Cyclic codes are the most studied subclass of linear codes and widely used in data storage and communication systems. Many cyclic codes have optimal parameters or the best parameters known. They are divided into simple-root cyclic codes and repeated-root cyclic codes. Although there are a huge number of references on cyclic codes, few of them are on repeated-root cyclic codes. Hence, repeated-root cyclic codes are rarely studied. There are a few families of distance-optimal repeated-root binary and $p$-ary cyclic codes for odd prime $p$ in the literature. However, it is open whether there exists an infinite family of distance-optimal repeated-root cyclic codes over $\bF_q$ for each even $q \geq 4$. In this paper, three infinite families of distance-optimal repeated-root cyclic codes with minimum distance 3 or 4 are constructed; two other infinite families of repeated-root cyclic codes with minimum distance 3 or 4 are developed; four infinite families of repeated-root cyclic codes with minimum distance 6 or 8 are presented; and two infinite families of repeated-root binary cyclic codes with parameters $[2n, k, d \geq (n-1)/\log_2 n]$, where $n=2^m-1$ and $k \geq n$, are constructed. In addition, 27 repeated-root cyclic codes of length up to $254$ over $\bF_q$ for $q \in \{2, 4, 8\}$ with optimal parameters or best parameters known are obtained in this paper. The results of this paper show that repeated-root cyclic codes could be very attractive and are worth of further investigation.

翻译：循环码是线性码中研究最深入的子类，广泛应用于数据存储与通信系统。许多循环码具有最优参数或已知最佳参数。循环码分为单根循环码和重复根循环码两类。尽管关于循环码的文献浩如烟海，但针对重复根循环码的研究寥寥无几，因此该领域鲜有深入探索。现有文献中仅存在少量针对奇素数$p$的距离最优重复根二元及$p$元循环码族。然而，对于每个偶特征域$\bF_q$（$q \geq 4$）是否存在无穷族距离最优重复根循环码仍是开放性问题。本文构造了三类最小距离为3或4的无穷族距离最优重复根循环码；发展了另外两类最小距离为3或4的无穷族重复根循环码；提出了四类最小距离为6或8的无穷族重复根循环码；并建立了参数为$[2n, k, d \geq (n-1)/\log_2 n]$（其中$n=2^m-1$且$k \geq n$）的两类无穷族二元重复根循环码。此外，本文在域$\bF_q$（$q \in \{2, 4, 8\}$）上得到27个长度不超过254且具有最优参数或已知最佳参数的重复根循环码。研究结果表明，重复根循环码极具研究价值，值得进一步探索。