标题： 反原始BCH码的最小距离，其设计距离为3 显示英文标题

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

摘要： 设$\mathcal{C}_{(q,q^m+1,3,h)}$表示具有设计距离3的反原始BCH码。本文证明了当且仅当$\gcd(2h+1,q+1,q^m+1)\ne1$时，$\mathcal{C}_{(q,q^m+1,3,h)}$的最小距离$d$等于3。当$q$和$m$都是奇数时，我们确定了$d=4$的充分必要条件，并在此情况下完全刻画了最小距离。 基于这些条件，我们研究了某些$h$的$\mathcal{C}_{(q,q^m+1,3,h)}$参数。 此外，提出了两个距离最优码的无限族和若干个具有已知最佳参数的线性码。 显示英文摘要

摘要： 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.