Quantum and LCD codes from skew constacyclic codes over a finite non-chain ring

Prakash, Om; Verma, Ram Krishna; Singh, Ashutosh

doi:10.1007/s11128-023-03951-0

Cited by 5 publications

(2 citation statements)

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“…Lemma 15. ( [28], [54], [65]) Let C be skew (θ t , β)-cyclic code of length n over F q such that ord(θ)|n and β ∈ F * q is such that β 2 = 1. If f (y) is the generator polynomial of C such that g(y)f (y) = y n −β.…”

Section: Units Of Tmentioning

confidence: 99%

“…Recently, Bharadwaj et al [14] obtained many new quantum codes from Euclidean dual-containing constacyclic codes over a general non-chain ring [15], studied skew constacyclic codes over F q [u, v]/⟨f (u), g(v), uv−vu⟩ and provide many properties of them. Being inspired by this study, Prakash et al [65] provided methods to construct quantum codes from Euclidean and Hermitian dual-containing skew constacyclic codes over the same ring [15]. They also obtained Euclidean and Hermitian LCD codes and provided many good examples.…”

Section: Introductionmentioning

confidence: 99%

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Quantum and LCD codes from Skew Constacyclic Codes over a General Class of Non-chain Rings

Rai,

Gupta,

Singh

2023

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In this paper, we study skew constacyclic codes over a class of non-chain rings $\mathcal{T}=\mathbb{F}_q[u_1,u_2,\ldots, u_r]/ \langle f_i(u_i), u_iu_j-u_ju_i\rangle_{i,j=1}^r$, where $q=p^m$, $p$ is some odd prime, $m$ is a positive integer and $f_i(u_i)$ are non-constant polynomials which split into distinct linear factors. We discuss the structural properties of skew constacyclic codes over $\mathcal{T}$ 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 $\mathcal{T}$, 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 $\mathbb{F}_q$. MSC Classification: MSC 11T06, MSC 81P70, MSC94B05, MSC 94B15, MSC 94B99

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Section: Units Of Tmentioning

confidence: 99%