Let k,m be positive integers and F2m be a finite field of order 2m of characteristic 2. The primary goal of this paper is to study the structural properties of cyclic codes over the ring Sk=F2m[v1,v2,…,vk]⟨vi2−αivi,vivj−vjvi⟩, for i,j=1,2,3,…,k, where αi is the non-zero element of F2m. As an application, we obtain better quantum error correcting codes over the ring S1 (for k=1). Moreover, we acquire optimal linear codes with the help of the Gray image of cyclic codes. Finally, we present methods for reversible DNA codes.

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

The key objective of this paper is to study the cyclic codes over mixed alphabets on the structure of FqPQ, where P=Fq[v]⟨v3−α22v⟩ and Q=Fq[u,v]⟨u2−α12,v3−α22v⟩ are nonchain finite rings and αi is in Fq/{0} for i∈{1,2}, where q=pm with m≥1 is a positive integer and p is an odd prime. Moreover, with the applications, we obtain better and new quantum error-correcting (QEC) codes. For another application over the ring P, we obtain several optimal codes with the help of the Gray image of cyclic codes.