Uncorrectable Errors of Weight Half the Minimum Distance for Binary Linear Codes

TL;DR

This paper derives bounds on the number of uncorrectable errors at half the minimum distance for certain binary linear codes, providing insights into their error correction limits and extending to larger error weights.

Contribution

It introduces new lower bounds on uncorrectable errors for binary linear codes, including Reed-Muller and BCH codes, and generalizes the bounds to larger error weights.

Findings

Bounds asymptotically match upper bounds for Reed-Muller codes

Lower bounds apply to primitive and extended primitive BCH codes

Generalized bounds are weak for large error weights

Abstract

A lower bound on the number of uncorrectable errors of weight half the minimum distance is derived for binary linear codes satisfying some condition. The condition is satisfied by some primitive BCH codes, extended primitive BCH codes, Reed-Muller codes, and random linear codes. The bound asymptotically coincides with the corresponding upper bound for Reed-Muller codes and random linear codes. By generalizing the idea of the lower bound, a lower bound on the number of uncorrectable errors for weights larger than half the minimum distance is also obtained, but the generalized lower bound is weak for large weights. The monotone error structure and its related notion larger half and trial set, which are introduced by Helleseth, Kl{\o}ve, and Levenshtein, are mainly used to derive the bounds.