BCH codes are important error correction codes, widely utilized due to their robust algebraic structure, multi-error correcting capability, and efficient decoding algorithms. Despite their practical importance and extensive study, their parameters, including dimension, minimum distance and Bose distance, remain largely unknown in general. This paper addresses this challenge by investigating the dimension and Bose distance of BCH codes of length $(q^m - 1)/λ$ over the finite field $\mathbb{F}_q$, where $λ$ is a positive divisor of $q - 1$. Specifically, for narrow-sense BCH codes of this length with $m \geq 4$, we derive explicit formulas for their dimension for designed distance $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ + 1$. We also provide explicit formulas for their Bose distance in the range $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ$. These ranges for $δ$ are notably larger than the previously known results for this class of BCH codes. Furthermore, we extend these findings to determine the dimension and Bose distance for certain non-narrow-sense BCH codes of the same length. Several optimal linear codes can be obtained from these BCH codes.

翻译：BCH 码是一类重要的纠错码，因其具有坚实的代数结构、多差错纠正能力以及高效的译码算法而被广泛应用。尽管 BCH 码具有重要的实用价值并得到了广泛研究，但其参数，包括维数、最小距离和 Bose 距离，在一般情况下大多仍是未知的。本文通过研究有限域 $\mathbb{F}_q$ 上长度为 $(q^m - 1)/λ$ 的 BCH 码的维数与 Bose 距离来应对这一挑战，其中 $λ$ 是 $q - 1$ 的一个正因子。具体而言，对于此类长度且 $m \geq 4$ 的狭义 BCH 码，我们推导了其设计距离 $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ + 1$ 范围内维数的显式公式。我们还给出了其在 $2 \leq δ\leq (q^{\lfloor (2m - 1)/3 \rfloor + 1} - 1)/λ$ 范围内 Bose 距离的显式公式。这些 $δ$ 的取值范围明显大于此前已知的关于此类 BCH 码的结果。此外，我们将这些发现推广到确定相同长度的某些非狭义 BCH 码的维数与 Bose 距离。从这些 BCH 码中可以获得若干最优线性码。