A Criterion for Decoding on the BSC

TL;DR

This paper introduces a new criterion for decoding linear codes on the binary symmetric channel, utilizing weight distribution bounds and Krawtchouk polynomial estimates to improve decoding and list-decoding results for transitive and Reed-Muller codes.

Contribution

It provides a novel decoding criterion based on weight distribution and Krawtchouk polynomial bounds, advancing decoding capabilities for specific code families.

Findings

Established a tight weight distribution bound for transitive codes

Developed a decoding criterion linking weight distribution of dual codes to error resilience

Achieved near-optimal list-decoding bounds for Reed-Muller codes

Abstract

We present an approach to showing that a linear code is resilient to random errors. We use this approach to obtain decoding results for both transitive codes and Reed-Muller codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular. 1) We give a tight bound on the weight distribution of every transitive linear code $C \subseteq \mathbb{F}_2^N$: $\Pr_{c \in C}[|c| = \alpha N] \leq 2^{-(1-h(\alpha)) \mathsf{dim}(C)}$. 2) We give a criterion that certifies that a linear code $C$ can be decoded on the binary symmetric channel. Let $K_s(x)$ denote the Krawtchouk polynomial of degree $s$, and let $C^\perp$ denote the dual code of $C$. We show that bounds on $\mathbb{E}_{c \in C^{\perp}}[ K_{\epsilon N}(|c|)^2]$ imply that $C$ recovers from errors on the binary symmetric channel with parameter $\epsilon$. Weaker bounds can be used to obtain list-decoding results using similar methods. One consequence of our criterion is that whenever the weight distribution of $C^\perp$ is sufficiently close to the binomial distribution in some interval around $\frac{N}{2}$, $C$ is resilient to $\epsilon$-errors. 3) We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for Reed-Muller codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for Reed-Muller codes achieve the information-theoretic optimal trade-off between rate and list size.