1328,1261,993,2262,1614,2235,1699,2405,2588-
,1371,966,3111,93
9,2208,885,1672,3330,2351,1591,1457,3140,912,2637,1810,1267,831,750,2380,-
3209,1620,1430,50
7,3114,2695,1240,860,1689,2600,1383,2466,1594,480,3101,2833,2276,1214,946-
,2643,2453,1805,7
56,2358,2263,1615,2698,1481,2168,2617,1386,967,2237,872,1701,3331,2590,13-
73,1444,3169,941,
2266,1811,1730,751,2409,1649,1431,3142,970,3115,1336,1269,889,1676,2382,1-
622,2467,1595,509,
3088,2820,2753,862,1215,947,2658,2440,1568,1434,2359,2278,2699,1244,920,2-
645,2604,1387,75
8,2388,3217,2224,3122,2591,1360,2457,1921,704,2362,2267,1826,1731,3333,21-
72,1636,1446,3143,
971,1813,1256,876,1705,2383,1623,2706,1489,496,3117,2849,1394,863,1678,26-
59,2469,1597,143
5,3146,2822,2279,1340,1273,949,2632,2498,1626,759,2417,3204,2701,1246,978-
,3123,2606,1389,2
444,1684,2390,2363,2282,1827,3096,924,2649,1447,762,2336,2269,1800,1733,2-
174,1638,2707,147
6,973,2612,1395,878,1679,3337,2232,1584,1450,3147,2823,1368,1817,1260,936-
,2661,2499,1627,1
437,2404,3233,1342,1247,979,3586,2607,1152,2473,884,1713,2391,3206,2826,2-
283,504,3125,953,
2636,2446,1686,763,2365,2256,1829,3154,926,2175,1639,2722,1505,736,2394,1-
858,879,3100,2709,
1478,1451,2614,1397,2452,1804,1737,2290,1642,1424,3149,2881,1343,1262,994-
,3587,2501,1629,
2236,1700,2406,3207,2827,1372,981,3112,940,2665,2447,886,1687,3345,2352,2-
285,1592,1458,315
5,927,2638,2723,1268,832,765,2395,3210,1859,508,3129,2696,1479,1690,2615,-
1384,2481,1833,31
02,2834,2291,1643,1453,2644,2454,1806,1263,995,2264,1616,2713,1482,2407,2-
618,1401,968,3589,
2238,887,1702,3332,3265,2829,1374,1459,3170,942,2639,2505,1812,1745,752,2-
410,3211,1650,14
32,3157,985,3116,2725,1270,890,1691,2397,1861,2468,1596,510,3103,2835,275-
4,1216,948,2673,2
455,1807,2360,2293,1645,2700,1483,2646,2619,1388,997,3218,2239,1703,3361,-
2592,1375,2458,19
22,3171,943,2268,1841,1732,3334,2411,1651,1461,3144,972,3593,1814,1271,89-
1,1706,2384,3213,
1624,2945,1490,511,3118,2850,2755,864,1693,2660,2470,1598,1436,3161,2837,-
2280,2729,1274,95
0,2647,2513,1865,760,2418,3219,2702,1485,3124,2621,1390,2459,1923,2364,22-
97,1828,1761,3335,
2650,1448,3173,1001,2270,1815,1734,1707,2413,1653,2708,1491,974,3119,2851-
,1396,893,1680,3
338,2471,1599,1465,3148,2824,2757,1818,1275,951,2662,2500,1628,1438,2419,-
3234,2703,1248,98
0,3601,2608,1391,2474,1714,2392,3221,2841,2284,3126,954,2651,2461,1925,76-
4,2366,2271,1830,
1735,3393,2176,1640,2737,1506,975,1873,880,1709,3339,2710,1493,1452,3177,-
2853,1398,1819,17
38,2663,2529,1657,1439,3150,2882,3235,1344,1277,1009,3588,2502,1630,2475,-
1715,2421,3208,28
28,2761,982,3127,955,2666,2448,1688,3346,2367,2286,1831,3156,928,2653,272-
4,1507,766,2396,3
225,1860,3130,2711,1480,1929,2616,1399,2482,1834,1739,3341,2292,1644,1454-
,3151,2883,1821,1
264,996,3617,2503,1631,2714,1497,2408,3237,2857,1402,983,3590,2667,2477,8-
88,1717,3347,3266,
