Construction of quantum codes from (γ,Δ)-cyclic codes

Let 𝔽_q be the finite field of q=p^m elements where p is a prime and m is a positive integer. This paper considers (γ,Δ)-cyclic codes over a class of finite commutative non-chain rings ℛ_q,s=𝔽_q[v_1,v_2,…,v_s]/⟨ v_i-v_i^2,v_iv_j=v_jv_i=0⟩ where γ is an automorphism of ℛ_q,s, Δ is a γ-derivation of ℛ_q,s and 1≤ i≠ j≤ s for a positive integer s. Here, we show that a (γ,Δ)-cyclic code of length n over ℛ_q,s is the direct sum of (θ,)-cyclic codes of length n over 𝔽_q, where θ is an automorphism of 𝔽_q and is a θ-derivation of 𝔽_q. Further, necessary and sufficient conditions for both (γ,Δ)-cyclic and (θ,)-cyclic codes to contain their Euclidean duals are established. Finally, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. The obtained codes have better parameters than those available in the literature.