CN110336640B - Decoding method for LT code of short code length system - Google Patents

Decoding method for LT code of short code length system Download PDF

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CN110336640B
CN110336640B CN201910566140.8A CN201910566140A CN110336640B CN 110336640 B CN110336640 B CN 110336640B CN 201910566140 A CN201910566140 A CN 201910566140A CN 110336640 B CN110336640 B CN 110336640B
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袁磊
郑植
季芸
袁浩
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    • H—ELECTRICITY
    • H03—ELECTRONIC CIRCUITRY
    • H03M—CODING; DECODING; CODE CONVERSION IN GENERAL
    • H03M13/00—Coding, decoding or code conversion, for error detection or error correction; Coding theory basic assumptions; Coding bounds; Error probability evaluation methods; Channel models; Simulation or testing of codes
    • H03M13/37—Decoding methods or techniques, not specific to the particular type of coding provided for in groups H03M13/03 - H03M13/35
    • H03M13/3761—Decoding methods or techniques, not specific to the particular type of coding provided for in groups H03M13/03 - H03M13/35 using code combining, i.e. using combining of codeword portions which may have been transmitted separately, e.g. Digital Fountain codes, Raptor codes or Luby Transform [LT] codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
    • H04L1/0045—Arrangements at the receiver end
    • H04L1/0047—Decoding adapted to other signal detection operation
    • H04L1/005—Iterative decoding, including iteration between signal detection and decoding operation
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
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    • H04L1/0057—Block codes
    • H—ELECTRICITY
    • H04—ELECTRIC COMMUNICATION TECHNIQUE
    • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
    • H04L1/00—Arrangements for detecting or preventing errors in the information received
    • H04L1/004—Arrangements for detecting or preventing errors in the information received by using forward error control
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Abstract

A decoding method of an LT code of a short code length system adopts an improved decoding method based on Belief Propagation (BP) algorithm. The process comprises the following steps: the method comprises the steps that an information source bit information sequence is subjected to system LT code encoding and Binary Phase Shift Keying (BPSK) modulation, a modulation symbol sequence is wirelessly transmitted to a receiver, a receiver demodulator obtains channel log-likelihood ratio information of system LT code encoding bits by adopting a traditional demodulation method, a receiver LT decoder is subjected to decoding by adopting a BP decoding algorithm, after the BP algorithm is completed, the receiver calculates soft information of the encoding bits, the receiver reorders and judges the soft information, bit inversion is carried out on the bit sequence obtained by judgment, the minimum weighted Hamming distance is found out, and therefore the finally received information sequence is obtained. Compared with the traditional BP decoding algorithm, the improved decoding method provided by the invention can further reduce the Frame Error Rate (FER) due to fully utilizing the soft information of the coded bits, thereby achieving the purpose of reliable transmission.

Description

Decoding method for LT code of short code length system
Technical Field
The invention relates to a decoding method of an LT code of a short code length system, which is suitable for a wireless noise channel in a wireless communication system, and belongs to the technical field of communication coding.
Background
Fountain codes are a new type of error correcting code technology without rate, and typical applications thereof include multicast and broadcast services, distributed network storage, and the like. LT (Luby transform) codes and Raptor codes are two main fountain codes at present, wherein the Raptor codes are formed by high-code-rate precoding and LT codes in a cascade manner. Fountain Codes were originally applied in erasure Channels to combat packet loss, and in recent years the academy has demonstrated that fountain Codes also have excellent performance in wireless noise Channels (e.g., AWGN Channels, Fading Channels) (see "Fixed-Rate radar Codes Over social noise Channels", IEEE Transactions on Vehicular Technology, vol.56, No.6, November 2008).
When the transmitter transmits signals using LT codes, how to design a high-performance decoding algorithm at the receiver is one of the key techniques for achieving reliable reception. The traditional decoding method is that firstly, a demodulator is adopted to obtain the information of the channel Log-likelihood Ratio (LLR) of the LT code encoding bit (see 'Bitwise Log-likelihood Ratios For Quadrature Amplitude modules', IEEE Communications Letters, Vol.19, No.6, June 2015), and then a fountain code decoder adopts a soft decoding algorithm to recover the transmitted information. The performance of the traditional Belief Propagation (BP) decoding algorithm approaches the maximum likelihood decoding performance and has acceptable complexity of the decoding algorithm when the LT code length is longer. However, when the code length is short (e.g. the code word length is less than 500), the decoding performance is deteriorated seriously, so that it is necessary to research a decoding algorithm with better performance in the short code length.
