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

Abstract

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.

Quantum and LCD codes from Skew Constacyclic Codes over a General Class of Non-chain Rings

“…Quantum Codes from Skew constacyclic codes over Fq[u 1 , u 2 , u 3 ]/⟨u3 1 − u 1 , u 2 2 − u 2 , u 3 3 − 1, u i u j − u j u i ⟩, α = 3 i 3 η i 1 i 2 i 3 α i 1 i 2 i 3 , β = (α 111 , α 112 , .. . , α 323 ) Euclidean and Hermitian LCD skew (Θ, α)-Constacyclic Codes over T and their Gray images This section deals with the construction of Euclidean and Hermitian LCD codes over T from skew constacyclic codes.…”