Apparatus and method for generating a frozen set associated with a polar code
An apparatus for generating a frozen set associated with a polar code of length ‘N’ and dimension ‘K’ comprises a processing unit configured to take in input the polar code length ‘N’, the dimension ‘K’, and a profile of a structure of a block lower triangular affine (BLTA) group. The BLTA group structure is associated with an affine transformation matrix of size ‘n×n’ and the profile is an ordered set of a plurality of values corresponding to block sizes of blocks. The blocks are sub-matrices of the affine transformation matrix with all the diagonals of blocks in the same order as the ordered block sizes, forming the diagonal of the affine transformation matrix, each of the block sizes is such that ‘n’ is equal to the sum of block sizes and ‘n’ is equal to log 2 (N). The processing unit generates the frozen set.
1 . An apparatus for generating a frozen set associated with a polar code of length N and dimension K, the frozen set being of size N−K and comprising bit indices, the apparatus comprising a processing unit configured to:
obtain as an input the polar code length N, the polar code dimension K, and a profile;
generate the frozen set so that the associated polar code admits all affine automorphisms belonging to a block lower triangular affine (BLTA) group consistent with the profile, wherein:
said BLTA group being associated with an affine transformation matrix of size n×n,
said profile being defined by an ordered set of a plurality of values corresponding to block sizes of a plurality of blocks,
said blocks being sub-matrices of said affine transformation matrix,
all the diagonals of said blocks forming the diagonal of said affine transformation matrix, and
each of the block sizes being such that n is equal to the sum of said block sizes and n is equal to log 2 (N);
generate a generating monomials set so that the associated polar code admits all the affine automorphisms belonging to the BLTA group, and generate the frozen set from said generating monomials set;
encoding data using the polar code associated with the frozen set; and
sending the data encoded using the polar code associated with the frozen set.
2 . The apparatus according to claim 1 , wherein the blocks are upper diagonal sub-matrices.
3 . The apparatus according to claim 1 , wherein the processing unit is further configured to determine an initial generating monomials set, define a current generating monomial set as being the initial generating monomials set, and perform a loop, said loop comprising repeating, for each given block of the ordered blocks corresponding to the ordered block sizes in the profile:
a determination of a set of generating monomials corresponding to said given block, and
a modification of the current generating monomials set by concatenation with the determined generating monomials corresponding to said given block.
4 . The apparatus according to claim 3 , wherein the processing unit is further configured to determine the initial generating monomials set by using a predefined polar code of length N and dimension K, having a predefined signal-to-noise ratio, the initial generating monomials set containing the K−1 most reliable monomials of the generating monomials set of the predefined polar code.
5 . The apparatus according to claim 3 , wherein the processing unit is further configured to check, at each stage of the loop, the dimension of the modified current generating monomials set, and if said dimension is greater than K, stop the loop and restart the loop from the first block of the ordered blocks corresponding to the ordered block sizes in the profile, with a new initial generating monomials set.
6 . The apparatus according to claim 3 , wherein the processing unit is further configured to check, at the last stage of the loop, the dimension of the modified current generating monomials set, and
if said dimension is equal to K, determine the generating monomials set as being the current generating monomials set,
if said dimension is not equal to K, stop the loop and restart the loop from the first block of the ordered blocks corresponding to the ordered block sizes in the profile, with a new initial generating monomials set.
7 . The apparatus according to claim 6 , wherein the processing unit is further configured to:
when the dimension of the modified current generating monomials set is greater than 1, determine the new initial generating monomials set by using a predefined polar code of length N and having a new dimension equal to the previous dimension minus 1, and having the predefined signal-to-noise ratio, the initial generating monomials set containing a number of most reliable monomials of the generating monomials set of the predefined polar code corresponding to the new dimension minus 1; and
when the dimension of the modified current generating monomials set is equal to 1, determine the new initial generating monomials set by using a predefined polar code of length N and of dimension K, and having a new predefined signal-to-noise ratio corresponding to the previous predefined signal-to-noise ratio incremented by a given incremental value, the initial generating monomials set containing the K−1 most reliable monomials of the generating monomials set of the predefined polar code.
8 . The apparatus according to claim 1 , wherein the processing unit further comprises a processor configured to generate a polar code codeword of the information word using the polar code associated with the frozen set generated by said apparatus.
9 . A method for generating a frozen set associated with a polar code of length N and dimension K, the frozen set being of size N−K and comprising bit indices, performed by a processing unit, the method comprising:
obtaining as an input the polar code length N, the polar code dimension K, and a profile; and
generating the frozen set so that the associated polar code admits all the affine automorphisms belonging to a block lower triangular affine (BLTA) group consistent with the profile, wherein:
said BLTA group being associated with an affine transformation matrix of size n×n,
said profile being defined by an ordered set of a plurality of values corresponding to block sizes of a plurality of blocks,
said blocks being sub-matrices of said affine transformation matrix,
all the diagonals of said blocks forming the diagonal of said affine transformation matrix, and
each of the block sizes being such that n is equal to the sum of said block sizes and n is equal to log 2 (N);
generating a generating monomials set so that the associated polar code admits all the affine automorphisms belonging to the BLTA group, and generating the frozen set from the said generating monomials set;
encoding data using the polar code associated with the frozen set; and
sending the data encoded using the polar code associated with the frozen set.
10 . The method according to claim 9 , wherein the blocks are upper diagonal sub-matrices.
11 . The method according to claim 9 , further comprising determining an initial generating monomials set, defining a current generating monomial set as being the initial generating monomials set, and performing a loop, said loop comprising repeating, for each given block of the ordered blocks corresponding to the ordered block sizes in the profile:
determining a set of generating monomials corresponding to said given block, and
modifying the current generating monomials set by concatenation with the determined generating monomials corresponding to said given block.
12 . The method according to claim 11 , further comprising determining the initial generating monomials set by using a predefined polar code of length N and dimension K, having a predefined signal-to-noise ratio, the initial generating monomials set containing the K−1 most reliable monomials of the generating monomials set of the predefined polar code.
13 . The method according to claim 11 , further comprising checking, at each stage of the loop, the dimension of the modified current generating monomials set, and if said dimension is greater than K, stopping the loop and restarting the loop from the first block of the ordered blocks corresponding to the ordered block sizes in the profile, with a new initial generating monomials set.
14 . The method according to claim 11 , further comprising checking, at the last stage of the loop, the dimension of the modified current generating monomials set, and:
if said dimension is equal to K, determining the generating monomials set as being the current generating monomials set;
if said dimension is not equal to K, stopping the loop and restarting the loop from the first block of the ordered blocks corresponding to the ordered block sizes in the profile, with a new initial generating monomials set.
15 . The method according to claim 14 , further comprising:
when the dimension of the modified current generating monomials set is greater than 1, determining the new initial generating monomials set by using a predefined polar code of length N and having a new dimension equal to the previous dimension minus 1, and having the predefined signal-to-noise ratio, the initial generating monomials set containing a number of most reliable monomials of the generating monomials set of the predefined polar code corresponding to the new dimension minus 1; and
when the dimension of the modified current generating monomials set is equal to 1, determining the new initial generating monomials set by using a predefined polar code of length N and of dimension K, and having a new predefined signal-to-noise ratio corresponding to the previous predefined signal-to-noise ratio incremented by a given incremental value, the initial generating monomials set containing the K−1 most reliable monomials of the generating monomials set of the predefined polar code.