On the Construction of Quantum and LCD Codes from Cyclic Codes over the Finite Commutative Rings

Abstract

Let F-q be a field of order q, where q is a power of an odd prime p, and alpha and beta are two non-zero elements of F-q. The primary goal of this article is to study the structural properties of cyclic codes over a finite ring R = F-q[u(1), u(2)]/(u(1)(2) - alpha(2), u(2)(2) - beta(2), u(1)u(2)- u(2)u(1)). We decompose the ring R by using orthogonal idempotents Delta(1), Delta(2), Delta(3), and Delta(4) as R = Delta R-1 circle plus Delta R-2 circle plus Delta R-3 circle plus Delta R-4, and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.