Two-weight and three-weight linear codes based on Weil sums

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

Recently, linear codes over finite fields with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. In this paper, for an odd prime p, the complete weight enumerator of a class of p-ary linear codes based on defining sets are determined. Furthermore, from the explicit complete weight enumerator of linear codes, a new class of optimal constant composition codes and several classes of asymptotically optimal systematic authentication codes are obtained.

The paper presents methods for designing functions having many applications in particular to construct linear codes with few weights. The former codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. We firstly provide new secondary constructions of bent functions generalizing the well-known Rothaus' constructions as well as their dual functions. From our generalization, we show that we are able to compute the dual function of a bent function built from Rothaus' construction. Next we present a result leading to a new method for constructing semi-bent functions and few Walsh transform values functions built from bent functions.

As a special type of linear codes, minimal linear codes have important applications in secret sharing. Up to now, only a few infinite families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ were reported in the literature, while vast knowledge exists on the ones with $$w_{\min }/w_{\max }> 1/2$$ . Herein, $$w_{\min }$$ and $$w_{\max }$$ respectively denote the minimum and maximum nonzero Hamming weights in a linear code. Recently, several classes of linear codes with certain properties were constructed by Zhou et al. from a generic construction. The objective of this paper is to obtain four families of minimal binary linear codes with $$w_{\min }/w_{\max }\le 1/2$$ from those linear codes proposed by Zhou et al. The parameters of our minimal linear codes are quite different from known ones. Based on the properties of Krawtchouk polynomials, the weight distributions of all these four families of binary linear codes are established.

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 characterisation of highly nonlinear Boolean functions is presented.

How to construct good linear codes is an important problem in coding theory. This paper considers the construction of linear codes from functions with two variables, presents a class of two-weight and three-weight ternary linear codes and employs the Gauss sums and exponential sums to determine the parameters and weight distribution of these codes. Linear codes with few weights have applications in consumer electronics, communication and date storage systems. Linear codes with two weights have applications in strongly regular graphs and linear codes with three weights can be applied in association schemes.

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.

Several classes of linear codes with few weights derived from Weil sums

In the paper, we explore the simplex and MacDonald codes over the finite ring $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$. Our investigation focuses on the unique properties of these codes, with the particular attention to their weight distributions and Gray images. The weight distribution is a crucial aspect as it provides insights into the error-detection and error-correction capabilities of the codes. Gray images play a significant role in understanding the structure and behavior of these codes. By examining the dual Gray images of simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$, we aim to develop efficient secret sharing schemes. These schemes benefit from the inherent properties of the codes, such as minimal weight and redundancy, which are essential for secure and reliable information sharing. Understanding the access structure of these schemes is vital, as it determines which subsets of participants can reconstruct the secret. Our study draws on various properties to elucidate this access structure, ensuring that the schemes are secure and efficient. Through this comprehensive analysis, we contribute to the field of coding theory by demonstrating how simplex and MacDonald codes over $\mathbb{Z}_{p}\mathcal{R}_{1} \mathcal{R}_{2}$ can be effectively utilized in cryptographic applications, particularly in designing robust and reliable secret sharing mechanisms.

Deep holes are very important in the decoding of generalized RS codes, and deep holes of RS codes have been widely studied, but there are few works on constructing general linear codes based on deep holes. Therefore, we consider constructing binary linear codes by combining deep holes with binary BCH codes. In this article, we consider the 2-error-correcting binary primitive BCH codes and the extended codes to construct new binary linear codes by combining them with deep holes, respectively. Furthermore, three classes of binary linear codes are constructed, and then we determine the parameters and the weight distributions of these new binary linear codes.

The bilateral minimum distance of a binary linear code is the maximum $d$ such that all nonzero codewords have weights between $d$ and $n-d$ . Let $Q\subset \{0,1\}^{n}$ be a binary linear code whose dual has bilateral minimum distance at least $d$ , where $d$ is odd. Roughly speaking, we show that the average $L_\infty $ -distance—and consequently, the $L_{1}$ -distance—between the weight distribution of a random cosets of $Q$ and the binomial distribution decays quickly as the bilateral minimum distance $d$ of the dual of $Q$ increases. For $d = \Theta (1)$ , it decays like $n^{-\Theta (d)}$ . On the other $d=\Theta (n)$ extreme, it decays like and $e^{-\Theta (d)}$ . It follows that, almost all cosets of $Q$ have weight distributions very close to the to the binomial distribution. In particular, we establish the following bounds. If the dual of $Q$ has bilateral minimum distance at least $d=2t+1$ , where $t\geq 1$ is an integer, then the average $\vphantom {\sum ^{R^{R}}} L_\infty $ -distance is at most $\min \{ (e\ln {({n}/{2t})})^{t}({2t}/{n})^{({t}/{2})}, \sqrt {2} e^{-({t}/{10})}\}$ . For the average $L_{1}$ -distance, we conclude the bound $\min \{ (2t+1)(e\ln {({n}/{2t})})^{t} ({2t}/{n})^{({t}/{2})-1}, \sqrt {2}(n+1)e^{-({t}/{10})}\}$ , which gives nontrivial results for $t\geq 3$ . We give applications to the weight distribution of cosets of extended Hadamard codes and extended dual BCH codes. Our argument is based on Fourier analysis, linear programming, and polynomial approximation techniques.

The binary code spanned by the rows of the point by block incidence matrix of a Steiner triple system STS(v) is studied. A sufficient condition for such a code to contain a unique equivalence class of STS(v)'s of maximal rank within the code is proved. The code of the classical Steiner triple system defined by the lines in PG(n-1,2) ( n\geq 3), or AG(n,3) ( n\geq 3) is shown to contain exactly v codewords of weight r = (v-1)/2, hence the system is characterized by its code. In addition, the code of the projective STS(2^{n}-1) is characterized as the unique (up to equivalence) binary linear code with the given parameters and weight distribution. In general, the number of STS(v)'s contained in the code depends on the geometry of the codewords of weight r. It is demonstrated that the ovals and hyperovals of the definingSTS(v) play a crucial role in this geometry. This relation is utilized for the construction of some infinite classes of Steiner triple systems without ovals.

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.

Pseudocyclic association schemes and strongly regular graphs

We introduce and characterize pseudo-triple-sum-sets as a natural extension of triple-sum-sets. We Show that if Ω is the set of coordinate forms of a linear projective code C(n,k) over F=GF(q) and if \(\widehat\Omega\)=F. Ω is a pseudo-triple-sum-set then C can be considered as a three classes association subscheme of the Hamming scheme H(n,q).