On quantum and LCD Codes Over The Ring $$F_q+vF_q+v^2F_q$$

Ali, Shakir; Mohammad, Ghulam; Jeelani, Mohammad; Khan, Noor Mohammad; Sharma, Pushpendra

doi:10.1007/s11128-022-03654-y

Cited by 3 publications

(1 citation statement)

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“…A characterization of a cyclic code over a finite field to be an LCD code was discovered by Yang and Massey [77] in 1994. Later this criterion worked as a foundation for many researchers to characterize cyclic and constacylic LCD codes over various chain and non-chain rings [2,6,44,80]. Hermitian LCD codes from cyclic codes over a finite field have been studied in [50].…”

Section: Introductionmentioning

confidence: 99%

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

Rai,

Gupta,

Singh

2023

Preprint

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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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