Polar Codes' Simplicity, Random Codes' Durability

Abstract

Over any discrete memoryless channel, we build codes such that: for one, their block error probabilities and code rates scale like random codes'; and for two, their encoding and decoding complexities scale like polar codes'. Quantitatively, for any constants $\pi ,\rho >0$ such that $\pi +2\rho <1$, we construct a sequence of error correction codes with block length $N$ approaching infinity, block error probability $\exp (-N^{\pi })$, code rate $N^{-\rho }$ less than the Shannon capacity, and encoding and decoding complexity $O(N\log N)$ per code block. The putative codes take uniform $\varsigma$-ary messages for sender's choice of prime $\varsigma$. The putative codes are optimal in the following manner: Should $\pi +2\rho >1$, no such codes exist for generic channels regardless of alphabet and complexity.

55 pages, 8 figures, 2 tables