Class Of Binary Linear Codes Research Articles

Several Families of Binary Minimal Linear Codes From Two-to-One Functions

Minimal linear codes have important applications in secure communications, including in the framework of secret sharing schemes and secure multi-party computation. A lot of research have been carried out to derive codes with few weights (but more importantly, being minimal) using algebraic or geometric approaches. One of the main power and fructify algebraic methods is based on the design of those codes by employing functions over finite fields. K. Li, C. Li, T. Helleseth, and L. Qu have recently identified in [Binary linear codes with few weights from two-to-one functions, IEEE Trans. Inf. Theory, 2021] some binary linear codes with few weights from two classes of two-to-one functions. In this paper, our ultimate objective is to expand the class of codes derived from the paper of Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . by proposing larger classes of binary linear codes with few weights via generic constructions involving other known families of two-to-one functions over the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^<i>n</i></sub> of order 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>n</i></sup> . We succeed in constructing such codes, and we also completely determine their weight distributions. The linear codes presented in this paper differ in parameters from those known in the literature. Besides, some of them are optimal concerning the well-known Griesmer bound. Notably, we prove that our codes are either optimal or almost optimal with respect to the online Database of Grassl. We next observe that the derived binary linear codes also have the minimality property for most cases. We then describe the access structures of the secret-sharing schemes based on their dual codes. Finally, we solve two problems left open in the paper by Li <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al</i> . (more specifically, a complete solution to Problem 2 and a partial solution to Problem 1).

Two-weight or three-weight binary linear codes from cyclotomic mappings

Recently, linear codes with few weights have been studied extensively. These linear codes have wide applications in secret sharing schemes and authentication codes. In this paper, we introduce a new construction of defining sets using cyclotomic mappings and construct three new classes of binary linear codes with two or three weights. We also explicitly determine the weight distributions of these codes.

Binary linear codes with few weights from Boolean functions

Boolean functions have very nice applications in coding theory and cryptography. In coding theory, Boolean functions have been used to construct linear codes in different ways. The objective of this paper is to construct binary linear codes with few weights using the defining-set approach. The defining sets of the codes presented in this paper are defined by some special Boolean functions and some additional restrictions. First, two families of binary linear codes with at most three or four weights from Boolean functions with at most three Walsh transform values are constructed and the parameters of their duals are also determined. Then several classes of binary linear codes with explicit weight enumerators are produced. Some of the binary linear codes are optimal or almost optimal according to the tables of best codes known maintained at http://www.codetables.de , and the duals of some of them are distance-optimal with respect to the sphere packing bound.

Weight distributions and weight hierarchies of two classes of binary linear codes

Linear codes with a few weights can be applied to communication, consumer electronics and data storage system. In addition, the weight hierarchy of a linear code has many applications such as on the type II wire-tap channel, dealing with t-resilient functions and trellis or branch complexity of linear codes and so on. In this paper, we present a formula for computing the weight hierarchies of linear codes constructed by the generalized method of defining sets. Then, we construct two classes of binary linear codes with a few weights and determine their weight distributions and weight hierarchies completely. Some codes of them can be used in secret sharing schemes.

Boolean functions with six-valued Walsh spectra and their application

Boolean functions play an important role in coding theory and symmetric cryptography. In this paper, three classes of Boolean functions with six-valued Walsh spectra are presented and their Walsh spectrum distributions are determined. They are derived from three classes of bent functions by complementing the values of the functions at three different points, where the bent functions are the Maiorana-McFarland types, Dillon $\mathcal {PS}_{ap}$ types and the monomial form $T{r^{n}_{1}}(\lambda x^{r(2^{m}-1)})$ , respectively. As an application, we construct some classes of binary linear codes and it turns out that these codes can be used in secret sharing schemes with interesting access structure.

