Systems and methods for using not perfectly polarized bit channels in parallel polar codes

1. A method for encoding information bits for transmission across a communication channel, the method comprising:

distributing information bits between m parallel polar codes such that each of the m parallel polar codes includes a subset of the information bits, wherein m is the number of parallel polar codes;

splitting the subset of information bits in each of the m parallel polar codes into a protected information section and a full rate information section, information bits in the protected information section of each of the m parallel polar codes being arranged in positions in a plurality of first bit channels in the respective parallel polar code of the m parallel polar codes and grouped into a plurality of L blocks, wherein each of the plurality of L blocks comprises a subset of the information bits in the protected information section for each of the m parallel polar codes, wherein L is the number of blocks;

protecting the information bits in the protected information section of each of the m parallel polar codes;

arranging a plurality of frozen bits in each of the m parallel polar codes, with the plurality of frozen bits being arranged in positions in a plurality of second bit channels that have a capacity near zero; and

generating a polar encoded codeword for each of the m parallel polar codes.

2. The method of claim 1 , wherein the plurality of first bit channels are bit channels that have a capacity between 0 and 1.

3. The method of claim 1 , wherein the plurality of first bit channels are non-perfectly polarized bit channels.

4. The method of claim 1 , wherein the information bits in the protected information section of each of the m parallel polar codes are protected using a repetition code.

5. The method of claim 4 , wherein the information bits in a block of the plurality of L blocks of a first parallel polar code from the m parallel polar codes are repeated in a block of the plurality of L blocks of a second parallel polar code from the m parallel polar codes.

6. The method of claim 5 , wherein the information bits in a block of the plurality of L blocks of a first parallel polar code from the m parallel polar codes are repeated d i times in the m parallel polar codes, d i being a factor of m and greater than one.

7. The method of claim 1 , wherein a total number of the plurality of blocks in the non-perfectly polarized bit channels in each of the m parallel polar codes is determined based on a total number of greater-than-one factors of m.

8. A decoding method, comprising:

receiving m polar coded codewords over a communication channel, each of the m polar coded codewords encoding a subset of information bits in a plurality of nodes and a frozen bit, wherein the information bits are distributed between the m polar coded codewords and m is the number of polar coded codewords; and

for each of the m polar coded codewords, decoding the polar coded codewords to generate a decoded message comprising a decoded bit for the frozen bit, a first part for a node in a full rate information section of the polar coded codeword and a second part for a node in a protected information section of the polar coded codeword, wherein the decoded bit is generated based on a predetermined value, the first part is generated based on a log-likelihood ratio (LLR) decoding algorithm and the second part is generated based on an averaged LLR decoding algorithm.

9. The method of claim 8 , wherein the decoding of the node is performed by a successive-cancellation (SC) or a successive-cancellation-list (SCL).

10. The method of claim 8 , wherein the averaged LLR decoding algorithm is based on:

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 d i is a total number of repetitions for the node in the m polar coded codewords, n represents that LLR k n are new modified LLRs, and LLR j o is the original LLR of the node in the respective codeword j in the m polar coded codewords, j=1 . . . m, K represents that the number of LLRs that are involved in averaging, and “*” here indicates the mathematical operation of multiplication.

11. An encoder that encodes information bits for transmission across a communication channel, the encoder comprising at least one processor and a memory storing programmed instructions that, when executed by the at least one processor, cause the encoder to perform the following steps:

distributing the information bits between m parallel polar codes such that each of the m parallel polar codes includes a subset of the information bits, wherein m is the number of parallel polar codes;

splitting the subset of information bits in each of the m parallel polar codes into a protected information section and a full rate information section, information bits in the protected information section of each of the m parallel polar codes being arranged in positions in a plurality of first bit channels in the respective parallel polar code of the m parallel polar codes and grouped into a plurality of L blocks, wherein each of the plurality of L blocks comprises a subset of the information bits in the protected information section, wherein L is the number of blocks;

protecting the information bits in the protected information section of each of the m parallel polar codes;

arranging a plurality of frozen bits in each of the m parallel polar codes, with the plurality of frozen bits being arranged in positions in a plurality of second bit channels that have a capacity near zero; and

generating a polar encoded codeword for each of the m parallel polar codes.

12. The encoder of claim 11 , wherein the plurality of first bit channels are bit channels that have a capacity between 0 and 1.

13. The encoder of claim 11 , wherein the plurality of first bit channels are non-perfectly polarized bit channels.

14. The encoder of claim 11 , wherein the information bits in the protected information section of each of the m parallel polar codes are protected using a repetition code.

15. The encoder of claim 11 , wherein the information bits in a block of the plurality of L blocks of a first parallel polar code from the m parallel polar codes are repeated in a block of the plurality of L blocks of a second parallel polar code from the m parallel polar codes.

16. The encoder of claim 15 , wherein the information bits in a block of the plurality of L blocks of a first parallel polar code from the m parallel polar codes are repeated d i times in the m parallel polar codes, d i being a factor of m and greater than one.

17. A decoder comprising:

at least one processor and a memory storing programmed instructions that, when executed by the at least one processor, cause the decoder to perform the following steps:

receiving m polar coded codewords over a communication channel, each of the m polar coded codewords encoding a subset of information bits in a plurality of nodes and a frozen bit, wherein the information bits are distributed between the m polar coded codewords and m is the number of polar coded codewords; and

for each of the m polar coded codewords, decoding the polar coded codeword to generate a decoded message comprising a decoded bit for the frozen bit, a first part for a node in a full rate information section of the polar coded codeword and a second part for a node in a full rate information section of the polar coded codeword, wherein the decoded bit is generated based on a predetermined value, the first part is generated based on a log-likelihood ratio (LLR) decoding algorithm and the second part is generated based on an averaged LLR decoding algorithm.

18. The decoder of claim 17 , wherein the averaged LLR decoding algorithm is based on:

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where

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 d i is a total number of repetitions for the node in the m polar coded codewords, n represents that LLR k n are new modified LLRs, and LLR j o β is the original LLR of the node in the respective codeword j in the m polar coded codewords, j=1 . . . m, K represents that the number of LLRs that are involved in averaging, and “*” here indicates the mathematical operation of multiplication.