WO2017107761A1 - 一种极化码与重复码或多比特偶校验码级联的纠错编码方法 - Google Patents

一种极化码与重复码或多比特偶校验码级联的纠错编码方法 Download PDF

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WO2017107761A1
WO2017107761A1 PCT/CN2016/108511 CN2016108511W WO2017107761A1 WO 2017107761 A1 WO2017107761 A1 WO 2017107761A1 CN 2016108511 W CN2016108511 W CN 2016108511W WO 2017107761 A1 WO2017107761 A1 WO 2017107761A1
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bit
code
bits
check
repeated
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PCT/CN2016/108511
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French (fr)
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屈代明
王涛
江涛
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华中科技大学
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Priority claimed from CN201510995761.XA external-priority patent/CN105680883B/zh
Priority claimed from CN201610847488.0A external-priority patent/CN106452460B/zh
Application filed by 华中科技大学 filed Critical 华中科技大学
Priority to EP16877572.4A priority Critical patent/EP3364542A4/en
Publication of WO2017107761A1 publication Critical patent/WO2017107761A1/zh
Priority to US15/965,855 priority patent/US20180248567A1/en

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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
    • H03M13/29—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 combining two or more codes or code structures, e.g. product codes, generalised product codes, concatenated codes, inner and outer codes
    • H03M13/2906—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 combining two or more codes or code structures, e.g. product codes, generalised product codes, concatenated codes, inner and outer codes using block codes
    • 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
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/11—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits using multiple parity bits
    • 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
    • H03M13/03—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words
    • H03M13/05—Error detection or forward error correction by redundancy in data representation, i.e. code words containing more digits than the source words using block codes, i.e. a predetermined number of check bits joined to a predetermined number of information bits
    • H03M13/13—Linear codes

Definitions

  • the invention belongs to the technical field of error correction coding, and more particularly to an error correction coding method in which a polarization code is cascaded with a repetition code or a multi-bit even parity code.
  • Polarization coding is a new coding scheme close to Shannon's limit.
  • the complexity of the coding and decoding algorithm is low, which is beneficial to engineering implementation.
  • the simulation results show that the SCL (Successive Cancellation List) decoding algorithm is With lower complexity O(L ⁇ Nlog(N)) (L is the number of paths, N is the code length), the error correction capability can reach the error correction capability of the maximum likelihood decoder.
  • the error correction performance still has a large gap with the Shannon limit, and this difference cannot be compensated by increasing the number of paths alone.
  • the LDPC Low-Density Parity-Check
  • the CRC Cyclical Redundancy Check
  • the present invention provides an error correction coding method in which a polarization code is overlapped with a repetition code or a multi-bit even parity code, and the purpose is to use an outer code with a lower coding complexity.
  • the polarization code is cascaded, and the error correction performance of the polarization code under the SCL decoding algorithm is improved without increasing the decoding complexity and storage complexity.
  • the error correction coding method of code or multi-bit even parity code concatenation includes the following steps:
  • the repeated bits may be repeated one or more times.
  • the channel capacity of the non-fixed bit channel mapped by the repeated bits is lower than the non-repeated bits mapped.
  • the channel capacity of a fixed bit channel is lower than the non-repeated bits mapped.
  • the sequence number of the non-fixed bits mapped by the repeated bits of the outer code codeword is greater than the repeated bit corresponding to the repeated bits. The sequence number of the non-fixed bits that are mapped.
  • the repeated bits of the outer code codeword are evenly distributed or approximately uniformly distributed.
  • the above error correction coding method maps the repeated bits of the outer code code word to the K h non-fixed bit channels with the lowest channel capacity in each segment by dividing the non-fixed bit sequence into S segments by sequence number.
  • the above error correction coding method replaces the outer code codeword with an inverted repetition code; when the repeated bit is 1, the repeated bit of the inverted repetition code is 0; when the repeated bit is 0, the reverse The repetition bit of the repetition code is 1;
  • the K repeated bits obtained by the repetition code encoding are The number of inverted repeat bits is 0 to K.
  • the error correction coding method has a partial bit of a tail portion of the outer code codeword as a check bit, and each check bit forms an even parity or an odd parity relationship with a plurality of information bits before the check bit.
  • the decoding of the error correction coding method described above adopts the following method:
  • bit decision is performed according to the SCL decoding algorithm; for the repeated bits, the decision is directly made according to the decision result of the repeated bits.
  • the decoding method comprises the following steps:
  • step (4.1) determining whether i is less than or equal to N; if yes, proceeding to step (4.2), and if not, proceeding to step (4.7);
  • N is the length of the concatenated code code
  • i is the index number of the i-th bit currently decoded, and the initial value is 1, and the value is a positive integer from 1 to N;
  • step (4.2) determining whether u i is a fixed bit, if yes, proceeding to step (4.3); if not, proceeding to step (4.4); wherein u i is a polarization code input sequence
  • step (4.4) determining whether u i is the repeating bit of the jth (1 ⁇ j ⁇ K) repeating code, if yes, proceeding to step (4.5); if not, proceeding to step (4.6); wherein K is the outer code The number of repeated bits;
  • T j is a set of all bit position numbers of the jth repetition code in the outer code code word; Refers to the polarization code input sequence after all the bits in the jth repetition code are mapped by the outer code codeword. a collection of serial numbers, Representation collection The smallest element in the value corresponding to the repeated bits in the jth repetition code;
  • min(X) represents the minimum value of the set X
  • the metric of the 2L' path is the probability that the path takes a value of 0 or 1 at u i . or L is the maximum number of paths of the SCL decoding algorithm;
  • the check bits in the outer code code word obtained by performing multi-bit even check coding on the information bit sequence in step (1) are concentrated on the tail of the outer code code word;
  • M is the number of information bits and K is the number of check equations.
  • the check bits in the outer code codeword obtained by performing multi-bit even check coding on the information bit sequence in step (1) are equally spaced in the outer code code word;
  • M is the number of information bits
  • K is the number of check equations
  • M+K is the outer code length. It is the lower integer of x.
  • the check bits in the outer code code words obtained by performing multi-bit even check coding on the information bit sequence in step (1) are unequally distributed in the outer code code words.
  • step (1) the closer to the front part of the outer code codeword, the more dispersed the check bit distribution, and the closer to the outer code code word, the more concentrated the check bit distribution is, according to the check bit.
  • This feature of the position distribution results in a set P of check bit position numbers.
  • the number of check bits concentrated in the tail of the outer code code word is K 1
  • the number of check bits distributed in the front interval is KK 1
  • the unequal interval is distributed.
  • the set of position numbers of the check bits is:
  • P ⁇ P 1 ⁇ P 2
  • P 1 ⁇ M+KK 1 +1, M+KK 1 +2,...,M+K ⁇ , Where M is the number of information bits and K is the number of check equations.
  • the parity bit when multi-bit even parity coding is performed on the information bit sequence in the step (1), the parity bit only checks the bits before the bit, and is independent of the subsequent bits.
  • the outer code can also be replaced with a multi-bit odd parity code.
  • the decoding of the above coding method is performed by using a modified SCL decoding algorithm, that is, bit decoding is performed according to the SCL decoding algorithm when decoding information bits, and when the parity bits are decoded, according to the even parity equation in which the parity bits are located The decision result of the information bit is verified.
  • a modified SCL decoding algorithm that is, bit decoding is performed according to the SCL decoding algorithm when decoding information bits, and when the parity bits are decoded, according to the even parity equation in which the parity bits are located The decision result of the information bit is verified.
  • Step 1 Determine whether i is less than or equal to N, if yes, go to Step2; if not, go to Step7; where N is the code length of the concatenated code; i is the index number of the i-th bit currently decoded, and the initial value is 1, the value is a positive integer from 1 to N;
  • Step 2 determining whether u i is a fixed bit, if yes, proceeding to Step 3; if not, proceeding to Step 4; wherein u i is the i-th bit of the polarization code input sequence;
  • max(X) represents the largest element of the set of values X
  • ⁇ X ⁇ Y ⁇ represents the difference between the sets X and Y, that is,
  • Step 6 Count the current number of paths L'.
  • each path can take a value of 0 or 1 at u i to obtain a 2L' alternative path.
  • the metric of the 2L' path is the value of the path at u i . Probability of 0 or 1: or
  • L is the maximum number of paths of the SCL decoding algorithm
  • Step 7 Output the corresponding decision sequence on the path with the largest path metric from the L paths
  • the SCL decoding algorithm uses a medium-short code length non-cascaded polarization code, and its error correction performance approaches the maximum likelihood decoder decoding performance, and the error correction capability is limited, and even if the SCL decoding algorithm path is increased The number and frame error rate performance will not be significantly improved. In addition, increasing the number of paths will linearly increase the algorithm storage complexity and decoding complexity, which is not conducive to engineering implementation. Under the same path number of SCL decoding, the present invention proposes Compared with the non-cascaded polarization code, the cascading scheme improves the error correction performance of the scheme significantly, and the error correction performance of the scheme can also significantly exceed the error correction performance of the polarization code of the cascaded CRC code.
  • the scheme adopts the modified SCL decoding algorithm. Compared with the original SCL decoding algorithm, the main difference between the scheme of cascading the polarization code and the multi-bit parity check code is that the algorithm directly passes the parity check bit. The decision result of the information bits in the check equation is verified. Since the number of check equations and the length of the check equation are much smaller than the code length of the concatenated code, there is no significant increase in decoding complexity.