2830,2287,1460,3185,957,2640,2506,1746,767,2425,3212,1889,3158,986,3131,2-
726,1509,1692,23
98,1862,2483,1835,3104,2836,2769,1455,2674,2456,1808,1741,2294,1646,2715,-
1484,2885,2620,14
03,998,3591,3457,2240,1704,3362,3267,2831,1376,1937,3172,944,2669,2507,18-
42,1747,3349,2412,
3241,1652,1462,3159,987,3594,2727,1272,892,1721,2399,3214,1863,2946,512,3-
133,2865,2756,16
94,2675,2485,1837,3162,2838,2295,1647,2730,1513,2648,2514,1866,999,3220,2-
717,1486,3363,262
2,1405,2460,1924,3649,2298,1843,1762,3336,3269,2889,1463,3174,1002,3595,2-
509,1816,1749,170
8,2414,3215,1654,2947,1492,989,3120,2852,2785,894,1695,3353,2472,1600,146-
6,3163,2839,2758,
2731,1276,952,2677,2515,1867,2420,3249,2704,1487,3602,2623,1392,2489,1953-
,3222,2842,2299,1
763,3365,2652,2462,1926,3175,1003,2272,1845,1736,3394,2415,1655,2738,1521-
,976,3597,1874,89
5,1710,3340,3273,2949,1494,1467,3178,2854,2759,1820,1753,2664,2530,1658,1-
440,3165,2897,323
6,2733,1278,1010,3603,2517,1869,2476,1716,2422,3223,2843,2762,3128,956,26-
81,2463,1927,2368,
2301,1832,1765,3395,2654,2739,1508,1005,3226,1875,1711,3369,2712,1495,193-
0,3179,2855,1400,
1849,1740,3342,2531,1659,1469,3152,2884,3713,1822,1279,1011,3618,2504,163-
2,2953,1498,2423,
3238,2858,2763,984,3605,2668,2478,1718,3348,3281,2845,2288,3186,958,2655,-
2521,1985,768,24
26,3227,1890,3397,3132,2741,1510,1931,1877,2484,1836,1769,3343,2770,1456,-
3181,2913,1823,17
42,3619,2533,1661,2716,1499,2886,3239,2859,1404,1013,3592,3458,2479,1719,-
3377,3268,2832,27
65,1938,3187,959,2670,2508,1748,3350,2427,3242,1891,3160,988,3609,2728,15-
11,1722,2400,3229,
1864,2961,3134,2866,2771,1933,2676,2486,1838,1743,3401,2296,1648,2745,151-
4,2887,1881,1000,
3621,3459,2718,1501,3364,3297,2861,1406,1939,3650,2671,2537,1844,1777,335-
1,3270,2890,3243,
1464,3189,1017,3596,2510,1750,1723,2429,3216,1893,2948,990,3135,2867,2786-
,1696,3354,2487,
1839,3164,2840,2773,2732,1515,2678,2516,1868,3250,2719,1488,3841,2624,140-
7,2490,1954,3651,
3461,2300,1764,3366,3271,2891,1941,3176,1004,3625,2511,1846,1751,3409,241-
6,3245,1656,2977,
1522,991,3598,2787,896,1725,3355,3274,2950,1468,3193,2869,2760,1754,2679,-
2545,1897,3166,28
98,3251,2734,1517,3604,2518,1870,2491,1955,3224,2844,2777,3367,2682,2464,-
1928,3653,2302,18
47,1766,3396,2893,2740,1523,1006,3599,3465,1876,1712,3370,3275,2951,1496,-
1945,3180,2856,27
89,1850,1755,3357,2532,1660,1470,3167,2899,3714,2735,1280,1012,3633,2519,-
1871,2954,2424,32
53,2873,2764,3606,2683,2493,1957,3282,2846,2303,1767,3425,2656,2522,1986,-
1007,3228,1905,33
98,3371,2742,1525,1932,3657,1878,1851,1770,3344,3277,3009,1471,3182,2914,-
3715,1824,1757,36
20,2534,1662,2955,1500,2901,3240,2860,2793,1014,3607,3473,2480,1720,3378,-
3283,2847,2766,31
88,960,2685,2523,1987,2428,3257,1892,3399,3610,2743,1512,1961,3230,1879,2-
962,1771,3373,277
2,1934,3183,2915,1853,1744,3402,2535,1663,2746,1529,2888,3717,1882,1015,3-
622,3460,2957,150
