Cyclic and LCD Codes over a Finite Commutative Semi-local Ring

Prakash, Om; Islam, Habibul; Ghosh, Arindam

doi:10.1007/978-981-19-3898-6_28

Cited by 3 publications

(3 citation statements)

References 31 publications

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“…One of the factorisation of y 6 − 1 ∈ F 27 [y; θ 2 ] is given as: 22 and v 2 (y)u 2 (y) = y 10 − 1, then v 2 (y) = (y + w 2 + 1)(y + w)(y + 2w 2 + w)(y + 2w)(y + 2w + 1) = y 5 + (w + 2)y 4 + (w 2 + 2w)y 3 + 2y be the matrix used Gray map φ : T → F 12 25 , where [72,67,4] code over F 27 . Hence by Theorem 2, there exists a [ [72,62,4]] 27 quantum code which is a new code as per database [9]. Now we conclude this section by enlisting the quantum codes constructed using Theorem 18.…”

Section: Proof 1 Let Us Assume Thatmentioning

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

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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Section: Proof 1 Let Us Assume Thatmentioning

confidence: 99%

“…Hermitian LCD codes from cyclic codes over a finite field have been studied in [50]. LCD codes over various classes of finite commutative rings have been thoroughly studied in [55,56,62,63]. Recently, Boulanour et al [19] provided a criterion for skew constacyclic codes to be LCD.…”

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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“…Initially, these codes were studied using the Euclidean inner product to obtain necessary and sufficient conditions for linear codes to be LCD along with their application. Subsequently, many works on LCD codes considered the Euclidean inner product, we refer ( [9,10,13,17,21,20,24,26]). In 2016, Li [11] considered the Hermitian inner product to obtain necessary and sufficient conditions for cyclic codes to be Hermitian LCD.…”

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confidence: 99%

Some constructions of $l$-Galois LCD codes

Yadav,

Debnath,

Prakash

2025

AMC

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Let Fq be the finite field with q elements where q = p m for some prime p and m > 0. In this article, we provide some constructions of Galois LCD codes over Fq in terms of their generator matrices. Here, we consider the Galois inner product instead of Euclidean or Hermitian inner products. For these constructions, we use the matrix A for which A[σ m−l (A)] t = I, 0 ≤ l ≤ m − 1 where σ is the Frobenius map. We also define the Galois self-dual basis and use it to show the existence of a Euclidean LCD code from a linear code over some finite field. Further, we present an application of Galois LCD codes in a multi-secret sharing scheme. Besides, we obtain several optimal and almost optimal Galois LCD codes to show the importance of our results.

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