  • the scheme has a slight improvement in storage complexity, that is, information such as the check equation corresponding to the outer code needs to be stored separately on the encoder and the decoder, but the number of the outer-code check equation and the length of the equation are much smaller than
  • the concatenated code code is long, so the storage amount of the information bits and the check bit position information of the check equation is weak relative to the entire system; for the scheme of the polarization code and the repeated code cascade, the main difference is: the algorithm When decoding the repeated bits, the decision is directly made according to the decision result of the repeated bits; compared with the conventional SCL decoding algorithm, the modified decoding method of the present invention only needs to determine the repeated bits as the repeated bits.
  • the value, its computational complexity is similar to the traditional SCL algorithm for a fixed bit decision of 0, no computational complexity.
  • the outer code of the scheme adopts a repetition code or a multi-bit even parity code.
  • the method of the invention can make the hardware circuit design of the outer encoder simple, and is beneficial to engineering implementation.
  • 1 is a schematic diagram of decoding a polarization code having a code length of 4 using an existing SCL decoding algorithm
  • FIG. 2 is a flowchart of a compiled code of an error correction coding method in which a polarization code and a repetition code or a multi-bit even parity code are cascaded according to the present invention
  • FIG. 3 is a schematic diagram of decoding of an error correction coding method in which a coded code length is 4 and a repetition code is cascaded;
  • FIG. 4 is a schematic diagram of coding of an outer code codeword according to an error correction coding method of a polarization code and a repetition code cascade according to an embodiment of the present disclosure
  • FIG. 5 is a schematic diagram of three types of determining parity check bit positions according to an embodiment of the present invention.
  • polarization coding is to polarize N independent and identically distributed channels into N bit channels by channel polarization operation, in which N bit channels are used.
  • the polarization code information bits are transmitted on M bit channels with the largest channel capacity.
  • the bit channel is called a non-fixed bit channel, and the bits transmitted on the non-fixed bit channel are non-fixed bits; the other (NM) bit channels are called fixed bits.
  • the channel, the bit transmitted on the fixed bit channel is a fixed bit.
  • Polarization code input sequence Bits u 1 to u N are sequentially transmitted on the 1st to Nth bit channels, and the set of polarization code non-fixed bit channel numbers is The elements in set A satisfy when a ⁇ i ⁇ j ⁇ M, a i ⁇ a j ; the set of fixed bit channel numbers is A c ;
  • Non-fixed bit sequence is Polarization code fixed bits are known at both ends of the transceiver, for symmetric channels, fixed bit sequences Set to an all-zero sequence; when M information bits are known, a non-fixed bit sequence u A can be determined to determine the polarization code input sequence Polarized codeword G N is a polarization code generation matrix.
  • Step 1 determine whether i is less than or equal to N; if yes, enter Step2; if not, then enter Step5; wherein, the initial value of i is 1;
  • Step 2 determine whether u i is a fixed bit, and if so, enter Step 3; if not, then enter Step 4;
  • Step 4 Count the current number of paths L'.
  • the metric of the 2L' path is the probability that the path takes a value of 0 or 1 at u i : or L is the maximum number of paths of the SCL decoder;
  • Step 5 output a corresponding decision sequence on a path with the largest path metric value from the L paths Decision sequence Non-fixed bit sequence That is, the decoding result of the polarization code information bit sequence.
  • the error correction coding method for cascading a polarization code and a repetition code or a multi-bit even parity code provided by the present invention is as shown in FIG. 2, and specifically includes the following steps:
  • An error correction coding method in which a polarization code and a repetition code are cascaded are as follows:
  • Step 1 External encoder coding
  • the outer encoder For a cascade system with a code length of N, a number of information bits of M, a number of repeated bits of K 1 (K 1 ⁇ M), and a number of repeated bits of K 2 (K 2 ⁇ K 1 ), the outer encoder input
  • the length of the end information sequence is M
  • the length of the output outer code code word is M+K 2 ;
  • the codeword obtained by the outer encoder is Containing M information bits and K 2 repetition bits; determining the outer encoder is equivalent to determining the outer code code word K 1 repeated bit position number, and the sequence number of the repeated bits corresponding to each repeated bit, the number of the jth ( 1 ⁇ j ⁇ K 1 ) repeated bits and the sequence number of the repeated bits corresponding thereto Described as a set T j ( 1 ⁇ j ⁇ K 1 ), it is easy to know that each set T j represents a
  • represents the number of elements of the set T j ,
  • 2 indicates that T j represents a two-repeat code; the smallest element in the set T j represents the sequence number of the repeated bits, and the other elements represent the sequence number of the repeated bits in the repetition code;
  • First outer code codeword number order bits divided according to S segments, evenly distributed bits in the second paragraph K 1 through S repeats, wherein the first h (2 ⁇ h ⁇ S-1) number of repetitive bits of segment allocation for ( Indicates that x pairs take the integer), and the number of repeated bits allocated in the h (h S) segment is (thus ensuring that the number of repeated bits per segment is an integer and the total number of repeated bits is K 1 );
  • the channel capacity of the non-fixed bit channel to which each bit of the outer code code word is mapped is known; the "bit with high channel capacity (low)” mentioned below refers to "after the outer code codeword is mapped” , an outer code codeword bit of a non-fixed bit channel transmission having a high (low) channel capacity";
  • the repeated bits that have been selected in the segment h' (1 ⁇ h' ⁇ S-2) are removed, and among the remaining bits, the channel capacity is the lowest.
  • the bits are used as the repeated bits in the segment;
  • the pairing method is specifically: the channel capacity is the lowest in the h (2 ⁇ h ⁇ S) segment
  • the repetition bit is paired with the repeated bit with the lowest channel capacity of the h-1th segment, and the corresponding repeated bit number and the repeated bit number form a set Tj ;
  • the channel capacity of the h (2 ⁇ h ⁇ S) segment is second.
  • the low repetition bit is paired with the second lowest repeating bit of the channel capacity of the h-1 segment, and so on, and the pairing of K 1 repeated bits is completed, and K 1 two repetition code sets T j (1 ⁇ are obtained).
  • Step 2 outer code codeword mapping
  • the outer code codeword mapping is a bit that inputs each bit in the outer code code word through the mapping relationship as a polarization code;
  • the non-fixed bit channel number set of the polarization code is The fixed bit channel number set is A c , and the polarization code input sequence is The non-fixed bit sequence of the polarization code is The fixed bit value is all 0s;
  • Outer code word The way to map to the non-fixed bit sequence u A is: the first bit x 1 to the last bit of the outer code word Map to the first bit of the non-fixed bit sequence in turn To the last bit Satisfy or
  • Non-fixed bit channel after mapping The upper is transmitted as the jth ( 1 ⁇ j ⁇ K 1 ) repeated bits in the outer code; the non-fixed bit channel The above is transmitted as a repeating bit corresponding to the jth ( 1 ⁇ j ⁇ K 1 ) repeated bits.
  • Step 3 encoding the inner encoder
  • step 2 Obtaining an inner code polarization code non-fixed bit sequence by step 2
  • the polarization code fixed bit sequence is all 0s, so that the polarization code input sequence is known
  • the codeword encoded by the cascading system is obtained as Where G N is a polarization code generation matrix.
  • Step 4 The receiving end decodes
  • the modified SCL decoding algorithm is used for decoding; the main correction is: when decoding the repeated bits, the decision is directly made according to the decision result of the repeated bits, instead of the decision according to the probability in the original SCL decoding algorithm; details as follows:
  • T j represents a set of all bit position numbers of the jth repetition code in the outer code code word, Representing that all bits in the jth repetition code are in the polarization code input sequence after mapping by the outer code codeword a collection of serial numbers, The smallest element Corresponding to the repeated bits in the jth repetition code, Corresponding to the repeated bits in the jth repetition code;
  • each path can take a value of 0 or 1 at u i to obtain a 2L' alternative path.
  • the metrics of the 2L' path are respectively taken at u i
  • the probability of a value of 0 or 1: or If 2L' ⁇ L (L is the maximum number of paths of the SCL decoding algorithm), 2L' paths are reserved; if 2L'>L, L paths with the largest metric value are retained; i i+1; return (4.1);
  • Figure 3 shows a concatenated code and a set of two repetition codes for a code length of four.
  • the receiving end decoder adopts a modified SCL decoding algorithm, and the number of paths is set to L; the error correction coding method of the polarization code and the repetition code cascade of the embodiment is as follows:
  • Step 1 External encoder coding
  • the bit numbers of the 8 information bits located in the outer code code word are as follows:
  • Step 2 outer code codeword mapping
  • the codeword obtained by the outer encoder is
  • the first to the 12th bits are sequentially mapped to the first to 12 non-fixed bits of the polarization code
  • the non-fixed bit sequence of the polarization code is
  • the non-fixed bit channel Transmitted as 4 repeated bits in the outer code; non-fixed bit channel
  • the upper transmission is 4 repetition bits in the outer code.
  • Step 3 encoding the inner encoder
  • step 2 Obtaining an inner code polarization code non-fixed bit sequence by step 2
  • the polarization code fixed bit sequence is all 0s, so that the polarization code input sequence is known
  • Obtaining the codeword encoded by the cascade system according to the polarization code encoding formula is Where G N is a polarization code generation matrix.
  • Step 4 decoding of the cascade system
  • Step 1 External encoder coding
  • the length of the outer code input end information sequence is M
  • the output outer code code word length is M+K.