Disclosure of Invention
Compared with the traditional BP decoding method, the improved system LT code decoding method has more excellent decoding performance.
The purpose of the invention is realized by the following technical scheme:
a decoding method of an LT code of a short code length system is an improved decoding method based on Belief Propagation (BP) algorithm, and the implementation steps are as follows:
step one, the length K of the transmitter pair1The method comprises the steps that firstly, Cyclic redundancy check (Cyclic redundancy check) is adopted for an original bit information sequence, CRC code coding is carried out to obtain an information source bit sequence with the length of K, then system LT coding is carried out on the information source bit sequence with the length of K to obtain a coded bit sequence, and Binary Phase Shift Keying (BPSK) modulation is adopted to obtain a modulation symbol sequence with the length of N; the receiver knows to generate a matrix G, which is a K × N binary matrix.
And step two, the modulation symbol sequence is transmitted to a receiver through a wireless channel.
And step three, the receiver demodulator obtains the channel Log-likelihood Ratio of the system LT code coded bits, namely the Log-likelihood Ratio and LLR information by adopting a traditional demodulation method.
Step four, a LT decoder of the receiver decodes by adopting a BP decoding algorithm, soft information of the information source bit sequence is calculated after a plurality of iterations, and hard decision on the soft information is carried out to obtain an information source bit sequence X; if the source bit sequence X is checked to be correct through the CRC code, the receiver finishes the whole decoding process; otherwise, the following steps are performed.
Step five, after BP decoding is finished, the receiver calculates the soft information L of the coding bit at the moment1To L for1Hard decision is carried out to obtain a sequence C1And combines the soft information L1Soft information L is obtained according to the sequence from big to small2Correspondingly, the matrix G is subjected to column transformation to obtain a matrix G1(ii) a Judgment matrix G1If the first K columns are not the maximum linearly independent group, then the matrix G is aligned again1Performing row-column transformation to obtain a matrix G2Make matrix G2Is linearly independent of the first K lines of the soft information L2The arrangement sequence is correspondingly changed to obtain the soft information L3。
Step six, the receiver firstly compares the soft information L3Hard decision is carried out on the first K bits to obtain a new information source bit sequence X1(ii) a Then to matrix G2Using Gaussian elimination method to make its front portion be unit matrix to obtain matrix G3And finally X1And matrix G3Multiplying to obtain a new coded bit sequence C2。
Step seven, according to the sequence C of the reverse process of the step five sorting2Reordering to obtain sequence C3。
Step eight, calculating a sequence C1And sequence C3The Weighted Hamming Distance, WHD, is denoted as WHD1。
Step nine, turning over the source bit sequence X1Get the sequence from the ith bit of
Figure GDA0002972397430000031
Figure GDA0002972397430000032
And matrix G3Multiplying to obtain a sequence
Figure GDA0002972397430000033
Repeating the seven steps to obtain the sequence
Figure GDA0002972397430000034
Computing sequences
Figure GDA0002972397430000035
And sequence C1Weighted hamming distance WHDiLet WHDopt＝min(WHD1,WHD2,...,WHDK,WHD1) While at the same time obtaining a signal with WHDoptCorresponding coded bits
Figure GDA0002972397430000036
Wherein the min function represents the minimum function.
Step ten, selecting a coding bit sequence
Figure GDA0002972397430000037
Front K of1The bits are used as the original bit information sequence.
Compared with the traditional BP decoding method, the improved decoding method provided by the invention has lower Frame Error Rate (FER) under the same Signal-to-Noise Ratio (SNR).
