76 research outputs found
Minimal linear codes from characteristic functions
Minimal linear codes have interesting applications in secret sharing schemes
and secure two-party computation. This paper uses characteristic functions of
some subsets of to construct minimal linear codes. By properties
of characteristic functions, we can obtain more minimal binary linear codes
from known minimal binary linear codes, which generalizes results of Ding et
al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By
characteristic functions corresponding to some subspaces of , we
obtain many minimal linear codes, which generalizes results of [IEEE Trans.
Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf.
Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic
functions to present a characterization of minimal linear codes from the
defining set method and present a class of minimal linear codes
Several classes of minimal binary linear codes violating the Aschikhmin-Barg\u27s bound
Minimal linear codes are a special class of codes which have important applications in secret sharing and secure two-party computation. These codes are characterized by the property that none of the codewords is covered by some other codeword. Denoting by and minimal and maximal weight of the codewords respectively, such codes are relatively easy to design when the ratio (known as Aschikhmin-Barg\u27s bound). On the other hand, there are few known classes of minimal codes violating this bound, hence having the property . In this article, we provide several explicit classes of minimal binary linear codes violating the Aschikhmin-Barg\u27s bound, at the same time achieving a great variety of the ratio . Our first generic method employs suitable characteristic functions of relatively low weight within the range . The second approach addresses a specification of characteristic functions covering the weights in and containing a skewed (removing one element) affine subspace of dimension . Finally, we also characterize an infinite family of such codes that utilize the class of so-called root Boolean functions of weight , which are useful in certain hardware testing applications. Consequently, many infinite classes of minimal codes crossing the Aschikhmin-Barg\u27s bound, with a wide range of the weight of their characteristic functions, are deduced. In certain cases we also completely specify the weight distribution of resulting codes
Infinite families of minimal binary codes via Krawtchouk polynomials
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, the general Maiorana-McFarland class. We employ a method first proposed by Ding et al. [7] 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 codes. 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. This study provides insights into the scope of this method
A Novel Application of Boolean Functions with High Algebraic Immunity in Minimal Codes
Boolean functions with high algebraic immunity are important cryptographic
primitives in some stream ciphers. In this paper, two methodologies for
constructing binary minimal codes from sets, Boolean functions and vectorial
Boolean functions with high algebraic immunity are proposed. More precisely, a
general construction of new minimal codes using minimal codes contained in
Reed-Muller codes and sets without nonzero low degree annihilators is
presented. The other construction allows us to yield minimal codes from certain
subcodes of Reed-Muller codes and vectorial Boolean functions with high
algebraic immunity. Via these general constructions, infinite families of
minimal binary linear codes of dimension and length less than or equal to
are obtained. In addition, a lower bound on the minimum distance of
the proposed minimal linear codes is established. Conjectures and open problems
are also presented. The results of this paper show that Boolean functions with
high algebraic immunity have nice applications in several fields such as
symmetric cryptography, coding theory and secret sharing schemes
Minimal binary linear codes - a general framework based on bent concatenation
Minimal codes are characterized by the property that none of the codewords is covered by some other linearly independent codeword. We first show that the use of a bent function in the so-called direct sum of Boolean functions , where is arbitrary, induces minimal codes. This approach gives an infinite class of minimal codes of length and dimension (assuming that h: \F_2^n \rightarrow \F_2), whose weight distribution is exactly specified for certain choices of . To increase the dimension of these codes with respect to their length, we introduce the concept of \textit{non-covering permutations} (referring to the property of minimality) used to construct a bent function in variables, which allows us to employ a suitable subspace of derivatives of and generate minimal codes of dimension instead. Their exact weight distribution is also determined. In the second part of this article, we first provide an efficient method (with easily satisfied initial conditions) of generating minimal linear codes that cross the so-called Ashikhmin-Barg bound. This method is further extended for the purpose of generating minimal codes of larger dimension , through the use of suitable derivatives along with the employment of non-covering permutations. To the best of our knowledge, the latter method is the most general framework for designing binary minimal linear codes that violate the Ashikhmin-Barg bound. More precisely, for a suitable choice of derivatives of , where is a bent function and satisfies certain minimality requirements, for any fixed , one can derive a huge class of non-equivalent wide binary linear codes of the same length by varying the permutation when specifying the bent function in the Maiorana-McFarland class. The weight distribution is given explicitly for any (suitable) when is an almost bent permutation
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