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South East Asian J of Mathematics and Mathematical Sciences

QUANTUM CODES THROUGH CONSTACYCLIC CODES OVER Fp + μFp + νFp + ϖFp + μνFp + νϖFp + μϖFp + μνϖFp

Rekha Kumawat, Monika Sangwan

Published Dec 30, 2025• 0 citations
Publisher

Abstract

The purpose of this paper is to identify the structural characteristics and construction of quantum codes over $\mathcal{F}_p$ using $\alpha_1+\alpha_2 {\mu}+\alpha_3 {\nu}+\alpha_4 {\varpi}+\alpha_5 {\mu}{\nu}+\alpha_6 {\nu}{\varpi}+\alpha_7 {\mu}{\varpi}+\alpha_8 {\mu}{\nu}{\varpi}- $constacyclic codes over ring $\mathcal{R} ~= ~\mathcal{F}_p[{\mu},{\nu},{\varpi}]/<{\mu}^2-{\mu}, {\nu}^2-{\nu}, {\varpi}^2-{\varpi}~, {\mu}{\nu}-{\nu}{\mu}, {\nu}{\varpi}-{\varpi}{\nu}, {\mu}{\varpi}-{\varpi}{\mu}>$, here $\mathcal F_{p}$ is a finite field with p elements. We define a Gray map from $\mathcal{R}$ to $\mathcal{F}_p^{8}$, a distance-preserving map. Breaking down constacyclic codes into cyclic and negacyclic codes results in the creation of quantum codes over the finite field $\mathcal{F}_p$. As an application, some examples are illustrated to obtain the quantum codes of different parameters.

DOI
10.56827/seajmms.2025.2103.20

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For an odd prime $ p $ and positive integers $ m $ and $ \ell $, let $ \mathbb{F}_{p^m} $ be the finite field with $ p^{m} $ elements and $ 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} $. Thus $ R_{\ell,m} $ is a finite commutative non-chain ring of order $ p^{2^{\ell} m} $ with characteristic $ p $. In this paper, we aim to construct quantum codes from skew constacyclic codes over $ R_{\ell,m} $. 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.