Drawings
FIG. 1 shows the length K of the original bit information sequence in AWGN channel1When the code rate R is 240 and 0.5, the simulation results of the traditional BP decoding and the improved decoding proposed by the present invention under different SNRs;
FIG. 2 shows the length K of the original bit information sequence in AWGN channel1240, noise variance σ2The iteration times are 20 times, and the simulation results of the traditional BP decoding and the improved decoding provided by the invention under different modulation symbol lengths are 0.5.
Detailed Description
The invention is further illustrated with reference to the following figures and examples.
A decoding method of LT codes of a short code length system comprises the following concrete steps:
step one, the length K of the transmitter pair1The original bit information sequence is firstly coded by a CRC code to obtain an information source bit sequence with the length K, then the information source bit sequence with the length K is subjected to systematic LT coding with the code rate R to obtain a coded bit sequence, the coded bit sequence is modulated by BPSK to obtain a modulation symbol sequence with the length N, and S represents a set S { -1, +1} of modulation symbols.
Step two, the modulation symbol sequence is subjected to mean value of 0 and variance of sigma2The AWGN channel of (a) is transmitted to the receiver.
Step three, the receiver demodulator utilizes the receiving sequence rj(j 1L N) calculating the channel LLR information of the system LT code coded bits, i.e.
Figure GDA0002972397430000041
And step four, calculating the soft information of the source bit sequence by the LT code decoder of the receiver by adopting a BP decoding algorithm, and judging. The specific calculation formula is as follows:
the message passed in the decoder is denoted m and the number of iterations is denoted l. From the ith coded bit ciTo the jth source bit djIs represented as an LLR message
Figure GDA0002972397430000042
Representing a slave source bit diTo coded bit cjThe LLR message of (1). Then, at the first iteration,
Figure GDA0002972397430000043
is composed of
Figure GDA0002972397430000044
Wherein N isiRepresents the ithSubscript set of source bits of the coded bit connection, L (c)i) Indicating the channel LLR information of the ith code bit obtained in step three. LLR messages
Figure GDA0002972397430000045
Dependent only on the delivery to the source bit diThe information of (1). Therefore, at the first iteration,
Figure GDA0002972397430000046
is shown as
Figure GDA0002972397430000047
Wherein epsiloniA set of indices representing the coded bits concatenated with the ith source bit. Finally, soft information transmitted by the ith information source bit after l iterations is calculated
Figure GDA0002972397430000048
Figure GDA0002972397430000049
To pair
Figure GDA00029723974300000410
Hard decision is made, the specific calculation formula is as follows,
Figure GDA00029723974300000411
to obtain the sequence X ═ d1,d2,...,di,...,dKIf the X passes through the CRC code check to be correct, the information source bit sequence can be completely recovered at the moment, and the decoding process is finished; otherwise, the following steps are continued.
Step five, the receiver calculates soft information of the ith coding bit, i is 1, 2.
Figure GDA0002972397430000051
First to L1Hard decision is carried out to obtain a sequence C1，
Figure GDA0002972397430000052
Then mixing L1L is obtained by arranging in the order from big to small2Correspondingly, the matrix G is subjected to column transformation to obtain G1(ii) a Judgment matrix G1If the first K columns are not the maximum linearly independent group, then the matrix G is aligned again1Performing row-column transformation to obtain G2Make matrix G2Is linearly independent of the first K column lines, while L is2Also correspondingly changing the arrangement sequence to obtain L3。
Step six, firstly aligning L3Hard decision is made on the first K bits, i 1, 2., K,
Figure GDA0002972397430000053
then to matrix G2Performing Gaussian elimination to obtain G3Make matrix G3The first K columns form a submatrix which is a unit matrix and finally X1And matrix G3Multiplying to obtain a new coded bit sequence C2。
Step seven and step five pairs L1Two successive orderings are carried out, and the sequence C is subjected to the inverse process2Reordering to obtain sequence C3。
Step eight, calculating a sequence C1And sequence C3The weighted hamming distance of (1, 2., N,
Figure GDA0002972397430000054
step nine, overturning X1Get the sequence from the ith bit of
Figure GDA0002972397430000055
Figure GDA0002972397430000056
And matrix G3Multiplying to obtain a sequence
Figure GDA0002972397430000057
Repeating the seven steps to obtain the sequence
Figure GDA0002972397430000058
Computing
Figure GDA0002972397430000059
And C1Weighted hamming distance WHDi，
Figure GDA0002972397430000061
Order to
WHDopt＝min(WHD1,WHD2,...,WHDK,WHD1)，
To obtain a reaction product with WHDoptCorresponding coded bit sequence
Figure GDA0002972397430000062
Step ten, selecting a coding bit sequence
Figure GDA0002972397430000063
Front K of1The bits are used as the original bit information sequence.
