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BCH codes with efficient encoding and decoding algorithms have many applications in communications, cryptography and combinatorial 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 of quadratic forms, we determine the weight enumerators of these codes. 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 $$1\le i\le \lfloor \frac{m+11}{6} \rfloor $$.

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