New quantum codes from self-orthogonal cyclic codes over ${\mathbb {F}}_{q^{2}}[u]/\langle u^k \rangle$

Abstract

In this paper, we propose a construction of q-ary quantum codes from Hermitian self-orthogonal cyclic codes over the finite chain ring $R={\mathbb {F}}_{q^{2}}[u]/\langle u^{k}\rangle$, with $u^{k}=0$, where ${\mathbb {F}}_{q^{2}}$ is a finite field with $q^2$ elements, $q=p^m$, p a prime. A Gray map from R to ${\mathbb {F}}^{k}_{q^{2}}$ is defined. Some characterizations of cyclic codes over R have been given in terms of their different types of generators. The structure of their dual codes has also been determined, and a necessary and sufficient condition for these codes to be self-orthogonal is presented. The construction of quantum codes is derived by applying Hermitian construction to the Gray images of self-orthogonal cyclic codes over R. From this construction, we have been able to obtain some new quantum codes with better parameters than some presently known comparable best codes available in the literature. Some of these codes have been given as examples, and the others have been given in the form of two tables.