Several classes of minimal binary linear codes violating the Ashikhmin-Barg bound

Pašalić, Enes; Rodriguez, R. S.; Zhang, Fengrong; Wei, Yongzhuang

doi:10.1007/s12095-021-00491-1

Cited by 6 publications

(4 citation statements)

References 29 publications

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“…AB condition is a sufficient but not necessary condition for a linear code to be minimal. Until now, many infinite families of minimal linear codes violating the AB condition have been found, see for instance [3,9,14,23,30,32,46,47,48]. The following result gives a necessary and sufficient condition for a binary linear code to be minimal.…”

Section: Minimal Linear Codesmentioning

confidence: 95%

“…It seems difficult to get the exact numbers of ν ∈ F n 2 in (37) such that W A μ (ν) = 0, 2 t , −2 t , 2 t+1 , −2 t+1 , respectively. If those numbers are obtained, then by Propositions 2 and 4, Relations (31), (32) and (36), the weight distribution of C F can also be obtained.…”

Section: The Case N Even and 2 < M ≤ T +mentioning

confidence: 99%

“…Thereafter, more families of minimal linear codes violating the AB condition were constructed by investigating the linear code of the form (1). For instance, some minimal linear codes violating the AB condition were obtained from cutting blocking sets [3,32]; from partial difference sets [38]; from characteristic functions [30]; from weakly regular plateaued/bent functions [31,43]; from Partial Spreads [41]; from Maiorana-McFarland functions [42,47]; and from the direct sum of Boolean functions [48], etc.…”

Section: Introductionmentioning

confidence: 99%

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Minimal Binary Linear Codes from Vectorial Boolean Functions

Li¹,

Peng²,

Kan³

et al. 2022

Preprint

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Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, we obtain a necessary and sufficient condition for another generic class of linear codes from vectorial Boolean functions to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.

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Section: Minimal Linear Codesmentioning

confidence: 95%

Section: The Case N Even and 2 < M ≤ T +mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Minimal Binary Linear Codes from Vectorial Boolean Functions

Li¹,

Peng²,

Kan³

et al. 2022

Preprint

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“…Very recently, with a similar idea as in [7], Du et al [8] constructed two classes of wide minimal codes, and determined their weight distributions. Currently, there are many more constructions of infinite families of wide minimal codes based on a large range of techniques and mathematical objects, see for instance [3,4,11,15,17,[19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

Infinite families of minimal binary codes via Krawtchouk polynomials

Du,

Rodríguez,

2024

Des. Codes Cryptogr.

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Linear codes play a crucial role in various fields of engineering and mathematics, including data storage, communication, cryptography, and combinatorics. Minimal linear codes, a subset of linear codes, are particularly essential for designing effective secret sharing schemes. In this paper, we introduce several classes of minimal binary linear codes by carefully selecting appropriate Boolean functions. These functions belong to a renowned class of Boolean functions, namely, the general Maiorana–McFarland class. We employ a method first proposed by Ding et al. (IEEE Trans Inf Theory 64(10):6536–6545, 2018) to construct minimal codes violating the Ashikhmin–Barg bound (wide minimal codes) by using Krawtchouk polynomials. The lengths, dimensions, and weight distributions of the obtained codes are determined using the Walsh spectrum distribution of the chosen Boolean functions. Our findings demonstrate that a vast majority of the newly constructed codes are wide minimal. Furthermore, our proposed codes exhibit a significantly larger minimum distance, in some cases, compared to some existing similar constructions. Finally, we address this method, based on Krawtchouk polynomials, more generally, and highlight certain generic properties related to it. These general results offer insights into the scope of this approach.

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