The Sufficient and Necessary Conditions for the Minimum Distance of the BCH Code <i>C</i>(q,q+1,3,h) to Be 3 and 4

In this paper, the sufficient and necessary conditions for the minimum distance of the BCH code$\mathcal {C}_{(q,q+1,3,h)}$to be 3 and 4 are provided, respectively. Let d be the minimum distance of the BCH code$\mathcal {C}_{(q,q+1,3,h)}$. The following results are proved: 1) for any q,$d=3$if and only if$\gcd (2h+1,q+1)\gt 1$; 2) for q odd,$d=4$if and only if$\gcd (2h+1,q+1)=1$. By combining these conditions with the dimensions of these codes, the parameters of this BCH code are determined completely when q is odd. Moreover, several infinite families of almost maximum distance separable (almost MDS or AMDS for short) codes are derived. Furthermore, a sufficient condition for these almost MDS codes to be distance-optimal and dimension-optimal locally repairable codes is presented. Based on these conditions, several examples are also given.