Parameters of some BCH codes over $ \mathbb{F}_q $ of length $ \frac{q^m-1}{2} $

Abstract

BCH codes, known for their error-correcting capability and efficient encoding and decoding algorithms, are an important class of cyclic codes over finite fields. Currently, BCH codes have been widely applied in consumer devices, communication systems, and data storage systems. However, there is still a lot of unsolved work in exploring the parameters of BCH codes, such as the dimensions and minimum distances. The objective of this paper is to study narrow-sense BCH codes of length $ \frac{q^m-1}{2} $ over $ \mathbb{F}_q $. The parameters of narrow-sense BCH codes of such length with designed distances $ \delta_i = \frac{q^m-q^{m-1}-q^{\lfloor\frac{m-3}2\rfloor+i}-1}{2} $ and $ \delta_{\lfloor\frac{m+6}{4}\rfloor+t} = \frac{q^m-q^{m-1}-q^\frac{3m-1}{4}-q^\frac{m+1}{2}-q^{\frac{m-11}{4}+t}-1}{2} $ are presented, respectively, where $ 1\le i\le \lfloor\frac{m+6}{4}\rfloor $ and $ 2\le t\le 4 $.