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.
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.
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, where q=pm, p is some odd prime, m is a positive integer and fi(ui) are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over 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, 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. 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.
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.
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.
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⟩ where p is an odd prime and m≥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. Then a necessary and sufficient condition for skew constacyclic codes over 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.
New non-binary quantum codes from skew constacyclic codes over the ring 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⟩ where p is an odd prime and m≥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. Then a necessary and sufficient condition for skew constacyclic codes over 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.