Cyclic codes are an interesting family of linear codes since they have efficient decoding algorithms and contain optimal codes as subfamilies. Constructing infinite families of cyclic codes with good parameters is important in both theory and practice. Recently, Tang and Ding [IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] proposed an infinite family of binary cyclic codes with good parameters. Shi et al. [arXiv:2309.12003v1, 2023] developed the binary Tang-Ding codes to the $4$-ary case. Inspired by these two works, we study $2^s$-ary Tang-Ding codes, where $s\geq 2$. Good lower bounds on the minimum distance of the $2^s$-ary Tang-Ding codes are presented. As a by-product, an infinite family of $2^s$-ary duadic codes with a square-root like lower bound is presented.

翻译：循环码是一类有趣的线性码，因为它们具有高效的译码算法，并且包含最优码作为子族。构造具有良好参数的无限循环码族在理论和实践中都十分重要。近期，Tang和Ding [IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842--7849, 2022] 提出了一个具有良好参数的无限二元循环码族。Shi等人 [arXiv:2309.12003v1, 2023] 将二元Tang-Ding码推广到了四元情形。受这两项工作的启发，我们研究了 $2^s$ 元Tang-Ding码，其中 $s\geq 2$。给出了 $2^s$ 元Tang-Ding码最小距离的良好下界。作为副产品，提出了一个具有平方根类下界的无限 $2^s$ 元对偶码族。