  • the bit position is a check bit, that is, the outer encoder encodes the code word. Bit sequence For information bits, To check the bits.
  • Method 2 The parity bits are equally spaced in the outer codewords. As shown in Figure 5(b), the outer code length is M+K, and the spacing between adjacent check bits is among them, Indicates that the next integer of x is taken, and the set of check bit positions is set to
  • Method 3 The parity bits are unequally spaced in the outer codewords.
  • a feature that the parity bits are unequally spaced is that the closer to the front of the outer codeword, the more dispersed the parity bit distribution; the closer to the outer codeword tail The more concentrated the bit distribution is.
  • a parity bit position number set P is obtained.
  • the method 3 may adopt a joint form of the foregoing method 1 and method 2, that is, a part of the check bits are concentrated in the tail portion, and another part of the check bits are equally spaced.
  • the number of check bits concentrated in the tail is K 1
  • the number of check bits distributed in the same interval is KK 1
  • P 1 ⁇ M+KK 1 +1,M+KK 1 +2,...,M+K ⁇ , .
  • K 1 K, the set P 2 is an empty set, and the third method is equivalent to the second method.
  • T j The set of position numbers of all the bits in the equation in the outer code
  • T j satisfies
  • Check bit The value is: Complete the outer code encoding.
  • Step 2 Outer code codeword mapping
  • the outer code codeword mapping refers to a bit in which each bit in the outer code code word is encoded as a polarization code by the mapping relationship.
  • the set of non-fixed bit channel numbers of the polarization code be The fixed bit channel number set is A c , and the polarization code input sequence is recorded as The non-fixed bit sequence of the polarization code is The fixed bit value is all 0s.
  • Outer code word The way to map to the non-fixed bit sequence u A is that the first bit x 1 to the last bit x M+K of the outer code code word are sequentially mapped to the first bit of the non-fixed bit sequence. To the last bit Satisfy or
  • Step 3 Inner encoder coding
  • the codeword coded by the cascading system is Where G N is a polarization code generation matrix.
  • Step 4 Receiver decoding
  • the cascaded system decoder uses a modified SCL decoding algorithm.
  • the main correction is that when the parity bit is decoded, it is verified according to the decision result of the information bits in the even parity equation where the parity bit is located, instead of being judged according to the probability in the original SCL decoding algorithm.
  • Step1 Determine whether i is less than or equal to N, if yes, go to Step2; if not, go to Step7;
  • Step2 Determine whether u i is a fixed bit, if yes, go to Step3; if not, go to Step4;
  • Step4 Determine whether u i is the jth (1 ⁇ j ⁇ K) check bits, if yes, go to Step 5; if not, go to Step 6;
  • Step 5 The current decision value of u i on each path is obtained by checking the information bits of the decision on the path: Return to Step1;
  • Step 6 Count the current number of paths L'.
  • each path can take a value of 0 or 1 at u i to obtain a 2L' alternative path.
  • the metric of the 2L' path is the value of the path at u i .
  • Probability of 0 or 1: or If 2L' ⁇ L, keep 2L'path; if 2L'> L, keep L path with the largest metric value; i i+1; return Step1;
  • Step 7 Output the corresponding decision sequence on the path with the largest path metric from the L paths
  • the number of non-fixed bit channels of the inner code polarization code is 256, fixed.
  • the number of bit channels is 256
  • the polarization code structure is performed at 2 dB to obtain the sequence number sets of the non-fixed bit channels, respectively.
  • Step 1 External encoder coding
  • the number of information bits is 240, and the number of check equations is 16, the outer code length is 256, and the outer code code word is According to the three methods determined by the check bit position number set P described above and in combination with the embodiment, the set P is sequentially obtained as follows:
  • the parity bits are unequally spaced in the outer codewords.
  • the outer code is Among the bits, x 1 ⁇ 31 , x 2 ⁇ 31 , x 3 ⁇ 31 , ..., x 8 ⁇ 31 , x 249 , x 250 , x 251 , ..., x 256 are check bits, and the remaining 240 bits are information.
  • Bits, in combination with this example, then set P ⁇ 31, 62, 93, 124, 155, 186, 217, 248, 249, 250, 251, 252, 253, 254, 255, 256 ⁇ .
  • the set P obtained by the method 2 is taken as an example to illustrate the determination of the position of the information bits in each check equation, and the encoding of each even check equation.
  • the value of the other 15 check bits can be determined, thereby completing the outer code encoding.
  • Step 2 Outer code codeword mapping
  • i ⁇ P ⁇ ; the non-fixed bit channel The upper transmission is the jth (j 1,...,16) parity bits of the outer code; the non-fixed bit channel All bits on the jth check equation of the outer code are sent.
  • P ⁇ 16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256 ⁇ obtained by method 2 as an example,
