New non-binary quantum codes from skew constacyclic and additive skew constacyclic codes

Abstract

This paper discusses the structure of skew constacyclic codes and their Hermitian dual over finite commutative non-chain ring $\mathfrak {R}_\ell :=\mathbb {F}_{q^2}[v_1,v_2,\dots ,v_\ell ]/\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\rangle _{1\le i, j\le \ell },$ where q is odd prime power. We also extend our study over mixed alphabet $\mathbb {F}_{q^2}\mathfrak {R}_\ell$ codes. First, we find necessary and sufficient conditions for skew constacyclic codes to contain their duals over $\mathfrak {R}_\ell$ and $\mathbb {F}_{q^2}\mathfrak {R}_\ell$. Then, a Gray map $\Psi : \mathfrak {R_{\ell }} \longrightarrow \mathbb {F}_{q^2}^{2^\ell }$, is defined, and with the help of this map, we also define another Gray map $\Phi :\mathbb {F}_{q^2}\mathfrak {R}_{\ell }\longrightarrow \mathbb {F}_{q^2}^{2^{\ell }+1}$ and prove that both maps are $\mathbb {F}_{q^2}$-linear Hermitian dual preserving. Finally, by applying Hermitian construction on dual-containing skew constacyclic codes, we construct many new quantum codes that improve the best-known parameters.