Quantum codes from a class of constacyclic codes over finite commutative rings

Abstract

© 2021 World Scientific Publishing Company. Let p be an odd prime, and k be an integer such that gcd(k,p) = 1. Using pairwise orthogonal idempotents γ1,γ2,γ3 of the ring R = p[u]/(uk+1 - u), with γ1 + γ2 + γ3 = 1, R is decomposed as R = γR ⊕ γ2 R ⊕ γ3R, which contains the ring R = γ1p ⊕ γ2p ⊕ γ3p as a subring. It is shown that, for λ0,λk p, λ0 + ukλ k R, and it is invertible if and only if λ0 and λ0 + λk are units of p. In such cases, we study (λ0 + ukλ k)-constacyclic codes over R. We present a direct sum decomposition of (λ0 + ukλ k)-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a (λ0 + ukλ k)-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over p, with better parameters than existing ones, are constructed from cyclic and negacyclic codes over R.