  • a P ⁇ a i
  • i ⁇ P ⁇ ⁇ a 16 , a 32 , a 48 , a 64 , a 80 , a 96 , a 112 , a 128 , a 144 , a 160 , a 176 , a 192 , a 208 , a 224 , a 240 , a 256 ⁇ ; transmitting outer code check bits; non-fixed bit channel Sending the first parity bit of the outer code; non-fixed bit channel All bits of the first check equation of the outer code are transmitted. In turn, you can get The value of the equal.
  • Step 3 Inner encoder coding
  • the codeword coded by the cascading system is Where G N is a polarization code generation matrix.
  • Step 4 Decoding of the cascade system
  • the decoding of the cascade system uses the modified SCL decoding algorithm.
  • the main difference between the algorithm and the original SCL decoding algorithm is that when decoding the check bits, the check bit decision value is directly based on the information bit decision value of the check equation. Check it out.
  • the first check equation as an example, when 32 paths are decoded to Bit, for The bit decision is obtained by verifying the decision value of the information bits in the first check equation.
  • the specific formula is as follows:
  • the decision is made according to the decoding principle of the SCL decoding algorithm.
  • the decoding result of the concatenated code can be obtained.
  • the decoding result in That is, the outer code code word decoding result, so that the set then This is the decoded information bit decoding result.

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Abstract

一种极化码与重复码或多比特偶校验码级联的纠错编码方法，在发送端采用重复码或多比特偶校验码作为外码，极化码作为内码；接收端译码器采用修正的SCL译码算法进行译码；在纠错性能上，提供的编码方法的误帧率性能可显著超越采用SCL译码算法或者RC-aided SCL译码算法下的极化码误帧率性能；在工程实现上，外码采用重复码或多比特偶校验码，编码简单；译码采用修正的SCL译码算法，相对于原始SCL译码算法无译码复杂度的提升，有利于工程实现。

Description

一种极化码与重复码或多比特偶校验码级联的纠错编码方法 [技术领域]
本发明属于纠错编码技术领域，更具体地，涉及一种极化码与重复码或多比特偶校验码级联的纠错编码方法。
[背景技术]
极化编码作为一种新型的接近香农限的编码方案，其编译码算法的复杂度较低，有利于工程实现；仿真结果表明极化码SCL(Successive Cancellation List，连续消除列表)译码算法在较低复杂度O(L·Nlog(N))(L为路径数量，N为码长)下，纠错能力可以达到最大似然译码器的纠错能力。但是对于采用SCL译码算法的中短码长的非级联极化码，其纠错性能仍然与香农极限存在较大差距，并且这个差距无法单独从增大路径数量进行弥补。
传统的极化码级联方法中，级联LDPC(Low-Density Parity-Check，低密度奇偶校验)码由于得到的级联码的特性不适合SCL译码算法，相对于采用SCL译码算法的极化码而言，其纠错性能无明显提升；级联CRC(Cyclical Redundancy Check，循环冗余校验)码需要额外的CRC校验电路，会带来一定的硬件开销，并且纠错能力有限；因此，这些传统方案均在某种程度上限制了极化码的工程应用。
[发明内容]
针对现有技术的以上缺陷或改进需求，本发明提供了一种极化码与重复码或多比特偶校验码级联的纠错编码方法，其目的在于以较低编码复杂度的外码级联极化码，并且在不提升译码复杂度和存储复杂度的情况下，提升极化码在SCL译码算法下的纠错性能。
为实现上述目的，按照本发明的一个方面，提供了一种极化码与重复 码或多比特偶校验码级联的纠错编码方法，包括如下步骤：
(1)通过对信息比特序列进行重复码编码或多比特偶校验编码获得外码码字；
(2)通过将所述外码码字的第一个比特至最后一个比特依次映射到极化码的第一个至最后一个非固定比特来进行外码码字映射，获得非固定比特序列；
(3)通过对所述非固定比特序列进行极化码编码获得级联码码字。
优选地，上述的纠错编码方法，其步骤(1)中对信息比特序列进行重复码编码时，被重复比特可被重复一次或多次。
进一步优选地，上述的纠错编码方法，其步骤(2)的外码码字映射过程中，被重复比特所映射的非固定比特信道的信道容量，低于未被重复的比特所映射的非固定比特信道的信道容量。
进一步优选地，上述的纠错编码方法，在步骤(2)的外码码字映射过程中，外码码字的重复比特所映射的非固定比特的序号大于所述重复比特对应的被重复比特所映射的非固定比特的序号。
进一步优选地，上述的纠错编码方法，在步骤(2)所获得的非固定比特序列中,外码码字的重复比特均匀分布或近似均匀分布。
进一步优选地，上述的纠错编码方法，通过将非固定比特序列按序号顺序划分为S个段，将外码码字的重复比特映射到每段内信道容量最低的Kh个非固定比特信道上，每段内分配的重复比特数相等或近似相等，使得外码码字的重复比特在非固定比特中均匀分布或近似均匀分布；其中，h＝1,2,...,S。
进一步优选地，上述的纠错编码方法，其将外码码字替换为反相重复码；当被重复比特为1，反相重复码的重复比特为0；当被重复比特为0，反向重复码的重复比特为1；
当被重复比特的重复次数为K时，重复码编码所获得的K个重复比特中 反相重复比特的数量为0～K。
进一步优选地，上述的纠错编码方法，将外码码字尾部的部分比特作为校验比特，每个校验比特与该校验比特之前的多个信息比特构成偶校验或奇校验关系。
进一步优选地，上述的纠错编码方法的译码采用以下方法：
(4)对于原始信息比特，根据SCL译码算法进行比特判决；对于重复比特，根据被重复比特的判决结果直接进行判决。
进一步优选地，上述的纠错编码方法，其译码方法包括如下步骤：
(4.1)判断i是否小于等于N；若是，进入步骤(4.2)，若否，则进入步骤(4.7)；
其中，N是指级联码码长，i是指当前译码第i个比特的索引序号，其初始值为1，取值从1到N的正整数；
(4.2)判断ui是否为固定比特，若是，进入步骤(4.3)；若否，则进入步骤(4.4)；其中，ui是指极化码输入序列
Figure PCTCN2016108511-appb-000001
的第i个比特；
Figure PCTCN2016108511-appb-000002
为1×N的行向量(u1,u2,u3,...,uN)；
(4.3)将当前每条路径上ui的判决值设为已知的固定比特的取值，并令i＝i+1，返回步骤(4.1)；
(4.4)判断ui是否为第j(1≤j≤K)个重复码的重复比特，若是，进入步骤(4.5)；若否，则进入步骤(4.6)；其中，K为外码中被重复比特的个数；
(4.5)将当前每条路径上重复比特ui的判决值设置为该路径上与ui对应的被重复比特的判决值，具体地：
Figure PCTCN2016108511-appb-000003
并令i＝i+1，返回步骤(4.1)；
其中，Tj是指外码码字中第j个重复码的所有比特位置序号集合；
Figure PCTCN2016108511-appb-000004
是指经过外码码字映射后，第j个重复码中的所有比特在极化码输入序列
Figure PCTCN2016108511-appb-000005
中的序号集合，
Figure PCTCN2016108511-appb-000006
表示取集合
Figure PCTCN2016108511-appb-000007
中的最小元素，该值对应第j个重复码中被重复比特；
在本发明中，min(X)均表示取集合X的最小值；
(4.6)统计当前路径数量L′，通过将当前每条路径在ui处取值0或1获得2L′条备选路径；并判断是否满足2L′≤L，若是，保留2L′条路径；若否，则保留L条度量值最大的路径；并令i＝i+1，返回步骤(4.1)；
其中，2L′条路径的度量值分别为该路径在ui处取值0或1的概率
Figure PCTCN2016108511-appb-000008
或
Figure PCTCN2016108511-appb-000009
L为SCL译码算法的最大路径数量；
(4.7)从L条路径中输出路径度量值最大的一条路径上对应的判决序列
Figure PCTCN2016108511-appb-000010
获得译码结果。
优选地，上述的纠错编码方法，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特集中于外码码字尾部；
校验比特位置序号集合为P＝{M+1,M+2,M+3,...,M+K}，集合P中的元素表示在外码的这些比特位置上为校验比特，也即外编码器编码码字
Figure PCTCN2016108511-appb-000011
中，比特序列
Figure PCTCN2016108511-appb-000012
为信息比特，
Figure PCTCN2016108511-appb-000013
为校验比特；
其中，M为信息比特数量，K为校验方程数量。
优选地，上述的纠错编码方法，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特在外码码字中等间隔分布；
相邻校验比特的间距为
Figure PCTCN2016108511-appb-000014
校验比特位置序号集合为
Figure PCTCN2016108511-appb-000015
其中，M为信息比特数量，K为校验方程数量，M+K为外码码长，
Figure PCTCN2016108511-appb-000016
是指取x的下整数。
优选地，上述的纠错编码方法，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特在外码码字中不等间隔分布。
优选地，上述的纠错编码方法，步骤(1)中，越靠近外码码字前部，校验比特分布越分散，越靠近外码码字尾部校验比特分布越集中，按照校验比特位置分布的这种特征，得到校验比特位置序号集合P。
优选地，上述的纠错编码方法，假设集中在外码码字尾部的校验比特的数量为K1，则前面等间隔分布的校验比特的数量为K-K1，在不等间隔分布下，校验比特的位置序号集合为：
P＝{P1∪P2|P1＝{M+K-K1+1,M+K-K1+2,...,M+K},
