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The construction of self-dual codes over small fields such that their minimumdistances are as large as possible is a long-standing challenging problem inthe coding theory. In 2009, a family of binary self-dual cyclic codes withlengths $n_i$ 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, weconstruct a family of (repeated-root) binary self-dual cyclic codes withlengths $n$ and minimum distances at least $\sqrt{n}-2$. New families oflengths $n=q^m-1$, $m=3, 5, \ldots$, 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.

New Binary Self-Dual Cyclic Codes with Square-Root-Like Minimum Distances