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 $\pi,\rho&gt;0$ such that $\pi+2\rho&lt;1$, we construct a sequence of error correction codes with block length $N$ approaching infinity, block error probability $\exp(-N<sup>\pi)$, code rate $N<sup>{-\rho}$ 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 $\pi+2\rho&gt;1$, no such codes exist for generic channels regardless of alphabet and complexity.