Figure PCTCN2016108511-appb-000017
；其中，M为信息比特数量，K为校验方程数量。
进一步优选地，上述的纠错编码方法，步骤(1)中对信息比特序列进行多比特偶校验编码时，校验比特仅校验该比特之前的比特，而与之后的比特无关。
外码还可以替换为多比特奇校验码。
上述编码方法的译码采用修正的SCL译码算法进行译码，即译码信息比特时根据SCL译码算法进行比特判决，译码校验比特时，根据校验比特所在的偶校验方程中信息比特的判决结果进行校验得到。
进一步优选的，译码的具体过程为：
Step1:判断i是否小于等于N，是，则进入Step2；否，则进入Step7；其中，N是指级联码码长；i是指当前译码第i个比特的索引序号，其初始值为1，取值为从1到N的正整数；
Step2:判断ui是否为固定比特，是，则进入Step3；否，则进入Step4；其中，ui是指极化码输入序列的第i个比特；
Step3:将当前每条路径上ui的判决值设置为已知的固定比特的取值；令i＝i+1；返回Step1；
Step4:判断ui是否为第j(j＝1,2,...,K)个校验比特，是，则进入Step5；否，则进入Step6；其中，K为校验方程数量；
Step5:当前每条路径上ui的判决值通过该路径上判决的信息比特校验得到：
Figure PCTCN2016108511-appb-000018
令i＝i+1；返回Step1；其中，Tj是指第j个校验方程的所有比特在外码码字中的位置序号集合，
Figure PCTCN2016108511-appb-000019
表示经过外码码字映射后，第j个校验方程中的所有比特在极化码输入序列
Figure PCTCN2016108511-appb-000020
中的序号集合，
Figure PCTCN2016108511-appb-000021
表示取值集合
Figure PCTCN2016108511-appb-000022
中的最大元素，该元素为第j个校验方程中的校验比特映射到
Figure PCTCN2016108511-appb-000023
中的序号；h是该求和运算中的临时变量，依次指代集合
Figure PCTCN2016108511-appb-000024
中的每一个元素；
Figure PCTCN2016108511-appb-000025
表示取集合
Figure PCTCN2016108511-appb-000026
与
Figure PCTCN2016108511-appb-000027
的差，
Figure PCTCN2016108511-appb-000028
本发明中，max(X)均表示取值集合X的最大元素，{X\Y}均表示取集合X和Y的差，也即
Figure PCTCN2016108511-appb-000029
Step6:统计当前路径数量L′，当前每条路径在ui处可取值0或1，从而得到2L′条备选路径，2L′条路径的度量值分别为该路径在ui处取值0或1的概率：
Figure PCTCN2016108511-appb-000030
或
Figure PCTCN2016108511-appb-000031
若2L′≤L，保留2L′条路径；若2L′>L，保留L条度量值最大的路径；i＝i+1；返回Step1；
其中，L为SCL译码算法的最大路径数量；
Step7:从L条路径中输出路径度量值最大的一条路径上对应的判决序列
Figure PCTCN2016108511-appb-000032
Step8:结束。
总体而言，通过本发明所构思的以上技术方案与现有技术相比，能够取得下列有益效果：
(1)本方案显著提升极化码纠错性能；
采用SCL译码算法的中短码长非级联的极化码，其纠错性能趋近于最大似然译码器译码性能，纠错能力有限，并且即使增大SCL译码算法路径 数量，误帧率性能不会明显改善，另外增大路径数量会线性增大算法存储复杂度和译码复杂度，不利于工程实现；在相同路径数量的SCL译码算下，本发明提出的级联方案与非级联的极化码相比，本方案的纠错性能提升显著，并且本方案的纠错性能也可以显著超越级联CRC码的极化码的纠错性能。
(2)本方案无明显译码复杂度和存储复杂度的提升；
本方案采用修正的SCL译码算法，相对于原始SCL译码算法而言，对于极化码与多比特偶检验码级联的方案，其主要区别在于：算法译码校验比特时，直接通过该校验方程中信息比特的判决结果校验得到。由于校验方程的数量和校验方程的长度远小于级联码的码长，因此无显著译码复杂度的提升。本方案在存储复杂度上存在微弱的提升，也即需要在编码器和译码器上分别存储外码对应的校验方程等信息，但是外码的校验方程的数量、方程的长度远小于级联码码长，因此校验方程的信息比特和校验比特位置信息的存储量相对于整个系统而言是微弱的；对于极化码与重复码级联的方案，其主要区别在于：算法译码重复比特时，直接根据被重复比特的判决结果直接进行判决；相比采用传统的SCL译码算法，本发明的这种修正的译码方法仅需将重复比特判决为被重复比特的取值，其运算复杂度与传统SCL算法对固定比特判决为0类似，无运算复杂度提升。
(3)本方案外码编码简单，易于实现；
本方案外码采用重复码或者多比特偶校验码，相对于其它级联的外码方案，本发明方法可以使外编码器的硬件电路设计简单，有利于工程实现。
[附图说明]
图1为对码长为4的极化码采用现有SCL译码算法译码的示意图；
图2为本发明提供的极化码与重复码或多比特偶校验码级联的纠错编码方法的编译码流程图；
图3为码长为4的极化码与重复码级联的纠错编码方法的译码示意图；
图4为本发明实施例提供的极化码与重复码级联的纠错编码方法的外码码字编码示意图；
图5为本发明实施例提供的三种确定偶校验码校验比特位置的示意图。
[具体实施方式]
为了使本发明的目的、技术方案及优点更加清楚明白，以下结合附图及实施例，对本发明进行进一步详细说明。应当理解，此处所描述的具体实施例仅仅用以解释本发明，并不用于限定本发明。此外，下面所描述的本发明各个实施方式中所涉及到的技术特征只要彼此之间未构成冲突就可以相互组合。
对码长为N、发送信息比特数量为M的极化码，极化编码是通过信道极化操作将N个独立同分布的信道极化为N个比特信道，在这N个比特信道中，信道容量最大的M个比特信道上发送极化码信息比特，该比特信道称为非固定比特信道，非固定比特信道上发送的比特为非固定比特；其它(N-M)个比特信道称为固定比特信道，固定比特信道上发送的比特为固定比特。
极化码输入序列
Figure PCTCN2016108511-appb-000033
比特u1到uN依次在第1个到第N个比特信道上发送，极化码非固定比特信道序号集合为
Figure PCTCN2016108511-appb-000034
集合A中的元素满足当1≤i<j≤M时，ai<aj；固定比特信道序号集合为Ac；
非固定比特序列为
Figure PCTCN2016108511-appb-000035
极化码固定比特在收发两端已知，对于对称信道，固定比特序列
Figure PCTCN2016108511-appb-000036
设置为全0序列；当已知M个信息比特时，可确定非固定比特序列uA，从而确定极化码输入序列
Figure PCTCN2016108511-appb-000037
极化码码字为
Figure PCTCN2016108511-appb-000038
GN为极化码生成矩阵。
极化码SCL译码算法译码时从比特u1到uN依次进行判决，得到极化码输入序列
Figure PCTCN2016108511-appb-000039
的译码结果；SCL算法译码过程中最多可保留L条译码路径， 在译码ui时，每条路径上已经译码出的序列
Figure PCTCN2016108511-appb-000040
唯一表征该路径；图1所示为对码长为4的极化码采用SCL译码算法示意图，其中u2为固定比特，(u1,u3,u4)为非固定比特，路径数量L＝2，译码时，每条路径在固定比特u2处判决为0，其它非固定比特则根据路径度量值进行判决；具体过程如下：
Step1、判断i是否小于等于N；若是，进入Step2；若否，则进入Step5；其中，i的初始值为1；
Step2、判断ui是否为固定比特，若是，进入Step3；若否，则进入Step4；
Step3、将当前每条路径上ui的判决值设置为0，令i＝i+1，返回Step1；
Step4、统计当前路径数量L′，当前每条路径在ui处可取值0或1，从而得到2L′条备选路径；若2L′≤L，保留2L′条路径；若2L′>L，保留L条度量值最大的路径，令i＝i+1；返回Step1；
2L′条路径的度量值分别为该路径在ui处取值0或1的概率：
Figure PCTCN2016108511-appb-000041
或
Figure PCTCN2016108511-appb-000042
L为SCL译码器的最大路径数量；
Step5、从L条路径中输出路径度量值最大的一条路径上对应的判决序列
Figure PCTCN2016108511-appb-000043
判决序列
Figure PCTCN2016108511-appb-000044
中的非固定比特序列
Figure PCTCN2016108511-appb-000045
即为极化码信息比特序列译码结果。
本发明提供的极化码与重复码或多比特偶校验码级联的纠错编码方法，其流程如图2所示，具体包括如下步骤：
(1)通过对信息比特序列进行重复码编码或多比特偶校验编码获得外码码字；
(2)通过将所述外码码字的第一个比特至最后一个比特依次映射到极化码的第一个至最后一个非固定比特来进行外码码字映射，获得非固定比特序列；
(3)通过对所述非固定比特序列进行极化码编码获得级联码码字。
以下对极化码与重复码级联的纠错编码方法与极化码与多比特偶校验 码级联的纠错编码方法分别进行阐述。
极化码与重复码级联的纠错编码方法，具体如下：
步骤1：外编码器编码；
对于码长为N，信息比特数量为M，被重复比特数量为K1(K1≤M)、重复比特的数量为K2(K2≥K1)的级联系统，则外编码器输入端信息序列的长度为M，输出的外码码字长度为M+K2；
外编码器编码得到的码字为
Figure PCTCN2016108511-appb-000046
包含M个信息比特和K2个重复比特；确定外编码器等价于确定外码码字
Figure PCTCN2016108511-appb-000047
中的K1个被重复比特位置序号，以及与每一个被重复比特对应的重复比特的序号，将第j(1≤j≤K1)个被重复比特的序号和与其对应的重复比特的序号记为集合Tj(1≤j≤K1)，易知每一个集合Tj表征外码码字中的一个|Tj|重复码；|Tj|表示集合Tj元素的个数，|Tj|＝2表示Tj表征一个二重复码；集合Tj中最小的元素表示被重复比特的序号，其它元素表示该重复码中重复比特的序号；
以外码码字中每一个重复码为二重复码为例说明集合Tj(1≤j≤K1)的构造，也即K1＝K2；需要强调的是，此处的集合Tj(1≤j≤K1)的构造具体方式仅作为本发明的解释，并不限制本发明。
(1.1)分段并确定每段重复比特和被重复比特数量：
首先将外码码字按照比特序号顺序划分为S段，将K1个重复比特平均分配到第2至第S段中，其中第h(2≤h≤S-1)段分配的重复比特数为
Figure PCTCN2016108511-appb-000048
(
Figure PCTCN2016108511-appb-000049
表示x对取下整数)，第h(h＝S)段分配的重复比特数为
Figure PCTCN2016108511-appb-000050
(从而保证每段重复比特数为整数，并且总的重复比特数为K1)；
第h(2≤h≤S)段的重复比特校验第h-1段的被重复比特，第h′(1≤h′≤S-2)段的被重复比特数量为
Figure PCTCN2016108511-appb-000051
第h′(h′＝S-1)段的被重复比特数量为
Figure PCTCN2016108511-appb-000052
(1.2)根据每段中的重复比特数量和被重复比特数量确定每段中的重复比特和被重复比特位置序号；
根据外码码字映射，已知外码码字每一个比特映射到的非固定比特信道的信道容量；下文提到的“信道容量高(低)的比特”是指“外码码字映射后，具有高(低)信道容量的非固定比特信道传输的外码码字比特”；
