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Contemporary Mathematics

Construction of Quantum Codes from Constacyclic Codes Over the Class of Commutative Rings

Pushpendra Sharma, Amal S. Alali, Shakir Ali, M.F. Azeem

Published Aug 19, 2025• 0 citations
Publisher

Abstract

This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.

DOI
10.37256/cm.6420257258

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This paper primarily investigates the structural properties of constacyclic codes over the ring , defined as where i, j = 1, 2, 3, ..., k, and k, m are positive integers. Here, denotes a finite field of order pm with characteristic p, an odd prime. Furthermore, we determine the necessary and sufficient conditions for the duals of constacyclic codes to exist. These findings make it easier to create new Quantum Error-Correcting (QEC) codes throughout the ring (i.e., when k = 1), as well as optimal linear codes that make use of the Gray images of constacyclic codes. Additionally, Table 4 presents several Linear Complementary Dual (LCD) codes obtained using the Gray map.
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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
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Consider the ring Be=Fq+uFq+⋯+ue−1Fq, ue=u (e≥2), where Fq denotes the finite field having q=pm elements (for m≥1 and a prime p), and q≡1(mode−1). Skew constacyclic codes over Be are studied in this paper. We present their generator polynomial and describe the criteria for their complementary duality. Moreover, we derive criteria for these codes to contain their dual and obtain quantum codes. Additionally, we establish a Gray map that preserves duality and investigate its properties. We also take into account additive skew constacyclic codes for this purpose and also derive criteria for the complementary duality of these codes. Finally, we provide several LCD and quantum codes (MDS/ near MDS). The latter are compared with the quantum codes obtained in the recent literature.
Quantum codes from constacyclic codes over SkS_{k}Sk​
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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.