Apparatus and methods for polar code construction and coding

1. A method for error-correction enabled communication, the method comprising:

encoding input bits according to a chained generator matrix to generate a codeword, the chained generator matrix being constructed recursively by, in each iteration:

constructing the chained generator matrix for a current iteration by, using a first base generator matrix from a previous iteration, replacing a first subset of the nonzero entries of the first base generator matrix with entries of a second base generator matrix, and replacing a second subset of the nonzero entries of the first base generator matrix with entries of a third base generator matrix that is different from the second base generator matrix, the third base generator matrix being constructed by applying a chaining matrix to the second base generator matrix; and

repeating the iterations until the chained generator matrix of a final iteration is of at least a target size; and

transmitting the codeword.

2. The method of claim 1 , wherein the first base generator matrix is based on a z-by-z kernel, wherein the second base generator matrix is a N/z-by-N/z matrix, and wherein the chained generator matrix is an N-by-N matrix.

3. The method of claim 2 , wherein z=2, N=2 n , and the chaining matrix is (F ⊗(n−1) ) T , where F is the kernel.

4. The method of claim 1 , further comprising:

selecting the second subset of the nonzero entries based on a target row weight for the chained generator matrix of the final iteration.

5. The method of claim 1 , further comprising:

constructing a further generator matrix based on the chained generator matrix;

applying, to a subset of entries in the further generator matrix, the chaining matrix to construct a further chained generator matrix that is different from the further generator matrix.

6. The method of claim 1 , further comprising:

storing to a memory the chained generator matrix from the final iteration.

7. The method of claim 6 , wherein the storing further comprises storing to the memory the chained generator matrix from each iteration before the final iteration.

8. The method of claim 1 , wherein non-zero entries in the first base generator matrix comprise a common matrix, wherein the encoding comprises:

applying to the common matrix respective parts of an input vector that includes the input bits;

applying each of a subset of the respective parts of the input vector to the chaining matrix and the common matrix.

9. The method of claim 8 , further comprising:

switching the common matrix and the chaining matrix between:

a non-chained polar code matrix as the common matrix and an identity matrix as the chaining matrix; and

a chained polar code matrix as the common matrix and a non-identity matrix as the chaining matrix.

10. A non-transitory processor-readable medium storing instructions which, when executed by one or more processors, cause the one or more processors to perform a method according to claim 1 .

11. An apparatus comprising:

a processor;

a memory coupled to the processor, the memory storing instructions which, when executed by the processor, cause the processor to perform a method according to claim 1 .

12. An apparatus for error-correction enabled communication, the apparatus comprising:

an encoder to encode input bits according to a chained generator matrix to generate a codeword, the chained generator matrix being constructed recursively by, in each iteration:

constructing the chained generator matrix for a current iteration by, using a first base generator matrix from a previous iteration, replacing a first subset of the nonzero entries of the first base generator matrix with entries of a second base generator matrix, and replacing a second subset of the nonzero entries of the first base generator matrix with entries of a third base generator matrix that is different from the second base generator matrix, the third base generator matrix being constructed by applying a chaining matrix to the second base generator matrix; and

repeating the iterations until the chained generator matrix of a final iteration is of at least a target size; and

a transmitter, coupled to the encoder, to transmit the codeword.

13. The apparatus of claim 12 , wherein the first base generator matrix is based on a z-by-z kernel, wherein the second base generator matrix is a N/z-by-N/z matrix, and wherein the chained generator matrix is an N-by-N matrix.

14. The apparatus of claim 13 , wherein z=2, N=2 n , and the chaining matrix is (F ⊗(n−1) ) T , where F is the kernel.

15. The apparatus of claim 12 , wherein the encoder is further configured to select the second subset of the nonzero entries based on a target row weight for the chained generator matrix of the final iteration.

16. The apparatus of claim 12 , wherein the encoder is further configured to:

construct a further generator matrix based on the chained generator matrix; and

apply, to a subset of entries in the further generator matrix, the chaining matrix to construct a further chained generator matrix that is different from the further generator matrix.

17. The apparatus of claim 12 , further comprising:

a memory, coupled to the encoder,

wherein the encoder is further configured to store to the memory the chained generator matrix from the final iteration.

18. The apparatus of claim 17 , wherein the encoder is further configured to store to the memory the chained generator matrix from each iteration before the final iteration.

19. The apparatus of claim 12 , wherein non-zero entries in the first base generator matrix comprise a common matrix, wherein the encoder is further configured to:

apply to the common matrix respective parts of an input vector that includes the input bits; and

apply each of a subset of the respective parts of the input vector to the chaining matrix and the common matrix.

20. The apparatus of claim 19 , wherein the encoder is further configured to switch the common matrix and the chaining matrix between: a non-chained polar code matrix as the common matrix and an identity matrix as the chaining matrix; and a chained polar code matrix as the common matrix and a non-identity matrix as the chaining matrix.