Optimal Binary Linear Codes From Maximal Arcs

The binary Hamming codes with parameters $[{2^{m}-1, 2^{m}-1-m, 3}]$ are perfect. Their extended codes have parameters $[{2^{m}, 2^{m}-1-m, 4}]$ and are distance-optimal. The first objective of this paper is to construct a class of binary linear codes with parameters $[{2^{m+s}+2^{s}-2^{m},2^{m+s}+2^{s}-2^{m}-2m-2,4}]$ , which have better information rates than the class of extended binary Hamming codes, and are also distance-optimal. The second objective is to construct a class of distance-optimal binary codes with parameters $[{2^{m}+2, 2^{m}-2m, 6}]$ . Both classes of binary linear codes have new parameters.

Linear codes from vectorial Boolean power functions

In this paper, three classes of binary linear codes with few weights are proposed from vectorial Boolean power functions, and their weight distributions are completely determined by solving certain equations over finite fields. In particular, a class of simplex codes and a class of first-order Reed-Muller codes can be obtained from our construction by taking the identity map, whose dual codes are Hamming codes and extended Hamming codes, respectively.

Binary LCD Codes and Self-Orthogonal Codes From a Generic Construction

Linear codes with certain special properties have received renewed attention in recent years due to their practical applications. Among them, binary linear complementary dual (LCD) codes play an important role in implementations against side-channel attacks and fault injection attacks. Self-orthogonal codes can be used to construct quantum codes. In this paper, four classes of binary linear codes are constructed via a generic construction which has been intensively investigated in the past decade. Simple characterizations of these linear codes to be LCD or self-orthogonal are presented. Resultantly, infinite families of binary LCD codes and self-orthogonal codes are obtained. Infinite families of binary LCD codes from the duals of these four classes of linear codes are produced. Many LCD codes and self-orthogonal codes obtained in this paper are optimal or almost optimal in the sense that they meet certain bounds on general linear codes. In addition, the weight distributions of two sub-families of the proposed linear codes are established in terms of Krawtchouk polynomials.

At most three-weight binary linear codes from generalized Moisio’s exponential sums

In this paper, we evaluate two classes of new exponential sums, which are viewed as generalized Moisio’s exponential sums in contrast to the results by Heng and Yue (IEEE Commun Lett 19:1488–1491, 2015) and Moisio (Finite Fields Appl 15:644–651, 2009). As applications, we present two classes of binary linear codes with two wights or three weights. These codes can be used in secret sharing schemes and authentication codes.

Several classes of binary linear codes and their weight enumerators

In the past decades, linear codes with a few weights have been extensively studied for their applications in space communication, data storage and cryptography etc. We construct several classes of binary linear codes and determine their weight distributions. Most of these codes can be used in secret sharing schemes.

Binary linear codes from vectorial boolean functions and their weight distribution

Binary linear codes with good parameters have important applications in secret sharing schemes, authentication codes, association schemes, and consumer electronics and communications. In this paper, we construct several classes of binary linear codes from vectorial Boolean functions and determine their parameters, by further studying a generic construction developed by Ding et al. recently. First, by employing perfect nonlinear functions and almost bent functions, we obtain several classes of six-weight linear codes which contain the all-one codeword, and determine their weight distribution. Second, we investigate a subcode of any linear code mentioned above and consider its parameters. When the vectorial Boolean function is a perfect nonlinear function or a Gold function in odd dimension, we can completely determine the weight distribution of this subcode. Besides, our linear codes have larger dimensions than the ones by Ding et al.’s generic construction.

Binary linear codes with two or three weights from niho exponents

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes, date storage systems, strongly regular graphs and some other fields. In this paper, we present several classes of binary linear codes with two or three weights and study their weight distributions. Two classes of strongly regular graphs are constructed from binary linear codes with two weights. Numerical results show that some of the obtained codes are either optimal or near optimal with respect to certain bounds on linear codes.

Non-asymptotic estimates for the probability of correct messages recovering in the binary erasure wiretap channel

We consider an information transmission system which consists of an irredundant source of binary messages, an ideal main channel between the sender and the legitimate receiver, and an erasure wiretap channel between the sender and the adversary. A known method of random messages coding by binary linear codes is used to ensure reliable information security in the wiretap channel. We obtain the exact expression and non-asymptotic estimates for the probability of correct messages recovering in the wiretap channel. This parameter characterizes practical security of the above mentioned systems. Obtained estimates are applicable to a wide class of binary linear codes with known values of length, dimension and minimal (or dual) distance. They allow us to formulate criteria for choosing codes to construct practically secure and effective systems with ran-dom coding in the binary erasure channel and to set suffi-cient condition of asymptotic perfectness of such systems. The results can be applied for cryptographic applications based on the use of the wiretap channel model.

