Polar Codes' Simplicity, Random Codes' Durability

TL;DR

This paper constructs error correction codes that combine the low complexity of polar codes with the robustness of random codes, achieving near-capacity performance with scalable error probabilities and code rates.

Contribution

The paper introduces a new class of codes that simultaneously match the error probability and rate scaling of random codes and the encoding/decoding complexity of polar codes.

Findings

Codes achieve error probability $ ext{exp}(-N^ extpi)$ and rate $N^{- ho}$ with $O(N extlog N)$ complexity.

Such codes are optimal under the condition $ extpi + 2 ho < 1$.

No codes with these properties exist if $ extpi + 2 ho > 1$ for generic channels.

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.