2,3379,3298,2862,2767,1940,3665,2672,2538,1778,3352,3285,2905,3244,3190,1-
018,3611,2525,198
9,1724,2430,3231,1894,2963,3136,2868,2801,1935,2488,1840,1773,3403,2774,2-
747,1516,2917,188
3,3623,3489,2720,1503,3842,3299,2863,1408,1969,3652,3462,2539,1779,3381,3-
272,2892,3721,194
2,3191,1019,3626,2512,1752,3410,2431,3246,1895,2978,992,3613,2788,1726,33-
56,3289,2965,3194,
2870,2775,1993,2680,2546,1898,3405,3252,2749,1518,3843,1885,2492,1956,368-
1,3463,2778,3368,
3301,2921,1943,3654,3627,2541,1848,1781,3411,2894,3247,2979,1524,1021,360-
0,3466,1727,3385,
3276,2952,1946,3195,2871,2790,1756,3358,2547,1899,3168,2900,3729,2736,151-
9,3634,2520,1872,
2969,3254,2874,2779,3845,2684,2494,1958,3655,3521,2304,1768,3426,2895,200-
1,1008,3629,3467,
1906,3413,3372,3305,2981,1526,1947,3658,2791,1852,1785,3359,3278,3010,147-
2,3197,2929,3716,
1758,3635,2549,1901,2956,2902,3255,2875,2794,3608,3474,2495,1959,3284,284-
8,2781,3427,2686,
2524,1988,3258,1907,3400,3849,2744,1527,1962,3659,3469,1880,1772,3374,327-
9,3011,1949,3184,
2916,3745,1854,1759,3417,2536,1664,2985,1530,2903,3718,2795,1016,3637,347-
5,2958,3380,3313,
2877,2768,3666,2687,2553,2017,3286,2906,3259,3429,3612,2526,1990,1963,323-
2,1909,2964,3375,
2802,1936,3661,1855,1774,3404,3013,2748,1531,2918,3719,1884,3624,3490,295-
9,1504,3857,3300,
2864,2797,1970,3667,3477,2540,1780,3382,3287,2907,3722,3192,1020,3641,252-
7,1991,2432,3261,
1896,2993,3614,2803,1965,3290,2966,1775,3433,2776,1994,2919,1913,3406,349-
1,2750,1533,3844,
3777,1886,1971,3682,3464,3017,3383,3302,2922,3723,1944,3669,3628,2542,178-
2,3412,2909,3248,
2980,1022,3615,3481,1728,3386,3291,2967,3196,2872,2805,1995,2548,1900,340-
7,3730,2751,1520,
3873,1887,2970,3683,3493,2780,3846,3303,2923,1973,3656,3522,2543,1783,344-
1,2896,3725,2002,
1023,3630,3468,3414,3387,3306,2982,1948,3673,2792,1786,3360,3293,3025,319-
8,2930,3731,1997,
3636,2550,1902,2971,3256,2876,2809,3847,2496,1960,3685,3523,2782,3428,292-
5,2003,3631,3497,
1908,3415,3850,3307,2983,1528,1977,3660,3470,1787,3389,3280,3012,1950,319-
9,2931,3746,1760,
3418,2551,1903,2986,2904,3733,2796,3638,3476,2973,3314,2878,2783,3905,268-
8,2554,2018,3525,
3260,3430,3851,2005,1964,3689,3471,1910,3376,3309,3041,1951,3662,3747,185-
6,1789,3419,3014,
2987,1532,2933,3720,3639,3505,2960,3858,3315,2879,2798,3668,3478,2555,201-
9,3288,2908,3737,
3431,3642,2528,1992,3262,1911,2994,3853,2804,1966,3663,3529,1776,3434,301-
5,2009,2920,3749,
1914,3421,3492,2989,1534,3859,3778,2799,1972,3697,3479,3018,3384,3317,293-
7,3724,3670,3643,
2557,2021,2910,3263,2995,3616,3482,1967,3292,2968,3435,2806,1996,1915,340-
8,3969,2752,1535,
3874,3779,1888,3684,3494,3019,3861,3304,2924,3753,1974,3671,3537,2544,178-
4,3442,2911,3726,
1024,3645,3483,3388,3321,2997,3674,2807,2025,3294,3026,3437,3732,1998,387-
5,1917,2972,3495,
2810,3848,3781,1975,3686,3524,3021,3443,2926,3727,2004,3632,3498,3416,386-
5,3308,2984,1978,
