Channel polarization with higher order memory

We introduce the design of a class of code sequences {e_n^(m), n = 1, 2,... } with memory level m = 1,2,... based on the channel polarization idea, where {e_n⁽¹⁾} coincides with the polar codes presented by Arıkan in [1]. The new codes achieve the symmetric capacity of arbitrary binary-input discrete memoryless channels. We derive bounds on the polarization performance as scaled with m. We show that {e_n^(m)} offers monotonically decreasing encoding and decoding complexity with growing m.