Quantum codes from a class of constacyclic codes over finite commutative rings
Quantum codes from a class of constacyclic codes over finite commutative rings
Let p be an odd prime, and kk be an integer such that gcd(k, p) = 1. Using pairwise orthogonal idempotents γ1, γ2, γ3 of the ring ℛ = 𝔽p[u]/〈uk+1−u〉, with γ1 + γ2 + γ3 = 1, ℛ is decomposed as ℛ = γ1ℛ ⊕ γ2ℛ ⊕ γ3ℛ, which contains the ring R = γ1𝔽p ⊕ γ2𝔽p ⊕ γ3𝔽p as a subring. It is shown that, for λ0, λk ∈ 𝔽p, λ0+ukλk ∈ R, and it is invertible if and only if λ0 and λ0 + λkare units of 𝔽p. In such cases, we study (λ0+ukλk)-constacyclic codes over RR. We present a direct sum decomposition of (λ0 + ukλk)-constacyclic codes and their duals, which provides their corresponding generators. Necessary and sufficient conditions for a (λ0 + ukλk)-constacyclic code to contain its dual are obtained. As an application, many new quantum codes over 𝔽p, with better parameters than existing ones, are constructed from cyclic and negacyclic codes over R.