The invention is further illustrated by the following specific examples.
Embodiment 1, a decoding method for a short code length system LT code includes the following steps:
step one, the length K of the transmitter pair1Firstly, the original bit information sequence of 240 is coded by using CRC-16 code to obtain source bit sequence with length K of 256, then the source bit information sequence is coded by using system LT with code rate R of 1/2 to obtain coded sequenceCode bit sequence, the Systematic LT code adopts the degree distribution function of the literature "Optimization Design and Systematic Lunbit Transform Codes Over BIAWGN Channels" (IEEE Transactions on Communications, Vol.64, No.8, August 2016)
Figure GDA0002972397430000064
And the coded bit sequence is subjected to BPSK modulation to obtain a modulation symbol sequence with the length of N being 256/R.
Step two, the modulation symbol sequence is subjected to mean value of 0 and variance of sigma2The AWGN channel of (a) is transmitted to the receiver.
Step three, the receiver demodulator utilizes the receiving sequence rj(j ═ 1L N) channel LLR information for the system LT code coded bits is calculated.
And fourthly, calculating soft information L of the information source bit sequence by the LT code decoder of the receiver by adopting a BP decoding algorithm after a plurality of iterations, and carrying out hard decision on the soft information L to obtain an information source bit sequence X. If X passes the CRC-16 code check, the receiver ends the whole decoding process, otherwise, the following steps are carried out.
Step five, after BP decoding is finished, the receiver calculates the soft information L of the coding bit sequence at the moment1To L for1Hard decision is carried out to obtain a sequence C1And mixing L1L is obtained by arranging in the order from big to small2Correspondingly, the matrix G is subjected to column transformation to obtain G1(ii) a Judgment matrix G1If the first K columns are not the maximum linearly independent group, then the matrix G is aligned again1Performing row-column transformation to obtain G2Make matrix G2Is linearly independent of the first K lines of (L)2Also correspondingly changing the arrangement sequence to obtain L3。
Step six, the receiver firstly compares the soft information L3Hard decision is carried out on the first K bits to obtain a new information source bit sequence X1(ii) a Then to G2Using Gaussian elimination method to make its front portion be unit matrix to obtain matrix G3And finally X1And matrix G3Multiple of each otherNew coding bit sequence C2。
Step seven, according to the sequence C of the reverse process of the step five sorting2Reordering to obtain sequence C3。
Step eight, calculating a sequence C1And sequence C3Weighted Hamming distance of (1), denoted WHD1；
Step nine, overturning X1Get the sequence from the ith bit of
Figure GDA0002972397430000071
Figure GDA0002972397430000072
And matrix G3Multiplying to obtain a sequence
Figure GDA0002972397430000073
Repeating the seven steps to obtain the sequence
Figure GDA0002972397430000074
Computing
Figure GDA0002972397430000075
And C1Weighted hamming distance WHDiLet WHDopt＝min(WHD1,WHD2,...,WHDK,WHD1) While at the same time obtaining a signal with WHDoptCorresponding coded bit sequence
Figure GDA0002972397430000076
Step ten, selecting a coding bit sequence
Figure GDA0002972397430000077
Front K of1The bits are used as the original bit information sequence.
Following the procedure of example 1, the signal-to-noise ratio is defined as SNR P/σ2And P is the normalized signal power. According to the difference of iteration times, simulation is carried out according to different signal-to-noise ratios, and a simulation result is shown in figure 1. As can be seen from fig. 1, for the conventional BP decoding algorithmIn other words, FER curves of 30 iterations and 50 iterations respectively almost coincide with each other, that is, when the iteration count increases to a certain value, the iteration count is continuously increased, only the decoding complexity is increased, and the FER performance cannot be improved. The improved decoding method proposed by the present invention can reduce the FER on this basis.
