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

Let Fq be a field of order q, where q is power an odd prime p, and α β are two non-zero elements Fq. The primary goal this article to study the structural properties cyclic codes over finite ring R=Fq[u1,u2]/⟨u12−α2,u22−β2,u1u2−u2u1⟩. We decompose R by using orthogonal idempotents Δ1,Δ2,Δ3, Δ4 as R=Δ1R⊕Δ2R⊕Δ3R⊕Δ4R, construct quantum-error-correcting (QEC) R. As application, we some optimal LCD codes.