3675,3485,1788,3390,3295,3027,3200,2932,3761,1999,2552,1904,3001,3734,281-
1,3877,2974,3687,
3553,2784,3906,2927,2033,3526,3499,3445,3852,3785,2006,1979,3690,3472,339-
1,3310,3042,1952,
3677,3748,1790,3420,3029,2988,2934,3735,3640,3506,2975,3316,2880,2813,390-
7,2556,2020,3527,
3738,3432,3881,2007,3691,3501,1912,3854,3311,3043,1981,3664,3530,1791,344-
9,3016,2010,2935,
3750,3422,3507,2990,3860,3793,2800,3698,3480,3033,3318,2938,3739,3909,364-
4,2558,2022,3264,
2996,3855,1968,3693,3531,3436,3045,2011,3751,1916,3423,3970,2991,1536,388-
9,3780,3699,3509,
3020,3862,3319,2939,3754,3672,3538,2559,2023,2912,3741,3646,3484,3322,299-
8,3913,2808,2026,
3533,3438,3971,2013,3876,3809,1918,3496,3049,3863,3782,3755,1976,3701,353-
9,3022,3444,2941,
3728,3647,3513,3866,3323,2999,3676,3486,2027,3296,3028,3439,3762,2000,191-
9,3002,3973,2812,
3878,3783,3688,3554,3023,3921,2928,3757,2034,3541,3500,3446,3867,3786,198-
0,3705,3487,3392,
3325,3057,3678,3763,2029,3030,3003,3736,3879,2976,3555,2814,3908,2035,352-
8,3977,3447,3882,
3787,2008,3692,3502,3869,3312,3044,1982,3679,3545,1792,3450,3031,2936,376-
5,3508,3005,3794,
2815,3937,3034,3557,3740,3910,3883,2037,3503,3856,3789,1983,3694,3532,345-
1,3046,2012,3752,
3424,3985,2992,3890,3795,3700,3510,3035,3320,2940,3769,3911,2560,2024,374-
2,3885,3695,3561,
3914,3047,2041,3534,3453,3972,2014,3891,3810,3511,3050,3864,3797,3756,370-
2,3540,3037,2942,
3743,3648,3514,3324,3000,3915,2028,3535,3440,4001,2015,3811,1920,3974,305-
1,3893,3784,3703,
3569,3024,3922,2943,3758,3542,3515,3868,3801,3706,3488,3326,3058,3917,376-
4,2030,3004,3975,
3880,3813,3556,3053,3923,3759,2036,3543,3978,3448,3897,3788,3707,3517,387-
0,3327,3059,3680,
3546,2031,3032,3766,3006,4033,2816,3938,3558,3979,3925,3884,3817,2038,350-
4,3871,3790,1984,
3709,3547,3452,3061,3767,3986,3007,3796,3939,3036,3559,3770,3912,2039,398-
1,3886,3791,3696,
3562,3929,3048,2042,3549,3454,3987,3892,3825,3512,3065,3798,3771,3941,303-
8,3744,3887,3563,
3916,2043,3536,3455,4002,2016,3812,3989,3052,3894,3799,3704,3570,3039,294-
4,3773,3516,3802,
3945,3565,3918,4003,2045,3976,3895,3814,3571,3054,3924,3760,3544,3993,389-
8,3803,3708,3518,
3328,3060,3919,2032,4005,3815,4034,3055,3953,3573,3980,3926,3899,3818,351-
9,3872,3805,3710,
3548,3062,3768,3008,4035,3940,3560,4009,3927,3819,2040,3982,3901,3792,371-
1,3577,3930,3063,
3550,3988,3826,3066,4037,3772,3942,3983,3888,3821,3564,3931,2044,3551,345-
6,4017,3827,3990,
3067,3800,3943,3040,3774,4041,3946,3566,3933,4004,2046,3991,3896,3829,357-
2,3069,3775,3994,
3804,3947,3567,3920,2047,4006,3816,4049,3056,3954,3574,3995,3900,3833,352-
0,3806,3949,4007,
4036,3955,3575,4010,3928,3820,3997,3902,3807,3712,3578,3064,4065,4038,401-
1,3957,3984,3903,
3822,3579,3932,3552,4018,3828,3068,4039,3944,4013,3823,4042,3961,3581,393-
4,4019,3992,3830,
3070,3776,4043,3948,3568,3935,2048,4021,3831,4050,3071,3996,3834,4045,395-
0,4008,4051,3956,