Embodiment 2, a decoding method based on short code length LT code includes the following steps:
transmitter pair length K1The original bit information sequence of 240 is firstly coded by a CRC-16 code to obtain a source bit sequence of length K256, then the source bit information sequence is subjected to systematic LT coding of code rate R to obtain a coded bit sequence, the LT code adopts the same degree distribution function as that in embodiment 1, and the coded bit sequence is further modulated by BPSK to obtain a modulation symbol sequence of length N256/R. Assume noise variance σ2The number of iterations is 20, 0.5. The rest of the procedure was the same as in example 1.
According to the process of the embodiment 2, simulation is carried out for different N/K, and FIG. 2 shows the simulation result. Under the same N/K, the FER of the improved decoding method provided by the invention is lower than that of the traditional BP decoding method, and the improved decoding method has lower FER.

Claims (7)

1. A decoding method of short code length system LT code is characterized in that: the improved decoding method based on Belief Propagation (BP) algorithm is realized by the following steps:
step one, the length K of the transmitter pair1The method comprises the steps that firstly, Cyclic redundancy check (Cyclic redundancy check) is adopted for an original bit information sequence, CRC code coding is carried out to obtain an information source bit sequence with the length of K, then system LT coding is carried out on the information source bit sequence with the length of K to obtain a coded bit sequence, and Binary Phase Shift Keying (BPSK) modulation is adopted to obtain a modulation symbol sequence with the length of N; the receiver knows to generate a matrix G, which is a K × N binary matrix;
step two, the modulation symbol sequence is transmitted to a receiver through a wireless channel;
thirdly, the receiver demodulator obtains the channel Log-likelihood Ratio of the system LT code coded bits, namely Log-likelihood Ratio and LLR information, by adopting a traditional demodulation method;
step four, a LT decoder of the receiver decodes by adopting a BP decoding algorithm, soft information of the information source bit sequence is calculated after a plurality of iterations, and hard decision on the soft information is carried out to obtain an information source bit sequence X; if the source bit sequence X is checked to be correct through the CRC code, the receiver finishes the whole decoding process; otherwise, the following steps are carried out;
step five, after BP decoding is finished, the receiver calculates the soft information L of the coding bit at the moment1To L for1Hard decision is carried out to obtain a sequence C1And combines the soft information L1Soft information L is obtained according to the sequence from big to small2Correspondingly, the matrix G is subjected to column transformation to obtain a matrix G1(ii) a Judgment matrix G1If the first K columns are not the maximum linearly independent group, then the matrix G is aligned again1Performing row-column transformation to obtain a matrix G2Make matrix G2Is linearly independent of the first K lines of the soft information L2The arrangement sequence is correspondingly changed to obtain the soft information L3；
Step six, the receiver firstly compares the soft information L3Hard decision is carried out on the first K bits to obtain a new information source bit sequence X1(ii) a Then to matrix G2Using Gaussian elimination method to make its front portion be unit matrix to obtain matrix G3And finally X1And matrix G3Multiplying to obtain a new coded bit sequence C2；
Step seven, according to the sequence C of the reverse process of the step five sorting2Reordering to obtain sequence C3；
Step eight, calculating a sequence C1And sequence C3The Weighted Hamming Distance, WHD, is denoted as WHD1；
Step nine, turning over the source bit sequence X1Get the sequence from the ith bit of
Figure FDA0003066137450000021
Figure FDA0003066137450000022
And matrix G3Multiplying to obtain a sequence
Figure FDA0003066137450000023
Repeating the seven steps to obtain the sequence
Figure FDA0003066137450000024
Computing sequences
Figure FDA0003066137450000025
And sequence C1Weighted hamming distance WHDiLet WHDopt＝min(WHD1,WHD2,...,WHDK,WHD1) While at the same time obtaining a signal with WHDoptCorresponding coded bits
Figure FDA0003066137450000026
Wherein the min function represents a minimum function;
step ten, selecting a coding bit sequence
Figure FDA0003066137450000027
Front K of1The bits are used as the original bit information sequence.
