Polar Codes With Higher-Order Memory

We introduce the design of a set of code sequences tC n : n ě 1,m ě 1u, with memory order m and code-length N “ Opφq, where φ P p1, 2s is the largest real root of the polynomial equation F pm, ρq “ ρm ́ρm ́1 ́1 and φ is decreasing in m. tC n u is based on the channel polarization idea, where tC n u coincides with the polar codes presented by Arıkan in [1] and can be encoded and decoded with complexity OpN logNq. tC n u achieves the symmetric capacity, IpW q, of an arbitrary binary-input, discrete-output memoryless channel, W , for any fixed m and its encoding and decoding complexities decrease with growing m. We obtain an achievable bound on the probability of block-decoding error, Pe, of tC n u and showed that Pe “ Op2 ́N β q is achievable for β ă φ ́1 1`mpφ ́1q .