Construction of quantum codes from $(γ,Δ)$-cyclic codes (2404.01904v1)

Abstract: Let $\mathbb{F}q$ be the finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(\gamma,\Delta)$-cyclic codes over a class of finite commutative non-chain rings $\mathscr{R}{q,s}=\mathbb{F}q[v_1,v_2,\dots,v_s]/\langle v_i-v_i^2,v_iv_j=v_jv_i=0\rangle$ where $\gamma$ is an automorphism of $\mathscr{R}{q,s}$, $\Delta$ is a $\gamma$-derivation of $\mathscr{R}{q,s}$ and $1\leq i\neq j\leq s$ for a positive integer $s$. Here, we show that a $(\gamma,\Delta)$-cyclic code of length $n$ over $\mathscr{R}{q,s}$ is the direct sum of $(\theta,\Im)$-cyclic codes of length $n$ over $\mathbb{F}_q$, where $\theta$ is an automorphism of $\mathbb{F}_q$ and $\Im$ is a $\theta$-derivation of $\mathbb{F}_q$. Further, necessary and sufficient conditions for both $(\gamma,\Delta)$-cyclic and $(\theta,\Im)$-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.