Three classes of linear codes with two or three weights

In this paper, we mainly determine the weight distributions of three classes of linear codes. Firstly, we prove that two classes of ternary linear codes from the following two planar functions have two or three weights: f(x)=x3k+12,x∈F3m, where k,m are odd, gcd(m,k)=1, and f(x)=x3k+1,x∈F3m, where mgcd(m,k) is odd. They are exactly a part of the open problem in Ding and Ding (2015 Section IV). Secondly, we construct a new class of binary linear codes with three weights. In particular, the linear codes in this paper have applications in consumer electronics, communication and secret sharing schemes.

On Nonlinearity of S‐Boxes and Their Related Binary Codes

The nonlinearity of S-boxes and their related supercodes of the first order Reed-Muller code are discussed. Based on the properties of multi-output bent functions and almost bent functions, we determine the maximum size of linear supercodes of the first order Reed-Muller code which have optimal or suboptimal minimum distance, and we also give the weight distributions of these supercodes which achieve the best possible size. Furthermore, an upper bound on the minimum distance of a class of binary linear codes is presented, which yields a new upper bound on the nonlinearity of S-boxes. The new bound on nonlinearity improves a bound given by Carlet et al. in 2007.

A Class of Binary Linear Codes With at Most Three Weights

In this letter, we use Gauss sums of index 2 to construct a class of new binary linear codes with at most three weights. In special cases, they are optimal or near optimal codes. These codes can be used in authentication codes and secret sharing schemes.

Binary Linear Codes With Two Weights

Linear codes have wide applications in secret sharing schemes and authentication codes. In this letter, a class of binary linear codes with two weights is presented. The weight distribution of this class of binary linear codes is settled. The dual codes are also studied and proved to be optimal or almost optimal.

On separability of the classes of homogeneous and transitive perfect binary codes

By the example of perfect binary codes, we prove the existence of binary homogeneous nontransitive codes. Thereby, taking into account previously obtained results, we establish a hierarchical picture of extents of linearity for binary codes; namely, there is a strict inclusion of the class of binary linear codes in the class of binary propelinear codes, which are strictly included in the class of binary transitive codes, which, in turn, are strictly included in the class of binary homogeneous codes. We derive a transitivity criterion for perfect binary codes of rank greater by one than the rank of the Hamming code of the same length.

UNIVERSAL GROBNER BASES FOR BINARY LINEAR CODES

Each linear code can be described by a code ideal given as the sum of a toric ideal and a non-prime ideal. In this way, several concepts from the theory of toric ideals can be translated into the setting of code ideals. It will be shown that after adjusting some of these concepts, the same inclusion relationship between the set of circuits, the universal Gr\"obner basis and the Graver basis holds. Furthermore, in the case of binary linear codes, the universal Gr\"obner basis will consist of all binomials which correspond to codewords that satisfy the Singleton bound and a particular rank condition. This will give rise to a new class of binary linear codes denoted as Singleton codes.

Ring Theoretic Approach to Reversible Codes Based on Circulant Matrices

Recently, Haley and Grant introduced the concept of reversible codes — a class of binary linear codes that can reuse the decoder architecture as the encoder and encodable by the iterative message-passing algorithm based on the Jacobi method over $\\mathbb{F}_2$. They also developed a procedure to construct parity check matrices of a class of reversible codes named type-I reversible codes by utilizing properties specific to circulant matrices. In this paper, we refine a mathematical framework for reversible codes based on circulant matrices through a ring theoretic approach. This approach enables us to clarify the necessary and sufficient condition on which type-I reversible codes exist. Moreover, a systematic procedure to construct all circulant matrices that constitute parity check matrices of type-I reversible codes is also presented.