Polar Codes with Higher-Order Memory

We introduce a construction of a set of code sequences C n ( m ) : n ≥ 1, m ≥ 1 with memory order m and code length N ( n ). C n ( m ) is a generalization of polar codes presented by Arıkan in [1], where the encoder mapping with length N ( n ) is obtained recursively from the encoder mappings with lengths N ( n − 1) and N ( n − m ), and C n ( m ) coincides with the original polar codes when m = 1. We show that C n ( m ) achieves the symmetric capacity I ( W ) of an arbitrary binary-input, discrete-output memoryless channel W for any fixed m . We also obtain an upper bound on the probability of block-decoding error P e of C n ( m ) and show that P_e = O(2^ - N^β) is achievable for β < 1/[1+ m ( ϕ − 1)], where ϕ ∈ (1, 2] is the largest real root of the polynomial F ( m , ρ ) = ρ m − ρ m − 1 − 1. The encoding and decoding complexities of C n ( m ) decrease with increasing m , which proves the existence of new polar coding schemes that have lower complexity than Arıkan’s construction.