2. The decoding method of short code length system LT code as claimed in claim 1, characterized in that: the fourth step is as follows: the message passed in the decoder is denoted by m and the number of iterations is denoted by l; from the ith coded bit ciTo the jth source bit djIs represented as an LLR message
Figure FDA0003066137450000028
Figure FDA0003066137450000029
Representing a slave source bit diTo coded bit cjThe LLR message of (2); thenAnd at the time of the l-th iteration,
Figure FDA00030661374500000210
is composed of
Figure FDA00030661374500000211
Wherein N isiA set of indices, L (c), representing the source bits concatenated with the ith coded biti) Channel LLR information indicating the ith code bit obtained in step three; LLR messages
Figure FDA00030661374500000212
Dependent only on the delivery to the source bit diThe information of (a); therefore, at the first iteration,
Figure FDA00030661374500000213
is shown as
Figure FDA00030661374500000214
Wherein epsiloniA set of indices representing coded bits concatenated with the ith source bit; finally, soft information transmitted by the ith information source bit after l iterations is calculated
Figure FDA00030661374500000215
Is composed of
Figure FDA00030661374500000216
To pair
Figure FDA00030661374500000217
Hard decision is made, the specific calculation formula is as follows,
Figure FDA0003066137450000031
to obtain the sequence X ═ d1,d2,...,di,...,dKIf the X passes through the CRC code check to be correct, the information source bit sequence can be completely recovered at the moment, and the decoding process is finished; otherwise, the following steps are continued.
3. The decoding method of short code length system LT code as claimed in claim 1, characterized in that: the fifth step is as follows: the receiver calculates the soft information for the ith coded bit, i 1,2, N,
Figure FDA0003066137450000032
wherein the content of the first and second substances,
Figure FDA0003066137450000033
representing the slave source bit d at the ith iterationnTo coded bit ciLLR message of, NiA set of indices representing source bits concatenated with the ith coded bit, N representing the set NiAn inner one of the elements;
L(xi|ri) Channel LLR information representing coded bits of the systematic LT code;
first to L1Hard decision is made to obtain C1，
Figure FDA0003066137450000034
Then mixing L1L is obtained by arranging in the order from big to small2Correspondingly, the matrix G is subjected to column transformation to obtain G1(ii) a Judgment matrix G1If the first K columns are not the maximum linearly independent group, then the matrix G is aligned again1Performing row-column transformation to obtain G2Make matrix G2Is linearly independent of the first K column lines, while L is2Also phaseCorrespondingly changing the arrangement sequence to obtain L3。
4. The decoding method of short code length system LT code as claimed in claim 1, characterized in that: the sixth step is as follows: first pair soft information L3Hard decision is made on the first K bits, i 1, 2., K,
Figure FDA0003066137450000035
then to matrix G2Performing Gaussian elimination to obtain a matrix G3Make matrix G3The first K sub-matrix is unit matrix, and the last source bit sequence X1And matrix G3Multiplying to obtain a new coded bit sequence C2。
5. The decoding method of short code length system LT code as claimed in claim 1, characterized in that: the seventh step is as follows: step five pairs of soft information L1Two successive orderings are carried out, and the sequence C is subjected to the inverse process2Reordering to obtain sequence C3。
6. The decoding method of short code length system LT code as claimed in claim 1, characterized in that: the eighth step is as follows: calculating the sequence C1And sequence C3The weighted hamming distance of (1, 2., N,
Figure FDA0003066137450000041
7. the decoding method of short code length system LT code as claimed in claim 1, characterized in that: the ninth step is as follows: flipping source bit sequence X1Get the ith bit of
Figure FDA0003066137450000042
Figure FDA0003066137450000043
And matrix G3Multiplying to obtain a sequence
Figure FDA0003066137450000044
Repeating the seven steps to obtain the sequence
Figure FDA0003066137450000045
Computing sequences
Figure FDA0003066137450000046
And sequence C1Weighted hamming distance WHDi，j＝1,2,...,N，
Figure FDA0003066137450000047
Order to
WHDopt＝min(WHD1,WHD2,...,WHDK,WHD1)，
To obtain a reaction product with WHDoptCorresponding coded bits
Figure FDA0003066137450000048
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