重复比特选取方式具体为：将第h(2≤h≤S-1)段中信道容量最低的
Figure PCTCN2016108511-appb-000053
个比特选择为重复比特；将第h(h＝S)段中信道容量最低的
Figure PCTCN2016108511-appb-000054
个比特选择为重复比特；
对于被重复比特选取方式为：去除第h′(1≤h′≤S-2)段内已经被选择的重复比特，在剩余比特中，信道容量最低的
Figure PCTCN2016108511-appb-000055
个比特作为该段内的被重复比特；去除第h′(h′＝S-1)段内已经选择的重复比特，在剩余比特中，信道容量最低的
Figure PCTCN2016108511-appb-000056
个比特作为该段内的被重复比特；
(1.3)依次配对第h(2≤h≤S)段内的重复比特与第h-1段的被重复比特；配对方法具体为：第h(2≤h≤S)段内信道容量最低的重复比特与第h-1段的信道容量最低的被重复比特进行配对，将对应的重复比特序号与被重复比特序号构成集合Tj；第h(2≤h≤S)段内信道容量第二低的重复比特与第h-1段的信道容量第二低的被重复比特进行配对，以此类推，完成K1个被重复比特的配对，得到K1个二重复码集合Tj(1≤j≤K1)；
(1.4)根据集合Tj(1≤j≤K1)进行外码编码，外码码字中重复比特的位置{Tj\min(Tj)}(1≤j≤K1)处的编码取值等于对应的min(Tj)(1≤j≤K1)处的被重复比特的取值，也即
Figure PCTCN2016108511-appb-000057
根据信息比特和Tj(1≤j≤K1)确定K1个重复比特的取值即可确定外码码字
Figure PCTCN2016108511-appb-000058
步骤2：外码码字映射；
外码码字映射是将外码码字中的各个比特通过该映射关系作为极化码编码输入的比特；极化码的非固定比特信道序号集合为
Figure PCTCN2016108511-appb-000059
固定比特信道序号集合为Ac，极化码输入序列为
Figure PCTCN2016108511-appb-000060
极化码的非固定比特序列为固定比特取值为全0；
外码码字
Figure PCTCN2016108511-appb-000062
映射到非固定比特序列uA的方式为：外码码字的第一个比特x1至最后一个比特
Figure PCTCN2016108511-appb-000063
依次映射到非固定比特序列的第一个比特
Figure PCTCN2016108511-appb-000064
至最后一个比特
Figure PCTCN2016108511-appb-000065
也即满足
Figure PCTCN2016108511-appb-000066
或者
Figure PCTCN2016108511-appb-000067
映射后，非固定比特信道
Figure PCTCN2016108511-appb-000068
上发送为外码中的第j(1≤j≤K1)个被重复比特；非固定比特信道
Figure PCTCN2016108511-appb-000069
上发送为外码与第j(1≤j≤K1)个被重复比特对应的重复比特。
步骤3：内编码器编码；
通过步骤2得到内码极化码非固定比特序列
Figure PCTCN2016108511-appb-000070
极化码固定比特序列为全0，从而可知极化码输入序列
Figure PCTCN2016108511-appb-000071
根据极化码编码公式，获得级联系统编码的码字为
Figure PCTCN2016108511-appb-000072
其中GN为极化码生成矩阵。
步骤4：接收端译码；
采用修正的SCL译码算法进行译码；主要修正的是：译码重复比特时，根据被重复比特的判决结果直接进行判决，而不是根据原始SCL译码算法中的概率进行判决；译码步骤具体如下：
(4.1)判断i是否小于等于N；若是，进入(4.2)；若否，则进入(4.7)；
(4.2)判断ui是否为固定比特；若是，进入(4.3)；若否，则进入(4.4)；
(4.3)将当前每条路径上ui的判决值设为已知的固定比特取值，并令i＝i+1，返回步骤(4.1)；实施例中，上述已知的固定比特序列为全0序列；由此，将当前每条路径上ui的判决值设置为0；
(4.4)判断ui是否为第j(1≤j≤K1)个重复码的重复比特；若是，进入(4.5)；若否，则进入(4.6)；
(4.5)当前每条路径上重复比特ui的判决值直接设置为该路径上与ui 对应的被重复比特的判决值：
Figure PCTCN2016108511-appb-000073
令i＝i+1；返回(4.1)；
其中，Tj表示外码码字中第j个重复码的所有比特位置序号集合，
Figure PCTCN2016108511-appb-000074
表示经过外码码字映射后，第j个重复码中的所有比特在极化码输入序列
Figure PCTCN2016108511-appb-000075
中的序号集合，
Figure PCTCN2016108511-appb-000076
中最小的元素
Figure PCTCN2016108511-appb-000077
对应第j个重复码中的被重复比特，
Figure PCTCN2016108511-appb-000078
对应第j个重复码中的重复比特；
(4.6)统计当前路径数量L′，当前每条路径在ui处可取值0或1，从而得到2L′条备选路径，2L′条路径的度量值分别为该路径在ui处取值0或1的概率：
Figure PCTCN2016108511-appb-000079
或
Figure PCTCN2016108511-appb-000080
若2L′≤L(L为SCL译码算法的最大路径数量)，保留2L′条路径；若2L′>L，保留L条度量值最大的路径；i＝i+1；返回(4.1)；
(4.7)从L条路径中输出路径度量值最大的一条路径上对应的判决序列
Figure PCTCN2016108511-appb-000081
即为译码结果；
在译码结果
Figure PCTCN2016108511-appb-000082
中，
Figure PCTCN2016108511-appb-000083
即为外码码字译码结果；令集合
Figure PCTCN2016108511-appb-000084
即为发送的信息比特译码结果。
图3所示为对码长为4的级联码、二重复码集合
Figure PCTCN2016108511-appb-000085
采用本发明提供的修正SCL(路径数量L＝2)进行译码的示意图；两条路径在译码u3比特时，因为u3为信息比特u1的重复比特，因此u3的判决值直接根据被重复比特
Figure PCTCN2016108511-appb-000086
判决结果得到，而不根据SCL译码算法的路径度量值进行判决。
下面结合实施例来阐述极化码与重复码级联的纠错编码方法，具体如下：
实施例中，级联码码长为N＝16，信息比特数量为M＝8，被重复比特数量为K1＝4，重复比特数量为K2＝4；外码码字中一共有4个二重复码(Tj(1≤j≤4)，每个Tj集合的元素个数为2)；内码极化码非固定比特信道的数量为12，在一定信噪比下进行极化码构造，分别得到非固定比特信道的 序号集合
Figure PCTCN2016108511-appb-000087
和固定比特信道序号的集合Ac.接收端译码器采用修正的SCL译码算法，路径数量设置为L；实施例的极化码与重复码级联的纠错编码方法，具体如下：
步骤1：外编码器编码；
信息比特数量为8，信息比特序列为
Figure PCTCN2016108511-appb-000088
外码码字中重复比特的数量为4，则外码码长为12，外码码字为
Figure PCTCN2016108511-appb-000089
如图4所示，4个二重复码集合分别为：T1＝{1,5}，T2＝{2,8}，T3＝{6,9}，T4＝{7,11}；外编码器编码时，8个信息比特位于外码码字的比特序号如下：
Figure PCTCN2016108511-appb-000090
也即
Figure PCTCN2016108511-appb-000091
根据集合T1＝{1,5}，外码码字比特x5＝x1＝m1；根据T2、T3和T4依次确定x8＝x2＝m2、x9＝x6＝m5和x11＝x7＝m6的取值，从而确定外码码字
Figure PCTCN2016108511-appb-000092
完成外编码器编码。
步骤2：外码码字映射；
外编码器得到的码字为
Figure PCTCN2016108511-appb-000093
极化码非固定比特信道集合为A＝{a1,a2,a3,...,a12}，将码字
Figure PCTCN2016108511-appb-000094
的第1个至12个比特依次映射到极化码第1个至12个非固定比特，则极化码非固定比特序列为
Figure PCTCN2016108511-appb-000095
映射后，非固定比特信道
Figure PCTCN2016108511-appb-000096
上发送为外码中的第j个被重复比特；非固定比特信道
Figure PCTCN2016108511-appb-000097
上发送为外码与第j个被重复比特对应的重复比特；
对于步骤1中得到的4个集合Tj(1≤j≤4)，非固定比特信道
Figure PCTCN2016108511-appb-000098
上发送为外码中4个被重复比特；非固定比特信道
Figure PCTCN2016108511-appb-000099
上发送为外码中4个重复比特。
步骤3：内编码器编码；
通过步骤2得到内码极化码非固定比特序列
Figure PCTCN2016108511-appb-000100
极化码固定比特序列为全0，从而可知极化码输入序列
Figure PCTCN2016108511-appb-000101
根据极化码编码公式获得级联系统编码的码字为
Figure PCTCN2016108511-appb-000102
其中GN为极化码生成矩阵。
步骤4：级联系统的译码；
实施例中，级联系统的译码采用修正的SCL译码方法；对于第一个二重复码集合T1＝{1,5}，当L条路径译码到
Figure PCTCN2016108511-appb-000103
比特时，对于
Figure PCTCN2016108511-appb-000104
比特的判决则根据第一个二重复码中已判决的被重复比特
Figure PCTCN2016108511-appb-000105
得到，也即，
Figure PCTCN2016108511-appb-000106
而对于除重复比特之外的比特判决，则根据SCL译码算法的译码原理进行判决；按照这种修正的SCL译码算法，获得码的译码结果
Figure PCTCN2016108511-appb-000107
在译码结果
Figure PCTCN2016108511-appb-000108
中，
Figure PCTCN2016108511-appb-000109
为外码码字译码结果，令集合I＝{1,2,3,4,6,7,10,12}，则
Figure PCTCN2016108511-appb-000110
即为发送的信息比特译码结果。
以下对极化码与多比特偶校验码级联的纠错编码方法具体阐述如下：
步骤1：外编码器编码；
对于码长为N，信息比特数量为M，校验方程数量为K的级联系统，则外码输入端信息序列的长度为M，输出的外码码字长度为M+K。设外编码器编码得到的码字为
Figure PCTCN2016108511-appb-000111
包含M个信息比特和K个偶校验比特。如图5所示，根据外码码字中K个偶校验比特的位置分布，本发明提供三种确定校验比特位置的方法：
方法一：校验比特集中于外码码字尾部。如图5(a)所示，校验比特位置序号集合为P＝{M+1,M+2,M+3,...,M+K}，集合P中的元素表示在外码的这些比特位置上为校验比特，也即外编码器编码码字
Figure PCTCN2016108511-appb-000112
中，比特序列
Figure PCTCN2016108511-appb-000113
为信息比特，
