Cyclic codes over a semi-local ring and their applications to QEC and EAQEC codes

Abstract

Let \(R_{q,v}={\mathbb {F}}_q+v{\mathbb {F}}_q+ v^2{\mathbb {F}}_q\) where q is an odd prime power and \(v^3=v\). In this paper, we first provide structures of the Euclidean sums and hulls of cyclic codes of length n over \(R_{q,v}\). Then, we exhibit a method of constructing new quantum error-correcting (abbreviated to QEC) codes via the Euclidean sums of cyclic codes over \(R_{q,v}\) and CSS constructions. Finally, we construct two new classes of entanglement-assisted quantum error-correcting (abbreviated to EAQEC) codes by means of the Euclidean hulls of cyclic codes of length n over \(R_{q,v}\). In addition, to enrich the variety of available QEC and EAQEC codes, many new QEC and EAQEC codes are constructed to illustrate our results.