Binary self-dual cyclic codes have been studied since the classical work of Sloane and Thompson published in IEEE Trans. Inf. Theory, vol. 29, 1983. Twenty five years later, an infinite family of binary self-dual cyclic codes with lengths $n_i$ and minimum distances $d_i \geq \frac{1}{2} \sqrt{n_i+2}$ was presented in a paper of IEEE Trans. Inf. Theory, vol. 55, 2009. However, no infinite family of Euclidean self-dual binary cyclic codes whose minimum distances have the square-root lower bound and no infinite family of Euclidean self-dual nonbinary cyclic codes whose minimum distances have a lower bound better than the square-root lower bound are known in the literature. In this paper, an infinite family of Euclidean self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a square-root-like lower bound is constructed. An infinite subfamily of this family consists of self-dual binary cyclic codes with the square-root lower bound. Another infinite subfamily of this family consists of self-dual cyclic codes over the fields ${\bf F}_{2^s}$ with a lower bound better than the square-root bound for $s \geq 2$. Consequently, two breakthroughs in coding theory are made in this paper. An infinite family of self-dual binary cyclic codes with a square-root-like lower bound is also presented in this paper. An infinite family of Hermitian self-dual cyclic codes over the fields ${\bf F}_{2^{2s}}$ with a square-root-like lower bound and an infinite family of Euclidean self-dual linear codes over ${\bf F}_{q}$ with $q \equiv 1 \pmod{4}$ with a square-root-like lower bound are also constructed in this paper.

翻译：自Sloane与Thompson于1983年在IEEE Trans. Inf. Theory第29卷发表经典论文以来，二元自对偶循环码的研究持续开展。二十五年后，IEEE Trans. Inf. Theory第55卷（2009年）的一篇论文提出了一个具有长度$n_i$且最小距离满足$d_i \geq \frac{1}{2} \sqrt{n_i+2}$的二元自对偶循环码无限族。然而，现有文献中尚未发现最小距离具有平方根下界的欧几里得自对偶二元循环码无限族，也未发现最小距离下界优于平方根下界的欧几里得自对偶非二元循环码无限族。本文构造了域${\bf F}_{2^s}$上具有平方根类下界的欧几里得自对偶循环码无限族。该族的一个无限子族由具有平方根下界的自对偶二元循环码构成；另一个无限子族由域${\bf F}_{2^s}$（$s \geq 2$）上具有优于平方根下界的自对偶循环码构成。由此，本文实现了编码理论的两项突破。文中还提出了一个具有平方根类下界的自对偶二元循环码无限族，并构造了域${\bf F}_{2^{2s}}$上具有平方根类下界的Hermitian自对偶循环码无限族，以及域${\bf F}_{q}$（$q \equiv 1 \pmod{4}$）上具有平方根类下界的欧几里得自对偶线性码无限族。