Figure PCTCN2016108511-appb-000114
为校验比特。
方法二：校验比特在外码码字中等间隔分布。如图5(b)所示，外码码长为M+K，相邻校验比特的间距为
Figure PCTCN2016108511-appb-000115
其中，
Figure PCTCN2016108511-appb-000116
表示取x的下整 数，校验比特位置序号集合为
Figure PCTCN2016108511-appb-000117
方法三：校验比特在外码码字中不等间隔分布。如图5(c)所示，方法三中，一种校验比特不等间隔分布的特征为：越靠近外码码字前部，校验比特分布越分散；越靠近外码码字尾部校验比特分布越集中。按照校验比特位置分布的这种特征，得到校验比特位置序号集合P。
具体实施中，方法三可采用上述方法一和方法二的一个联合形式，也即一部分检验比特集中于尾部，另一部分校验比特等间隔分布。假设集中在尾部的校验比特的数量为K1，则前面等间隔分布的校验比特的数量为K-K1，可知在不等间隔分布下，校验比特的位置序号集合为：P＝{P1∪P2|P1＝{M+K-K1+1,M+K-K1+2,...,M+K},
Figure PCTCN2016108511-appb-000118
。当K1＝0时，集合P1为空集，此时方法三等效于方法一；当K1＝K时，集合P2为空集，此时方法三等效于方法二。
假设根据上述三种方法之一确定的校验比特位置集合记为
Figure PCTCN2016108511-appb-000119
pj表示第j(j＝1,2,3,...,K)个校验比特在外码码字比特中的序号，校验比特
Figure PCTCN2016108511-appb-000120
所在方程中的所有比特在外码中的位置序号集合记为Tj，Tj满足
Figure PCTCN2016108511-appb-000121
并且pj属于Tj,也即pj＝max(Tj)，从而保证校验比特仅校验该比特之前的比特而与该比特之后的比特无关。校验比特
Figure PCTCN2016108511-appb-000122
的取值为：
Figure PCTCN2016108511-appb-000123
完成外码编码。
步骤2：外码码字映射
外码码字映射是指将外码码字中的各个比特通过该映射关系作为极化码编码输入的比特。设极化码的非固定比特信道序号集合为
Figure PCTCN2016108511-appb-000124
固定比特信道序号集合为Ac，极化码输入序列记为
Figure PCTCN2016108511-appb-000125
极化码的非固定比特序列为
Figure PCTCN2016108511-appb-000126
固定比特取值为全0。外码码字
Figure PCTCN2016108511-appb-000127
映射到非固定比特序列uA的方式为：外码码字的 第一个比特x1至最后一个比特xM+K依次映射到非固定比特序列的第一个比特
Figure PCTCN2016108511-appb-000128
至最后一个比特
Figure PCTCN2016108511-appb-000129
也即满足
Figure PCTCN2016108511-appb-000130
或者
Figure PCTCN2016108511-appb-000131
通过映射后可知，非固定比特信道AP＝{ai|i∈P}上发送为外码的校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000132
上发送为外码第j(j＝1,...,K)个校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000133
上发送为外码第j个校验方程的所有比特。
步骤3：内编码器编码
通过步骤2得到内码极化码非固定比特序列
Figure PCTCN2016108511-appb-000134
极化码固定比特序列为全0，从而可知极化码输入序列
Figure PCTCN2016108511-appb-000135
根据极化码编码公式，从而可得级联系统编码的码字为
Figure PCTCN2016108511-appb-000136
其中GN为极化码生成矩阵。
步骤4：接收端译码
级联系统译码器采用修正的SCL译码算法。主要修正的是：译码校验比特时，根据校验比特所在的偶校验方程中信息比特的判决结果进行校验得到，而不是根据原始SCL译码算法中的概率进行判决。
具体译码步骤如下：
初始化输入：i＝1,路径数量L；
Step1:判断i是否小于等于N，是，则进入Step2；否，则进入Step7；
Step2:判断ui是否为固定比特，是，则进入Step3；否，则进入Step4；
Step3:将当前每条路径上ui的判决值设置为0；i＝i+1；返回Step1；
Step4:判断ui是否为第j(1≤j≤K)个校验比特，是，则进入Step5；否，则进入Step6；
Step5:当前每条路径上ui的判决值通过该路径上判决的信息比特校验得到：
Figure PCTCN2016108511-appb-000137
返回Step1；
Step6:统计当前路径数量L′，当前每条路径在ui处可取值0或1，从而得到2L′条备选路径，2L′条路径的度量值分别为该路径在ui处取值0或1的 概率：
Figure PCTCN2016108511-appb-000138
或
Figure PCTCN2016108511-appb-000139
若2L′≤L，保留2L′条路径；若2L′>L，保留L条度量值最大的路径；i＝i+1；返回Step1；
Step7:从L条路径中输出路径度量值最大的一条路径上对应的判决序列
Figure PCTCN2016108511-appb-000140
Step8:结束。
在译码结果
Figure PCTCN2016108511-appb-000141
中，
Figure PCTCN2016108511-appb-000142
即为外码码字译码结果，令集合
Figure PCTCN2016108511-appb-000143
则
Figure PCTCN2016108511-appb-000144
即为发送的信息比特译码结果。
下面结合一个实施例来阐述极化码与多比特偶校验码级联的纠错编码方法本发明方法，具体如下：
该实施例中，级联码码长为N＝512，信息比特数量为M＝240，校验方程的数量为K＝16，从而可知内码极化码非固定比特信道的数量为256，固定比特信道的数量为256，在2dB下进行极化码构造，分别得到非固定比特信道的序号集合
Figure PCTCN2016108511-appb-000145
和固定比特信道序号的集合Ac.接收端译码器采用修正的SCL译码算法，路径数量设置为L＝32。
步骤1：外编码器编码
信息比特数量为240，校验方程数量为16，则外码码长为256，记外码码字为
Figure PCTCN2016108511-appb-000146
根据上述说明的校验比特位置序号集合P确定的三种方法并结合实施例，依次得到集合P如下：
方法一，校验比特集中于外码码字尾部。这表明在外码
Figure PCTCN2016108511-appb-000147
的比特中，后16个比特，也即
Figure PCTCN2016108511-appb-000148
为偶校验比特；前240个比特，也即
Figure PCTCN2016108511-appb-000149
为信息比特，则集合P＝{241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256}。
方法二，校验比特在外码码字中等间隔分布。这表明在外码
Figure PCTCN2016108511-appb-000150
的比特中，x1×16，x2×16，x3×16，…，x16×16为校验比特，其余240个比特为信息比特，则集合P＝{16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256}。
方法三，校验比特在外码码字中不等间隔分布。假设在不等间隔分布 中，集中在尾部的校验比特的数量K1＝8，前面等间隔分布的校验比特数量为K-K1＝8，这表明在外码
Figure PCTCN2016108511-appb-000151
的比特中，x1×31，x2×31，x3×31，…，x8×31，x249，x250，x251，…，x256为校验比特，其余240个比特为信息比特，结合本例，则集合P＝{31,62,93,124,155,186,217,248,249,250,251,252,253,254,255,256}。
以方法二得到的集合P为例说明每个校验方程中信息比特位置的确定，以及每个偶校验方程的编码。
已知P＝{16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256}，也即第一个校验方程的校验比特为x16，该方程中所有比特的位置将从集合S1＝{1,2,3,...,16}中挑选，并且校验比特的位置p1＝16一定被选择，集合S1\max(S1)中的元素按照概率α＝0.5选择是否参与该校验方程，从而得到第一个校验方程所有比特位置集合为T1＝{6,8,10,11,16}，因此第一个校验比特x16的编码公式如下：
Figure PCTCN2016108511-appb-000152
按照上述方式可以确定另外15个校验比特的取值，从而完成外码编码。
步骤2：外码码字映射
已知外编码器得到的码字为
Figure PCTCN2016108511-appb-000153
极化码非固定比特信道集合为A＝{a1,a2,a3,...,a256}，将码字
Figure PCTCN2016108511-appb-000154
的第1个至256个比特依次映射到极化码第1个至256个非固定比特，则极化码非固定比特序列为
Figure PCTCN2016108511-appb-000155
映射后可知，非固定比特信道AP＝{ai|i∈P}上发送外码的校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000156
上发送为外码第j(j＝1,...,16)个校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000157
上发送为外码第j个校验方程的所有比特。以方法二得到的P＝{16,32,48,64,80,96,112,128,144,160,176,192,208,224,240,256}为例，则
AP＝{ai|i∈P}＝{a16,a32,a48,a64,a80,a96,a112,a128,a144,a160,a176,a192,a208,a224,a240,a256}；发送外码校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000158
上发送外码第1个校验比特；非固定比特信道
Figure PCTCN2016108511-appb-000159
上发送外码第1个校验方程的 所有比特。依次可以得到
Figure PCTCN2016108511-appb-000160
等的取值。
步骤3：内编码器编码
通过步骤2得到内码极化码非固定比特序列
Figure PCTCN2016108511-appb-000161
极化码固定比特序列为全0，从而可知极化码输入序列
Figure PCTCN2016108511-appb-000162
根据极化码编码公式，从而可得级联系统编码的码字为
Figure PCTCN2016108511-appb-000163
其中GN为极化码生成矩阵。
步骤4：级联系统的译码
级联系统的译码采用修正的SCL译码算法，该算法与原始SCL译码算法的主要区别在于，译码校验比特时，校验比特判决值直接根据该校验方程的信息比特判决值进行校验得到。以第一个校验方程为例，当32条路径译码到
Figure PCTCN2016108511-appb-000164
比特时，对于
Figure PCTCN2016108511-appb-000165
比特的判决则根据第一个校验方程中信息比特的判决值进行校验得到，具体公式如下：
