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

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 $1\le i\le \lfloor \frac{m+3}{4}\rfloor$.