A Criterion for Decoding on the BSC

2022-02-01

AI Summary

This work investigates the random-error robustness of linear codes over the binary symmetric channel (BSC), focusing on the (list) decoding performance of transitive codes and Reed–Muller (RM) codes. We introduce a universal decodability criterion based on the second moment of Krawtchouk polynomials of the dual code, establishing for the first time a quantitative link between this moment and BSC decoding/list-decoding capability. Leveraging this criterion, we derive tight upper bounds on the weight distribution of transitive codes and prove information-theoretic optimality of RM codes—specifically, optimal rate–list-size trade-offs—in certain parameter regimes. Our approach integrates Krawtchouk analysis, dual-weight estimation, and information-theoretic bound derivation. The resulting criterion applies broadly to classical linear code families and achieves optimal performance in multiple asymptotic regimes, significantly extending the theoretical decoding limits of algebraic codes under random noise.

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.

Problem

• Decoding resilience in binary symmetric channels
• Weight distribution bounds for transitive linear codes
• Optimal list-decoding for Reed-Muller codes

Innovation

• Uses Krawtchouk polynomials analysis
• Bounds weight for transitive codes
• Optimizes Reed-Muller code decoding