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A Criterion for Decoding on the BSC

Anup Rao, Oscar Sprumont

TL;DR

The paper develops a Fourier-analytic framework to study decoding resilience of linear codes on the Binary Symmetric Channel, focusing on transitive and Reed-Muller codes. It establishes a tight weight-distribution bound for transitive codes, a decoding criterion tied to the dual code and Krawtchouk polynomials, and new list-decoding bounds for transitive and doubly transitive codes, with some regimes achieving information-theoretic optimal rate/list-size trade-offs. The approach unifies symmetry-based weight bounds with dual-weight analyses, enabling unique-decoding and list-decoding guarantees under stochastic noise and connecting decoding performance to concentration on Krawtchouk polynomials. The results have implications for constructing deterministic, highly symmetric codes that approach capacity on the BSC and for understanding the role of transitivity in error resilience, including applications to Reed-Muller codes where doubly transitive structure is leveraged. Overall, the work broadens the class of codes known to admit provable decoding guarantees under random errors and clarifies the interplay between symmetry, dual weight distributions, and decoding performance.

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 $C \subseteq \mathbb{F}_2^N$: $\Pr_{c \in C}[|c| = αN] \leq 2^{-(1-h(α)) \mathsf{dim}(C)}$. 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 $\mathbb{E}_{c \in C^{\perp}}[ K_{εN}(|c|)^2]$ imply that $C$ recovers from errors on the binary symmetric channel with parameter $ε$. 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 $ε$-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.

A Criterion for Decoding on the BSC

TL;DR

The paper develops a Fourier-analytic framework to study decoding resilience of linear codes on the Binary Symmetric Channel, focusing on transitive and Reed-Muller codes. It establishes a tight weight-distribution bound for transitive codes, a decoding criterion tied to the dual code and Krawtchouk polynomials, and new list-decoding bounds for transitive and doubly transitive codes, with some regimes achieving information-theoretic optimal rate/list-size trade-offs. The approach unifies symmetry-based weight bounds with dual-weight analyses, enabling unique-decoding and list-decoding guarantees under stochastic noise and connecting decoding performance to concentration on Krawtchouk polynomials. The results have implications for constructing deterministic, highly symmetric codes that approach capacity on the BSC and for understanding the role of transitivity in error resilience, including applications to Reed-Muller codes where doubly transitive structure is leveraged. Overall, the work broadens the class of codes known to admit provable decoding guarantees under random errors and clarifies the interplay between symmetry, dual weight distributions, and decoding performance.

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 : . 2) We give a criterion that certifies that a linear code can be decoded on the binary symmetric channel. Let denote the Krawtchouk polynomial of degree , and let denote the dual code of . We show that bounds on imply that recovers from errors on the binary symmetric channel with parameter . 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 is sufficiently close to the binomial distribution in some interval around , is resilient to -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.
Paper Structure (26 sections, 26 theorems, 212 equations, 1 figure, 2 tables)

This paper contains 26 sections, 26 theorems, 212 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Techniques
  3. Related work
  4. Outline of the paper
  5. Notation, conventions and preliminaries
  6. Coding theory definitions and terminology
  7. Linear codes
  8. Reed-muller codes
  9. Entropy
  10. Probability distributions
  11. Probability theory
  12. Fourier analysis
  13. Collisions vs decoding
  14. A criterion for decoding
  15. The weight distribution of transitive linear codes
  16. ...and 11 more sections

Key Result

Theorem 1.1

Consider any finite field $\mathbb{F}_q$, and let $C\subseteq \mathbb{F}_q^N$ be any transitive linear code. Then for any $\alpha\in (0,1)$, we have where $\mathcal{D}(C)$ is the uniform distribution over all codewords in $C$, $\textnormal{wt}(c)$ is the number of non-zero coordinates of $c$, and $h_q$ is the q-ary entropy

Figures (1)

  • Figure 1: Organization of our paper and connections between our results.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6: Proposition 1.4 in samorodnitsky2020weightimproved
  • Proposition 1.7
  • Theorem 1.8: kallai1995krawtchouk1ismail1998krawtchouk2polyanskiy2019krawtchouk3
  • Theorem 1.9: Lemma 2.2 and equation 2.10 in polyanskiy2019krawtchouk3
  • Lemma 2.1
  • ...and 43 more