The construction of self-dual codes over small fields such that their minimum distances are as large as possible is a long-standing challenging problem in the coding theory. In 2009, a family of binary self-dual cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i}$, $n_i$ goes to the infinity for $i=1,2, \ldots$, was constructed. In this paper, we construct a family of (repeated-root) binary self-dual cyclic codes with lengths $n$ and minimum distances at least $\sqrt{n}-2$. New families of lengths $n=q^m-1$, $m=3, 5, \ldots$, self-dual codes over ${\bf F}_q$, $q \equiv 1$ $mod$ $4$, with their minimum distances larger than or equal to $\sqrt{\frac{q}{2}}\sqrt{n}-q$ are also constructed.

翻译：在小域上构造具有尽可能大最小距离的自对偶码是编码理论中长期存在的挑战性问题。2009年，人们构造了一族长度为$n_i$、最小距离$d_i \geq \frac{1}{2} \sqrt{n_i}$（其中$n_i$趋向无穷，$i=1,2, \ldots$）的二元自对偶循环码。本文构造了一族（重根）二元自对偶循环码，其长度$n$的最小距离至少为$\sqrt{n}-2$。此外，还构造了在域${\bf F}_q$（其中$q \equiv 1$ $mod$ $4$）上的新型长度为$n=q^m-1$（$m=3, 5, \ldots$）的自对偶码族，其最小距离大于等于$\sqrt{\frac{q}{2}}\sqrt{n}-q$。