  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.
