On Bose distance of a class of BCH codes with two types of designed distances

BCH codes are an interesting class of cyclic codes with good error-correcting capability and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Let \(\mathbb {F}_q\) ${\mathbb{F}}_q$ be the finite field of size q and \(n=q^m-1\) $n=q^m-1$ , where m is a positive integer. Let \(\mathcal C_{(q, m, \delta )}\) ${\mathcal{C}}_{(q,m,\delta )}$ be the primitive narrow-sense BCH codes of length n over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ with designed distance \(\delta \) $\delta$ . Denote \(s = m - t\) $s=m-t$ , \(r = m \bmod s\) $r=mmods$ and \(\lambda = \lfloor t/s \rfloor \) $\lambda =\lfloor t/s\rfloor$ . In this paper, we mainly investigate the dimensions and Bose distances of the codes \({\mathcal {C}}_{(q, m, \delta )}\) ${\mathcal{C}}_{(q,m,\delta )}$ with designed distance of the following two types: 1.

\(\delta =q^t+h\) $\delta =q^t+h$ , \(\lceil \frac{m}{2} \rceil \le t < m\) $\lceil \frac{m}{2}\rceil \le t<m$ , \(0 \le h < q^s + \sum \limits _{i = 1}^{\lambda - 1} q^{r + is}\) $0\le h<q^s+\sum _{i=1}^{\lambda -1}{q}^{r+is}$ ;