Construction of Quantum Codes over the Class of Commutative Rings and Their Applications to DNA Codes

Abstract

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.