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

Let (Formula presented.) be a field of order q, where q is a power of an odd prime p, and (Formula presented.) and (Formula presented.) are two non-zero elements of (Formula presented.). The primary goal of this article is to study the structural properties of cyclic codes over a finite ring (Formula presented.). We decompose the ring R by using orthogonal idempotents (Formula presented.), and (Formula presented.) as (Formula presented.), and to construct quantum-error-correcting (QEC) codes over R. As an application, we construct some optimal LCD codes.