A family of cyclic codes with increasing dimensions and fixed minimum distances

Cyclic codes are widely employed in storage devices, communication systems, and consumer electronics. Their superiority lies in their clear algebraic structure, efficient coding, and decoding algorithms. BCH codes are a significant subclass of cyclic codes. We study a family of cyclic codes of length $ n $ over the finite field $ \mathbb{F}_q $ with $ n\mid (q+1) $. The dimensions of this family of cyclic codes gradually increase, whereas their minimum distances are kept constant at $ \frac{n}{2} $. This family of cyclic codes generates a family of non-narrow-sense BCH codes. The dual codes of non-narrow-sense BCH codes are both AMDS codes, distance-optimal and dimension-optimal locally repairable codes. Consequently, we determine the parameters of narrow-sense BCH codes with large designed distances and their dual codes. Meanwhile, we also give the parameters of the extended code of a classical family of linear complementary dual MDS codes.