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arXiv:2212.02821 · Discrete Mathematics Algorithms and Applications

New quantum codes from constacyclic codes over a general non-chain ring

Swati Bhardwaj, Mokshi Goyal, Madhu Raka

Published Aug 18, 2023• 11 citations
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Abstract

Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.

DOI
10.1142/s1793830923500726

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Let [Formula: see text] be a prime power and let [Formula: see text] be a finite non-chain ring, where [Formula: see text], are polynomials, not all linear, which split into distinct linear factors over [Formula: see text]. We characterize constacyclic codes over the ring [Formula: see text] and study quantum codes from these. As an application, some new and better quantum codes, as compared to the best known codes, are obtained. We also prove that the choice of the polynomials [Formula: see text], [Formula: see text], is irrelevant while constructing quantum codes from constacyclic codes over [Formula: see text], it depends only on their degrees.
Let [Formula: see text], where p is an odd prime and r is a positive integer. Consider a ring [Formula: see text], where [Formula: see text] with [Formula: see text] and [Formula: see text] with [Formula: see text]. In this paper, we examine the algebraic structure of constacyclic codes over the ring [Formula: see text] of block length [Formula: see text]. Further, a construction of quantum error-correcting codes (QECCs) from constacyclic codes over [Formula: see text] is also given. Moreover, we derive a number of new QECCs.
Abstract In this paper, we study skew constacyclic codes over a class of non-chain rings T=Fq[u1,u2,…,ur]/⟨fi(ui),uiuj−ujui⟩i,j=1r\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^rT=Fq​[u1​,u2​,…,ur​]/⟨fi​(ui​),ui​uj​−uj​ui​⟩i,j=1r​, where q=pmq=p^mq=pm, ppp is some odd prime, mmm is a positive integer and fi(ui)f_i(u_i)fi​(ui​) are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over T\mathcal{T}T and their dual. We characterize (Euclidean and Hermitian) dual-containing skew constacyclic codes. These characterizations serve as a foundational framework for the development of techniques to construct quantum codes. Consequently, we derive plenty of new quantum codes including many Maximum Distance Separable (MDS) quantum codes, and many quantum codes with better parameters than existing ones. Our work further extends to the characterization of skew constacyclic Euclidean and Hermitian Linear Complementary Dual (LCD) codes over T\mathcal{T}T, and we establish that their Gray images also preserve the LCD property. From this analysis, we derive numerous Maximum Distance Separable (MDS) codes and Best-Known Linear Codes (BKLC) over Fq\mathbb{F}_qFq​. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99
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.
In this paper, we considerR=Fq[u,v]/<u2−u,v2−v,uv−vu>R=\mathbb{F}_{q}[u, v]/ < u^{2}-u, v^{2}-v, uv - vu >R=Fq​[u,v]/<u2−u,v2−v,uv−vu>which is a finite non-chain ring, whereqqqis an odd prime power andu2=u,v2=v,uv=vuu^{2}=u, v^{2}=v, uv=vuu2=u,v2=v,uv=vu. We construct new quantum codes from(α+βu+γv+δuv)(\alpha+\beta u+\gamma v+\delta uv)(α+βu+γv+δuv)- constacyclic codes overRRR. We give the structure of(α+βu+γv+δuv)(\alpha+\beta u+\gamma v+\delta uv)(α+βu+γv+δuv)- constacyclic codes overRRRand obtain self-orthogonal codes overFq\mathbb{F}_{q}Fq​by Gray map.