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

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 $\gcd (2h+1,q+1,q^m+1)\mathrm{\ne }1$. 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.