The dimension and Bose distance of certain primitive BCH codes

Abstract

BCH codes are a significant class of cyclic codes that play an important role in both theoretical research and practical applications. Their strong error-correcting abilities and efficient encoding and decoding methods make BCH codes widely applicable in various areas, including communication systems, data storage devices, and consumer electronics. Although BCH codes have been extensively studied, the parameters of BCH codes are not known in general. Let $q$ be a prime power and $m$ be a positive integer. Denote by $\mathcal{C}_{\left(q,m,\delta)\right)}$ the narrow-sense primitive BCH code with length $q^m-1$ and designed distance $\delta$. As of now, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are fully understood only for $m \leq 2$. For $m \geq 4$, the dimensions of $\mathcal{C}_{(q,m,\delta)}$ are known only for the range $2 \leq \delta \leq q^{\lfloor (m+1)/2 \rfloor +1}$ and for a limited number of special cases. In this paper, we determined the dimension and Bose distance of $\mathcal{C}_{(q,m,\delta)}$ for $m\geq 4$ and $\delta\in [2, q^{\lfloor ( 2m-1)/{3}\rfloor+1}]. $ Additionally, we have also extended our results to some primitive BCH codes that are not necessarily narrow-sense.