Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes over $\F_q$ which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of $\mathbb{F}_q$. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how dual-containing and self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.

翻译：准循环码近年来被用于量子纠错码的构造。本文提出一种在有限域$\F_q$上构造无穷族准循环码的方法，该类码关于欧几里得内积和埃尔米特内积均为自正交的。特别地，通过利用定义在$\mathbb{F}_q$域扩张上的分量码，我们计算了该类码的维数及其最小距离的下界。同时证明该最小距离下界满足类平方根下界，并展示如何从本文构造中导出对偶包含及自对偶准循环码。基于CSS构造，我们证明了具有优良参数的量子纠错码的存在性。