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

Abstract

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 $\mathbb {F}_q$ be the finite field of size q and $n=q^m-1$, where m is a positive integer. Let $\mathcal C_{(q, m, \delta )}$ be the primitive narrow-sense BCH codes of length n over $\mathbb {F}_q$ with designed distance $\delta$. Denote $s = m - t$, $r = m \bmod s$ and $\lambda = \lfloor t/s \rfloor$. In this paper, we mainly investigate the dimensions and Bose distances of the codes ${\mathcal {C}}_{(q, m, \delta )}$ with designed distance of the following two types:

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

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

This extensively extends the results on Bose distance in Ding et al (IEEE Trans Inf Theory 61(5):2351–2356, 2015). Moreover, the parameters of the hulls of the BCH code ${\mathcal {C}}_{(q, m, q^t)}$ are studied in some cases.