Triple-Error-Correcting BCH-Like Codes

TL;DR

This paper introduces a new triple-error-correcting code with a novel zero set and proposes a method to discover additional such codes, expanding the landscape of BCH-like codes.

Contribution

The paper presents a new zero set for triple-error-correcting codes and a method for finding further similar codes, enhancing code construction techniques.

Findings

New zero set for triple-error-correcting codes

A novel method for discovering BCH-like codes

Potential for discovering more such codes

Abstract

The binary primitive triple-error-correcting BCH code is a cyclic code of minimum distance 7 with generator polynomial having zeros $\alpha$, $\alpha^3$ and $\alpha^5$ where $\alpha$ is a primitive root of unity. The zero set of the code is said to be {1,3,5}. In the 1970's Kasami showed that one can construct similar triple-error-correcting codes using zero sets consisting of different triples than the BCH codes. Furthermore, in 2000 Chang et. al. found new triples leading to triple-error-correcting codes. In this paper a new such triple is presented. In addition a new method is presented that may be of interest in finding further such triples.