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

TL;DR

This paper determines the minimum distance of antiprimitive BCH codes with designed distance $3$, showing $d=3$ exactly when $\gcd(2h+1,q+1,q^m+1)\ne 1$; for odd $q$ and $m$, it provides a complete $d=4$ characterization via $\gcd(2h+1,q+1)=1$ and derives two infinite families of distance-optimal codes. It further derives code parameters for selected $h$ and analyzes both odd and even $q$ cases to construct distance-optimal instances. The results extend prior work, improve known parameters, and offer practical constructions with strong minimum distance properties for antiprimitive BCH codes.

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.

Theorems & Definitions (32)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• Lemma 2
• Lemma 3
• Theorem 4
• Remark 1
• Corollary 1
• Lemma 4