Figure PCTCN2016108511-appb-000166
而对于集合P之外的非固定比特信道上的比特判决，则根据SCL译码算法的译码原理进行判决。按照这种修正的SCL译码算法，从而可以得到级联码的译码结果
Figure PCTCN2016108511-appb-000167
在译码结果
Figure PCTCN2016108511-appb-000168
中，
Figure PCTCN2016108511-appb-000169
即为外码码字译码结果，令集合
Figure PCTCN2016108511-appb-000170
则
Figure PCTCN2016108511-appb-000171
即为发送的信息比特译码结果。
本领域的技术人员容易理解，以上所述仅为本发明的较佳实施例而已，并不用以限制本发明，凡在本发明的精神和原则之内所作的任何修改、等同替换和改进等，均应包含在本发明的保护范围之内。

Claims (18)

  1. 一种极化码与重复码或多比特偶校验码级联的纠错编码方法，其特征在于，包括如下步骤：
    (1)通过对信息比特序列进行重复码编码或多比特偶校验编码获得外码码字；
    (2)通过将所述外码码字的第一个比特至最后一个比特依次映射到极化码的第一个至最后一个非固定比特来进行外码码字映射，获得非固定比特序列；
    (3)通过对所述非固定比特序列进行极化码编码获得级联码码字。
  2. 如权利要求1所述的纠错编码方法，其特征在于，所述步骤(1)中对信息比特序列进行重复码编码时，被重复比特可被重复一次或多次。
  3. 如权利要求2所述的纠错编码方法，其特征在于，在所述步骤(2)的外码码字映射过程中，被重复比特所映射的非固定比特信道的信道容量，低于未被重复的比特所映射的非固定比特信道的信道容量。
  4. 如权利要求2或3所述的纠错编码方法，其特征在于，在所述步骤(2)的外码码字映射过程中，所述外码码字的重复比特所映射的非固定比特的序号大于该重复比特对应的被重复比特所映射的非固定比特的序号。
  5. 如权利要求4所述的纠错编码方法，其特征在于，所述非固定比特序列中,外码码字的重复比特均匀分布或近似均匀分布。
  6. 如权利要求5所述的纠错编码方法，其特征在于，通过将所述非固定比特序列按序号顺序划分为S个段，将所述外码码字的重复比特映射到每段内信道容量最低的Kh个非固定比特信道上，每段内分配的重复比特数相等或近似相等，使得外码码字的重复比特在非固定比特中均匀分布或近似均匀分布；其中，h＝1,2,...,S。
  7. 如权利要求4所述的纠错编码方法，其特征在于，所述外码码字为 反相重复码；当被重复比特为1，所述反相重复码的重复比特为0；当被重复比特为0，所述反向重复码的重复比特为1；
    当被重复比特的重复次数为K时，重复码编码所获得的K个重复比特中反相重复比特的数量为0～K。
  8. 如权利要求4所述的纠错编码方法，其特征在于，将外码码字尾部的部分比特作为校验比特，每个校验比特与该校验比特之前的多个信息比特构成偶校验或奇校验关系。
  9. 如权利要求4所述的纠错编码方法，其特征在于，所述纠错编码方法的译码方法如下：
    (4)对于原始信息比特，根据SCL译码算法进行比特判决；对于重复比特，根据被重复比特的判决结果直接进行判决。
  10. 如权利要求9所述的纠错编码方法，其特征在于，所述步骤(4)包括如下步骤：
    (4.1)判断i是否小于等于N；若是，进入步骤(4.2)，若否，则进入步骤(4.7)；
    其中，N是指级联码码长，i是指当前译码第i个比特的索引序号，其初始值为1，取值从1到N的正整数；
    (4.2)判断ui是否为固定比特，若是，进入步骤(4.3)；若否，则进入步骤(4.4)；其中，ui是指极化码输入序列
    Figure PCTCN2016108511-appb-100001
    的第i个比特；
    Figure PCTCN2016108511-appb-100002
    为1×N的行向量(u1,u2,u3,...,uN)；
    (4.3)将当前每条路径上ui的判决值设为已知的固定比特的取值，并令i＝i+1，返回步骤(4.1)；
    (4.4)判断ui是否为第j个重复码的重复比特，若是，进入步骤(4.5)；若否，则进入步骤(4.6)；其中，K为外码中被重复比特的个数，1≤j≤K；
    (4.5)将当前每条路径上重复比特ui的判决值设置为该路径上与ui对 应的被重复比特的判决值，具体地：
    Figure PCTCN2016108511-appb-100003
    并令i＝i+1，返回步骤(4.1)；
    其中，Tj是指外码码字中第j个重复码的所有比特位置序号集合；
    Figure PCTCN2016108511-appb-100004
    是指经过外码码字映射后，第j个重复码中的所有比特在极化码输入序列
    Figure PCTCN2016108511-appb-100005
    中的序号集合，
    Figure PCTCN2016108511-appb-100006
    表示取集合
    Figure PCTCN2016108511-appb-100007
    中的最小元素；
    (4.6)统计当前路径数量L′，通过将当前每条路径在ui处取值0或1获得2L′条备选路径；并判断是否满足2L′≤L，若是，保留2L′条路径；若否，则保留L条度量值最大的路径；并令i＝i+1，返回步骤(4.1)；
    其中，2L′条路径的度量值分别为该路径在ui处取值0或1的概率
    Figure PCTCN2016108511-appb-100008
    或
    Figure PCTCN2016108511-appb-100009
    L为SCL译码算法的最大路径数量；
    (4.7)从L条路径中输出路径度量值最大的一条路径上对应的判决序列
    Figure PCTCN2016108511-appb-100010
    获得译码结果。
  11. 如权利要求1所述的纠错编码方法，其特征在于，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特集中于外码码字尾部；
    校验比特位置序号集合为P＝{M+1,M+2,M+3,...,M+K}，集合P中的元素表示在外码的这些比特位置上为校验比特，也即外编码器编码码字
    Figure PCTCN2016108511-appb-100011
    中，比特序列
    Figure PCTCN2016108511-appb-100012
    为信息比特，
    Figure PCTCN2016108511-appb-100013
    为校验比特；
    其中，M为信息比特数量，K为校验方程数量。
  12. 如权利要求1所述的纠错编码方法，其特征在于，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特在外码码字中等间隔分布；
    相邻校验比特的间距为
    Figure PCTCN2016108511-appb-100014
    校验比特位置序号集合为
    Figure PCTCN2016108511-appb-100015
    其中，M为信息比特数量，K为校验方程数量，M+K为外码码长，
    Figure PCTCN2016108511-appb-100016
    是指取x的下整数。
  13. 如权利要求1所述的纠错编码方法，其特征在于，步骤(1)中通过对信息比特序列进行多比特偶校验编码所获得的外码码字中的校验比特在外码码字中不等间隔分布。
  14. 如权利要求13所述的纠错编码方法，其特征在于，假设集中在外码码字尾部的校验比特的数量为K1，则前面等间隔分布的校验比特的数量为K-K1，在不等间隔分布下，校验比特的位置序号集合为：
    Figure PCTCN2016108511-appb-100017
    Figure PCTCN2016108511-appb-100018
    其中，M为信息比特数量，K为校验方程数量。
  15. 如权利要求1或11至14中任一项的纠错编码方法，其特征在于，步骤(1)中对信息比特序列进行多比特偶校验编码时，校验比特仅校验该比特之前的比特，而与之后的比特无关。
  16. 如权利要求15所述的纠错编码方法，其特征在于，外码替换为多比特奇校验码。
  17. 如权利要求15所述的纠错编码方法，其特征在于，该方法的译码采用修正的SCL译码算法进行译码，即译码信息比特时根据SCL译码算法进行比特判决，译码校验比特时，根据校验比特所在的偶校验方程中信息比特的判决结果进行校验得到。
  18. 如权利要求17所述的纠错编码方法，其特征在于，所述译码的具体过程为：
    Step1:判断i是否小于等于N，是，则进入Step2；否，则进入Step7；其中，N是指级联码码长；i是指当前译码第i个比特的索引序号，其初始值为1，取值为从1到N的正整数；
    Step2:判断ui是否为固定比特，是，则进入Step3；否，则进入Step4；其中，ui是指极化码输入序列
    Figure PCTCN2016108511-appb-100019
    的第i个比特；
    Step3:将当前每条路径上ui的判决值设置为接收端已知的固定比特的取值；令i＝i+1；返回Step1；
    Step4:判断ui是否为第j(j＝1,2,...,K)个校验比特，是，则进入Step5；否，则进入Step6；其中，K为校验方程数量；
    Step5:当前每条路径上ui的判决值通过该路径上判决的信息比特校验得到：
    Figure PCTCN2016108511-appb-100020
    令i＝i+1；返回Step1；其中，Tj是指第j个校验方程的所有比特在外码码字中的位置序号集合，
    Figure PCTCN2016108511-appb-100021
    表示经过外码码字映射后，第j个校验方程中的所有比特在极化码输入序列
    Figure PCTCN2016108511-appb-100022
    中的序号集合，
    Figure PCTCN2016108511-appb-100023
    表示取值集合
    Figure PCTCN2016108511-appb-100024
    中的最大元素，该元素为第j个校验方程中的校验比特映射到
    Figure PCTCN2016108511-appb-100025
    中的序号；
    Figure PCTCN2016108511-appb-100026
    是指取集合
    Figure PCTCN2016108511-appb-100027
    与
    Figure PCTCN2016108511-appb-100028
    的差，
    Figure PCTCN2016108511-appb-100029
    h为求和运算中的临时变量，依次指代集合
    Figure PCTCN2016108511-appb-100030
    中的每一个元素；
    Step6:统计当前路径数量L′，当前每条路径在ui处可取值0或1，从而得到2L′条备选路径，2L′条路径的度量值分别为该路径在ui处取值0或1的概率：
    Figure PCTCN2016108511-appb-100031
    或
    Figure PCTCN2016108511-appb-100032
    若2L′≤L，保留2L′条路径；若2L′>L，保留L条度量值最大的路径；i＝i+1；返回Step1；其中，L为SCL译码算法的最大路径数量；
    Step7:从L条路径中输出路径度量值最大的一条路径上对应的判决序列
    Figure PCTCN2016108511-appb-100033
    Step8:结束。
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