The minimum distance of the antiprimitive BCH code with designed distance 3

TL;DR

This paper characterizes the minimum distance of antiprimitive BCH codes with designed distance 3, providing conditions for when the minimum distance is 3 or 4, and constructs optimal and near-optimal codes.

Contribution

It offers a complete characterization of the minimum distance for these BCH codes and introduces new families of optimal and best-known linear codes.

Findings

Minimum distance equals 3 if and only if gcd condition holds.

When q and m are odd, the paper determines when the minimum distance is 4.

Two infinite families of distance-optimal codes are constructed.

Abstract

Let $\mathcal{C}_{(q,q^m+1,3,h)}$ denote the antiprimitive BCH code with designed distance 3. In this paper, we demonstrate that the minimum distance $d$ of $\mathcal{C}_{(q,q^m+1,3,h)}$ equals 3 if and only if $\gcd(2h+1,q+1,q^m+1)\ne1$. When both $q$ and $m$ are odd, we determine the sufficient and necessary condition for $d=4$ and fully characterize the minimum distance in this case. Based on these conditions, we investigate the parameters of $\mathcal{C}_{(q,q^m+1,3,h)}$ for certain $h$. Additionally, two infinite families of distance-optimal codes and several linear codes with the best known parameters are presented.