The Exact Parameters of A Family of BCH Codes

Despite the theoretical and practical significance of BCH codes, the exact minimum distance and dimension remain unknown for many families. This paper establishes the precise minimum distance and dimension of narrow-sense BCH codes ${\text{C}}_{(q,m,\lambda ,{\mathrm{\ell }}_0,{\mathrm{\ell }}_1)}$ over $\text{gf}(q)$ of length $\frac{q^m-1}{\lambda }$ and designed distance $\frac{(q-\lambda {\mathrm{\ell }}_0)q^{m-1-{\mathrm{\ell }}_1}-1}{\lambda }$, where $\lambda \mid (q-1)$, $0\le {\mathrm{\ell }}_0<\frac{q-1}{\lambda }$, and $0\le {\mathrm{\ell }}_1\le m-1$. These results conclusively resolve the three open problems posed by Li et al. (IEEE Trans. Inf. Theory, vol. 63, no. 11, pp. 7219-7236, Nov. 2017) while establishing complementary advances to Ding’s seminal framework (IEEE Trans. Inf. Theory, vol. 61, no. 10, pp. 5322-5330, Oct. 2015).