Euclidean and Hermitian LCD MDS codes

Carlet, Claude; Mesnager, Sihem; Tang, Chunming; Qi, Yanfeng

doi:10.1007/s10623-018-0463-8

Cited by 106 publications

(87 citation statements)

References 31 publications

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“…LCD codes also can be used for constructions of lattices [17]. Optimal and MDS codes that are LCD are studied in many papers (see [2], [8], [10], [12], [16], [20], [22], [23], [27], [19]).…”

Section: Introductionmentioning

confidence: 99%

Galois hulls of linear codes over finite fields

Liu

Pan

2019

Des. Codes Cryptogr.

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The ℓ-Galois hull h ℓ (C) of an [n, k] linear code C over a finite field F q is the intersection of C and C ⊥ ℓ , where C ⊥ ℓ denotes the ℓ-Galois dual of C which introduced by Fan and Zhang (2017). The ℓ-Galois LCD code is a linear code C with h ℓ (C) = 0. In this paper, we show that the dimension of the ℓ-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the ℓ-Galois hull by the generator matrix of the code. Moreover, we obtain that the dimension of the ℓ-Galois hulls of ternary codes are also invariant under monomial equivalence. We show that every [n, k] linear code over F q is monomial equivalent to an ℓ-Galois LCD code for any q > 4. We conclude that if there exists an [n, k] linear code over F q for any q > 4, then there exists an ℓ-Galois LCD code with the same parameters for any 0 ≤ ℓ ≤ e − 1, where q = p e for some prime p. As an application, we characterize the ℓ-Galois hull of matrix product codes over finite fields.

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Section: Introductionmentioning

confidence: 99%

Galois hulls of linear codes over finite fields

Liu

Pan

2019

Des. Codes Cryptogr.

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“…The following two lemmas characterize Euclidean and Hermitian LCD codes, as shown in [6,23]. Let O n and j n denote the all zeros and all ones column vectors of length n. The following simple result will be crucial so we state and prove it explicitly here.…”

Section: Euclidean Lcd Codes From Weighing Matricesmentioning

confidence: 99%

LCD codes from weighing matrices

Crnković

Egan

Rodrigues

et al. 2019

AAECC

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Linear codes with complementary duals are linear codes whose intersection with their duals are trivial, shortly named LCD codes. In this paper we outline a construction for LCD codes over finite fields of order q using weighing matrices and their orbit matrices. The LCD codes constructed can be of any length dimension according to the choice of matrices used in their construction. As a special case, LCD codes of length 2n and dimension n are constructed which also have the property of being formally self-dual. Alternatively, under a condition depending on q that the codes are not LCD, this method constructs self-dual codes. To illustrate the method we construct LCD codes from weighing matrices, including the Paley conference matrices and Hadamard matrices. We also extend the construction to Hermitian LCD codes over the finite field of order 4. In addition, we propose a decoding algorithm that can be feasible for the LCD codes obtained from some of the given methods.2010 Mathematics Subject Classification: 05B20, 05B30, 94B05, 12E20.

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“…Jin showed that some Reed-Solomon codes are equivalent to LCD codes [16]. In [6], the authors proved that any MDS code is equivalent to an LCD code and constructed LCD Maximum distance Separable codes. Jitman et.…”

Section: Introductionmentioning

confidence: 99%

Do non-free LCD codes over finite commutative Frobenius rings exist?

Bhowmick

Fotue-Tabue

Martı́nez-Moro

et al. 2020

Des. Codes Cryptogr.

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In this paper, we clarify some aspects on LCD codes in the literature. We first prove that a non-free LCD code does not exist over finite commutative Frobenius local rings. We then obtain a necessary and sufficient condition for the existence of LCD code over finite commutative Frobenius rings. We later show that a free constacyclic code over finite chain ring is LCD if and only if it is reversible, and also provide a necessary and sufficient condition for a constacyclic code to be reversible over finite chain rings. We illustrate the minimum Lee-distance of LCD codes over some finite commutative chain rings and demonstrate the results with examples. We also got some new optimal 4 codes of different lengths which are cyclic LCD codes over 4 .

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Euclidean and Hermitian LCD MDS codes

Cited by 106 publications

References 31 publications

Galois hulls of linear codes over finite fields

Galois hulls of linear codes over finite fields

LCD codes from weighing matrices

Do non-free LCD codes over finite commutative Frobenius rings exist?

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