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International Workshop Algebraic and Combinatorial Coding Theory

Quantum codes from skew constacyclic codes over Fp m + vFp m + v2Fp m

Ram Krishna Verma, Om Prakash, Ashutosh Singh

Published Oct 11, 2020• 4 citations
Publisher

Abstract

For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.

DOI
10.1109/acct51235.2020.9383402

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International Journal of Information and Coding Theory•Jan 1, 2018•1 citation
For an odd prime p and m ≥ 1, let R = Fpm[v]/{v3} = v}. Here our aim is to construct new non-binary quantum codes from skew constacyclic codes over this non-chain ring ℛ. In order to achieve our goal, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring ℛ. Then we determine a necessary and sufficient condition for these codes to contain their Euclidean duals. Finally, by applying CSS construction on these dual containing skew constacyclic codes and using Gray map, we obtain several new quantum codes.
New non-binary quantum codes from skew constacyclic codes over the ring Fpm+vFpm+v2Fpm\mathbb{F}_{p^m}+v\mathbb{F}_{p^m}+v^2 \mathbb{F}_{p^m}Fpm​+vFpm​+v2Fpm​
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
In this article, we construct new non-binary quantum codes from skew constacyclic codes over finite commutative non-chain ring R=Fpm[v]/⟨v3=v⟩\mathcal{R}= \mathbb{F}_{p^m}[v]/\langle v^3 =v \rangleR=Fpm​[v]/⟨v3=v⟩ where ppp is an odd prime and m≥1m \geq 1m≥1. In order to obtain such quantum codes, first we study the structural properties of skew constacyclic codes and their Euclidean duals over the ring R\mathcal{R}R. Then a necessary and sufficient condition for skew constacyclic codes over R\mathcal{R}R to contain their Euclidean duals is established. Finally, with the help of CSS construction and using Gray map, many new non-binary quantum codes are obtained over Fpm\mathbb{F}_{p^m}Fpm​.
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.
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