3576,4025,3835,3998,3808,3951,4066,4053,4012,3958,3999,3904,3837,3580,406-
7,4040,3959,4014,
3824,4057,3962,3582,4020,4069,4015,4044,3963,3583,3936,4022,3832,3072,407-
3,4046,3965,4023,
4052,4026,3836,4047,3952,4081,4054,4027,4000,3838,4068,4055,3960,4029,383-
9,4058,4070,4016,
4059,3964,3584,4071,4074,4061,3966,4024,4075,4048,3967,4082,4028,4077,408-
3,4056,4030,3840,
4085,4031,4060,4072,4089,4062,4076,4063,3968,4078,4084,4079,4086,4032,408-
7,4090,4091,4064, 4093,4080,4088,4092,4094,4095,4096.
[0214] It can be easily learned that the corresponding first
polarization weight vector in which the polarization weights are
sorted in ascending order and the corresponding first polarization
weight vector in which the polarization weights are sorted in
descending order are mutually in a reverse order. In other words,
the former can be obtained by sequentially reading the latter in
reverse order, or vice versa.
[0215] Both a first polarization weight vector that is
corresponding to any N or N.sub.max and in which polarization
weights are sorted in descending order and a first polarization
weight vector that is corresponding to any N or N.sub.max and in
which polarization weights are sorted in ascending order can be pre
stored.
[0216] FIG. 10 is a schematic structural diagram of an example of a
communications apparatus according to this application. The
communications apparatus may be a base station or a terminal. Based
on the same idea, this application provides a communications
apparatus, configured to perform the foregoing method procedure. As
shown in FIG. 10, the apparatus 1000 includes a processor 1001, a
memory 1002, and a communications interface 1003. The processor
1001, the memory 1002, and the communications interface 1003 are
mutually coupled.
[0217] The memory 1002 may include a volatile memory (volatile
memory), for example, a RAM. The memory may alternatively include a
non-volatile memory (non-volatile memory), for example, a flash
memory (flash memory), an HDD, or an SSD. The memory 1002 may
alternatively include a combination of the foregoing types of
memories.
[0218] The communications interface 1003 may be a wired
communications interface, a wireless communications interface, or a
combination thereof. The wired communications interface may be, for
example, an Ethernet interface. The Ethernet interface may be an
optical interface, an electrical interface, or a combination
thereof. The wireless communications interface may be a WLAN
interface.
[0219] The processor 1001 may be a CPU, an NP, or a combination of
a CPU and an NP.
[0220] The processor 1001 may further include a hardware chip. The
hardware chip may be an ASIC, a PLD, or a combination thereof. The
PLD may be a CPLD, an FPGA, a GAL, or any combination thereof. In
some embodiments, the memory 1002 may be further combined with the
processor 1001, for example, may be implemented by using a
circuit.
