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 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 non-perfectly polarized 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; and

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

2. 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.

3. The method of claim 2 , 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.

4. The method of claim 3 , 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.

5. 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.

6. A method for decoding m polar coded codewords received over a communication channel, each of the m polar coded codewords encoding information bits in a plurality of nodes, wherein m is the number of polar coded codewords, the method comprising:

for each node in each codeword from the m polar coded codewords:

when the node is in a full rate information section: decoding the node based on a log-likelihood ratio (LLR) decoding algorithm to generate a first part of a decoded message; and

when the node is in a protected information section: decoding the node based on an averaged LLR decoding algorithm to generate a second part of the decoded message.

7. The method of claim 6 , further comprising, for each of the m polar coded codewords: when a bit in the respective codeword is a frozen bit, generating a decoded bit in the decoded message based on a predetermined value.

8. The method of claim 6 , wherein the decoding of the node is performed by a successive-cancellation (SC) or a successive-cancellation-list (SCL).

9. The method of claim 6 , wherein the averaged LLR decoding algorithm is based on:

LLR

k

n

=

1

K

⁢

∑

j

=

(

k

-

1

)

*

d

i

+

1

k

*

d

i

LLR

j

o

where

≤

k

≤

m

d

i

,

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.

10. An encoder that encodes information bits for transmission across a communication channel, the encoder comprising circuitry configured to:

distribute 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;

split 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 non-perfectly polarized 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;

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

arrange a plurality of frozen bits in each of the m parallel polar codes; and

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

11. The encoder of claim 10 , wherein the circuitry comprises at least one processor and a memory storing programmed instructions that when executed by the at least one processor cause the at least one processor to encode the information bits.

12. The encoder of claim 10 , wherein the information bits in the protected information section of each of the m parallel polar codes are protected using a repetition code.

13. The encoder of claim 12 , 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.

14. The encoder of claim 13 , 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.

15. A decoder that decodes m polar coded codewords received over a communication channel, wherein m is the number of polar coded codewords, the decoder comprising circuitry configured to:

for each node in each codeword from the m polar coded codewords:

when the node is in a full rate information section: decode the node based on a log-likelihood ratio (LLR) decoding algorithm to generate a first part of a decoded message;

when the node is in a protected information section: decode the node based on an averaged LLR decoding algorithm to generate a second part of the decoded message; and

when a bit in the respective codeword is a frozen bit: generate a decoded bit in the decoded message based on a predetermined value.

16. The encoder of claim 10 , wherein a total number of the plurality of L 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.