Parameters of Several Classes of BCH Codes

Abstract

Because of their efficient encoding and decoding algorithms, cyclic codes—an interesting class of linear codes—are widely used in communication systems, storage devices, and consumer electronics. BCH codes form a special class of cyclic codes, and are usually among the best cyclic codes. A subclass of good BCH codes is the narrow-sense primitive BCH codes. However, the dimension and minimum distance of these codes are not known in general. The main objective of this paper is to study the dimension and minimum distances of a subclass of the narrow-sense primitive BCH codes with design distance $\delta =(q-\ell _{0})q^{m-\ell _{1}-1}-1$ for certain pairs $(\ell _{0}, \ell _{1})$ , where $0 \leq \ell _{0} \leq q-2$ and $0 \leq \ell _{1} \leq m-1$ . The parameters of other related classes of BCH codes are also investigated, and some open problems are proposed in this paper.