Cyclic codes are an interesting type of linear codes and have wide applications in communication and storage systems due to their efficient encoding and decoding algorithms. Inspired by the recent work on binary cyclic codes published in IEEE Trans. Inf. Theory, vol. 68, no. 12, pp. 7842-7849, 2022, and the arXiv paper arXiv:2301.06446, the objectives of this paper are the construction and analyses of four families of ternary cyclic codes with length $n=3^m-1$ for odd $m$ and dimension $k \in \{n/2, (n + 2)/2\}$ whose minimum distances have a square-root-like lower bound. Their duals have parameters $[n, k^\perp, d^\perp]$, where $k^\perp \in \{n/2, (n- 2)/2\}$ and $d^\perp$ also has a square-root-like lower bound. These families of codes and their duals contain distance-optimal cyclic codes.

翻译：循环码是一类有趣的线性码，因其高效的编码与译码算法而在通信和存储系统中具有广泛应用。受近期发表在《IEEE Trans. Inf. Theory》第68卷第12期第7842-7849页（2022年）关于二进制循环码的研究以及arXiv论文arXiv:2301.06446的启发，本文旨在构造并分析四类三元循环码，其长度为$n=3^m-1$（其中$m$为奇数），维数$k \in \{n/2, (n + 2)/2\}$，且最小距离具有类平方根下界。它们的对偶码的参数为$[n, k^\perp, d^\perp]$，其中$k^\perp \in \{n/2, (n- 2)/2\}$，且$d^\perp$同样具有类平方根下界。这些码族及其对偶码包含距离最优的循环码。