New constructions of semi-bent functions in polynomial forms

Dong, Deshuai; Qu, Longjiang; Fu, Shaojing; Li, Chao

doi:10.1016/j.mcm.2012.10.014

Cited by 14 publications

(6 citation statements)

References 11 publications

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“…There are a lot of constructions of semibent functions from GF(2 m ) to GF (2). We refer the reader to [16,21,37,54,55,57] for detailed constructions. All semibent functions can be plugged into Corollary 5 to obtain three-weight binary linear codes.…”

Section: Linear Codes From Semibent Functionsmentioning

confidence: 99%

A construction of binary linear codes from Boolean functions

Ding

2016

Discrete Mathematics

145

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Boolean functions have important applications in cryptography and coding theory. Two famous classes of binary codes derived from Boolean functions are the Reed-Muller codes and Kerdock codes. In the past two decades, a lot of progress on the study of applications of Boolean functions in coding theory has been made. Two generic constructions of binary linear codes with Boolean functions have been well investigated in the literature. The objective of this paper is twofold. The first is to provide a survey on recent results, and the other is to propose open problems on one of the two generic constructions of binary linear codes with Boolean functions. These open problems are expected to stimulate further research on binary linear codes from Boolean functions.Its length is q − 1, and its dimension is at most 2m and is equal to 2m in many cases. The dual of C * ( f ) has dimension at least q − 1 − 2m. This is a generic construction of linear codes, which has a long history and its importance is supported by Delsarte's Theorem [24]. It gives a coding-theory characterisation of APN monomials, almost bent functions, and semibent functions (see, for examples, [13], [8] and [44]) when q = 2. We will not deal with this construction in this paper. The second generic construction of linear codes from functionsIn this section, we present the second generic construction of linear codes over GF(p) with any subset D of GF(p m ), and introduce basic results about the linear codes. In Section 5, we will consider specific families of binary linear codes from Boolean functions obtained with this generic construction.3

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Section: Linear Codes From Semibent Functionsmentioning

confidence: 99%

A construction of binary linear codes from Boolean functions

Ding

2016

Discrete Mathematics

145

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“…Example 7: Let m = 7 and let f be a semibent function from GF(2 7 ) to GF(2) with |D f | = 2 7−1 −2 (7−1)/2 = 56. Then the code C D f has parameters [56,7,24], while the optimal binary code has parameters [56,7,26].…”

Section: ) Linear Codes From Bent Functionsmentioning

confidence: 99%

“…We refer the reader to [12], [14], [24], [39], [40], and [42] for detailed constructions. All semibent functions can be plugged into Corollary 11 to obtain three-weight binary linear codes.…”

Section: ) Linear Codes From Bent Functionsmentioning

confidence: 99%

“…Define n = (3 m − 1)/2 and D = α t : Tr 3 m /3 (α t + α t ) = 0, 0 ≤ t ≤ n − 1 , (24) where α is a generator of GF(3 m ) * . Then D is a difference set in (GF(3 m ) * /GF(3) * , ×) with the following parameters Our main result of this section is the following.…”

Section: B a Ternary Casementioning

confidence: 99%

“…Theorem 16: Let h be an odd positive integer and let m = 3h. Let D be defined as in (24). Then the ternary code C D has parameters…”

Section: B a Ternary Casementioning

confidence: 99%

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Linear Codes From Some 2-Designs

Ding

2015

IEEE Trans. Inform. Theory

240

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A classical method of constructing a linear code over GF(q) with a t-design is to use the incidence matrix of the t-design as a generator matrix over GF(q) of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.

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On Semi-bent Functions and Related Plateaued Functions Over the Galois Field $$\mathbb{F}_{2^{n}}$$

Mesnager

2014

Open Problems in Mathematics and Computational Science

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New constructions of semi-bent functions in polynomial forms

Cited by 14 publications

References 11 publications

A construction of binary linear codes from Boolean functions

A construction of binary linear codes from Boolean functions

Linear Codes From Some 2-Designs

On Semi-bent Functions and Related Plateaued Functions Over the Galois Field $$\mathbb{F}_{2^{n}}$$

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