New quantum codes from skew constacyclic codes

Abstract

<p style= text-indent:20px; >For an odd prime <inline-formula><tex-math id= M1 >\\begin{document}$ p $\\end{document}</tex-math></inline-formula> and positive integers <inline-formula><tex-math id= M2 >\\begin{document}$ m $\\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id= M3 >\\begin{document}$ \\ell $\\end{document}</tex-math></inline-formula>, let <inline-formula><tex-math id= M4 >\\begin{document}$ \\mathbb{F}_{p^m} $\\end{document}</tex-math></inline-formula> be the finite field with <inline-formula><tex-math id= M5 >\\begin{document}$ p^{m} $\\end{document}</tex-math></inline-formula> elements and <inline-formula><tex-math id= M6 >\\begin{document}$ R_{\\ell,m} = \\mathbb{F}_{p^m}[v_1,v_2,\\dots,v_{\\ell}]/\\langle v^{2}_{i}-1, v_{i}v_{j}-v_{j}v_{i}\\rangle_{1\\leq i, j\\leq \\ell} $\\end{document}</tex-math></inline-formula>. Thus <inline-formula><tex-math id= M7 >\\begin{document}$ R_{\\ell,m} $\\end{document}</tex-math></inline-formula> is a finite commutative non-chain ring of order <inline-formula><tex-math id= M8 >\\begin{document}$ p^{2^{\\ell} m} $\\end{document}</tex-math></inline-formula> with characteristic <inline-formula><tex-math id= M9 >\\begin{document}$ p $\\end{document}</tex-math></inline-formula>. In this paper, we aim to construct quantum codes from skew constacyclic codes over <inline-formula><tex-math id= M10 >\\begin{document}$ R_{\\ell,m} $\\end{document}</tex-math></inline-formula>. First, we discuss the structures of skew constacyclic codes and determine their Euclidean dual codes. Then a relation between these codes and their Euclidean duals has been obtained. Finally, with the help of a duality-preserving Gray map and the CSS construction, many MDS and better non-binary quantum codes are obtained as compared to the best-known quantum codes available in the literature.</p>