[0221] Optionally, the memory 1002 may be further configured to
store an instruction for performing various operations. The
processor 1001 invokes the instruction stored in the memory 1002,
to perform one or more steps or optional implementations in the
foregoing embodiments, so that the communications apparatus 1000
implements the foregoing methods, for example, calculates or stores
at least one first polarization weight vector that is corresponding
to N and in which polarization weights are sequentially sorted. A
polarization weight of each polarized channel may be calculated in
a manner including but not limited to the methods disclosed in the
foregoing embodiments.
[0222] All or some of the foregoing embodiments may be implemented
by using software, hardware, firmware, or any combination thereof.
When software is used to implement the embodiments, the embodiments
may be implemented completely or partially in a form of a computer
program product. The computer program product includes one or more
computer instructions. When the computer instructions are loaded
and executed on the computer, the procedures or functions according
to the embodiments of the present invention are all or partially
generated. The computer may be a general-purpose computer, a
dedicated computer, a computer network, or another programmable
apparatus. The computer instructions may be stored in a
computer-readable storage medium or may be transmitted from a
computer-readable storage medium to another computer-readable
storage medium. For example, the computer instructions may be
transmitted from a website, a computer, a server, or a data center
to another web site, computer, server, or data center in a wired
(for example, a coaxial cable, an optical fiber, or a digital
subscriber line (DSL)) or wireless (for example, infrared, radio,
or microwave) manner. The computer-readable storage medium may be
any usable medium accessible by a computer, or a data storage
device, such as a server or a data center, integrating one or more
usable media. The usable medium may be a magnetic medium (for
example, a floppy disk, a hard disk, or a magnetic tape), an
optical medium (for example, a DVD), a semiconductor medium (for
example, a solid state disk (SSD)), or the like.
[0223] A person skilled in the art should understand that the
embodiments of the present invention may be provided as a method or
a computer program product. Therefore, the present invention may
use a form of hardware only embodiments, software only embodiments,
or embodiments with a combination of software and hardware.
Moreover, the present invention may use a form of a computer
program product that is implemented on one or more computer-usable
storage media (including but not limited to a disk memory, a
CD-ROM, an optical memory, and the like) that include
computer-usable program code.
[0224] The present invention is described with reference to the
flowcharts and/or the block diagrams of the method, the device
(system), and the computer program product according to the
embodiments of the present invention. It should be understood that
computer program instructions may be used to implement each process
and/or each block in the flowcharts and/or the block diagrams, and
a combination of a process and/or a block in the flowcharts and/or
the block diagrams. These computer program instructions may be
provided for a general-purpose computer, a dedicated computer, an
embedded processor, or a processor of any other programmable data
processing device to generate a machine, so that the instructions
executed by the computer or the processor of any other programmable
data processing device generate an apparatus for implementing a
specified function in one or more processes in the flowcharts
and/or in one or more blocks in the block diagrams.
[0225] These computer program instructions may be stored in a
computer readable memory that can instruct the computer or any
other programmable data processing device to work in a specific
manner, so that the instructions stored in the computer readable
memory generate an artifact that includes an instruction apparatus.
The instruction apparatus implements a specified function in one or
more processes in the flowcharts and/or in one or more blocks in
the block diagrams.
[0226] These computer program instructions may alternatively be
loaded onto a computer or another programmable data processing
device, so that a series of operations and steps are performed on
the computer or the another programmable device, thereby generating
computer-implemented processing. Therefore, the instructions
executed on the computer or the another programmable device provide
steps for implementing a specified function in one or more
processes in the flowcharts and/or in one or more blocks in the
block diagrams.
[0227] Although some embodiments of the present invention have been
described, a person skilled in the art can make changes and
modifications to these embodiments once they learn of the basic
inventive concept. Therefore, the following claims are intended to
be construed as to cover the embodiments and all changes and
modifications falling within the scope of the present
invention.
[0228] Obviously, a person skilled in the art can make various
modifications and variations to the present invention without
departing from the scope of the present invention. The present
invention is intended to cover these modifications and variations
provided that they fall within the scope of protection defined by
the following claims and their equivalent technologies.
* * * * *