Coding and Decoding of Polar Codes Extended to Lengths which are not Powers of Two

A method, comprising:

• encoding data x of dimension k into a codeword c of length n using a C(n, k, d) code, wherein the code C(n, k, d) has a length n and a minimum distance d, wherein n=2m1+ . . . +2ms, wherein mh, h=1, . . . , s, is an integer number, on the basis of the equation:
• c=uA,
• wherein ui=xji, 0≤ji<k−1, if i∉F, wherein F is a set of n−k frozen bit indices of the code C(n, k, d), and ut=Σs=0i−1ϕi,sus, if i∈F, wherein is a constraint matrix given by a solution of the equation:
• T=0,
• wherein ϕi is an index of a row of the matrix , which has a last non-zero element in column i, wherein is a precoding matrix, and wherein A is defined as:
• A
• =
• (
• A
• m
• 1
• 0
• …
• 0
• 0
• A
• m
• 2
• …
• 0
• ⋮
• ⋮
• ⋱
• ⋮
• 0
• 0
• …
• A
• m
• s
• )
• ,
• and
• wherein
• A
• m
• h
• =
• (
• 1
• 0
• 1
• 1
• )
• ⊗
• m
• h
• ,
• and wherein Q⊗m denotes the m-times Kronecker product of a matrix Q with itself.

A decoding apparatus

• a processor;
• and
• a non-transitory computer-readable storage medium storing a program for decoding a codeword c of length n, the program to be executed by the processor, the program including instructions to:
• decode the codeword c using a C(n, k, d) code, wherein the code C(n, k, d) has a length n and a minimum distance d, wherein n=2m1+ . . . +2ms, wherein mh, h=1, . . . , s, is an integer number, wherein:
• c=uA,
• wherein ui=xji, 0≤ji<k−1, wherein x is data of dimension k, if i∉F, wherein F is a set of n−k frozen bit indices of the code C(n, k, d) and ui=Σs=0i−1ϕi,sus, if i∈F, wherein V is a constraint matrix given by a solution of the equation:
• T=0,
• wherein ϕi is an index of a row of the matrix V, which has a last non-zero element in column i, wherein is a precoding matrix, and wherein A is defined as:
• A
• =
• (
• A
• m
• 1
• 0
• …
• 0
• 0
• A
• m
• 2
• …
• 0
• ⋮
• ⋮
• ⋱
• ⋮
• 0
• 0
• …
• A
• m
• s
• )
• ,
• wherein
• A
• m
• h
• =
• (
• 1
• 0
• 1
• 1
• )
• ⊗
• m
• h
• ,
• and wherein Q⊗m denotes the m-times Kronecker product of a matrix Q with itself.

A method, comprising:

• decoding for decoding a codeword c of length n using a C(n, k, d) code, wherein the code C(n, k, d) has a length n and a minimum distance d, wherein n=2m1+ . . . +2ms, wherein mh, h=1, . . . , s, is an integer number, wherein:
• c=uA,
• wherein ui=xji, 0≤ji<k−1, wherein x is data of dimension k, if i∉F, wherein F is a set of n−k frozen bit indices of the code C(n, k, d) and ui=Σs=0i−1ϕi,sus, if i∈F, wherein V is a constraint matrix given by a solution of the equation:
• T=0,
• wherein ϕi is an index of a row of the matrix , which has a last non-zero element in column i, wherein is a precoding matrix, and wherein A is defined as:
• A
• =
• (
• A
• m
• 1
• 0
• …
• 0
• 0
• A
• m
• 2
• …
• 0
• ⋮
• ⋮
• ⋱
• ⋮
• 0
• 0
• …
• A
• m
• s
• )
• ,
• wherein
• A
• m
• h
• =
• (
• 1
• 0
• 1
• 1
• )
• ⊗
• m
• h
• ,
• and wherein Q⊗m denotes the m-times Kronecker product of a matrix Q with itself.