# A class of narrow-sense BCH codes over $\mathbb{F}_q$ of length $\frac{q^m-1}{2}$

## TL;DR

This paper characterizes a class of narrow-sense BCH codes over finite fields with length (q^m-1)/2, determining their weight enumerators, minimal distances, and trace representations using association schemes and cyclotomic cosets.

## Contribution

It provides explicit trace representations and weight enumerators for a new class of BCH codes of length (q^m-1)/2, linking cyclotomic coset leaders to code parameters.

## Key findings

- Determined weight enumerators of the codes.
- Proved minimal and Bose distances equal to designed distances.
- Established trace representations for the codes.

## Abstract

BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorics design. This paper studies a class of linear codes of length $ \frac{q^m-1}{2}$ over $\mathbb{F}_q$ with special trace representation, where $q$ is an odd prime power. With the help of the inner distributions of some subsets of association schemes from bilinear forms associated with quadratic forms, we determine the weight enumerators of these codes. From determining some cyclotomic coset leaders $\delta_i$ of cyclotomic cosets modulo $ \frac{q^m-1}{2}$, we prove that narrow-sense BCH codes of length $ \frac{q^m-1}{2}$ with designed distance $\delta_i=\frac{q^m-q^{m-1}}{2}-1-\frac{q^{ \lfloor \frac{m-3}{2} \rfloor+i}-1}{2}$ have the corresponding trace representation, and have the minimal distance $d=\delta_i$ and the Bose distance $d_B=\delta_i$, where $1\leq i\leq \lfloor \frac{m+3}{4} \rfloor$.