Constructing few-weight linear codes and strongly regular graphs

Linear codes with few weights have applications in data storage systems, secret sharing schemes, graph theory and so on. In this paper, we construct a class of few-weight linear codes by choosing defining sets from cyclotomic classes and we also establish few-weight linear codes by employing weakly regular bent functions. Notably, we get some codes that are minimal and we also obtain a class of two-weight optimal punctured codes with respect to the Griesmer bound. Finally, we get a class of strongly regular graphs with new parameters by using the obtained two-weight linear codes.     

Constructions of several classes of linear codes with a few weights

Recently, linear codes with a few weights have been constructed and extensively studied due to their applications in secret sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, we construct several classes of linear codes with a few weights over ${\mathbb{F}}_p$, where $p$ is an odd prime. The weight distributions of these constructed codes are also settled by applications of the theory of quadratic forms and Gauss sums over finite fields. Some of the linear codes obtained are optimal or almost optimal. The parameters of these linear codes are new in most cases. Moreover, two classes of MDS codes are obtained.     

The weight distributions of two classes of p-ary cyclic codes with few weights

Cyclic codes have attracted a lot of research interest for decades as they have efficient encoding and decoding algorithms. In this paper, for an odd prime p, we investigate two classes of p-ary cyclic codes for special cases and determine their weight distributions explicitly. The results show that both codes have at most five nonzero weights.     

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.     

Several classes of linear codes with few weights from the closed butterfly structure

Linear codes with few weights have applications in data storage systems, secret sharing schemes and authentication codes. In this paper, inspired by the butterfly structure [6], [29] and the works of Li, Yue and Fu [21] and Jian, Lin and Feng [19], we introduce a new defining set with the form of the closed butterfly structure and consequently we obtain three classes of 3-weight binary linear codes and a class of 4-weight binary linear codes whose dual is optimal. The lengths and weight distributions of these four classes of linear codes are completely determined by some detailed calculations on certain exponential sums. Computer experiments show that many (almost) optimal codes can be obtained from our construction.     

Complete weight enumerators of a class of linear codes with two or three weights

We construct a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. The results show that they have at most three weights and they are suitable for applications in secret sharing schemes. This is an extension of the results raised by Wang et al. (2017).     

A construction of weakly and non-weakly regular bent functions

In this article a technique for constructing p-ary bent functions from near-bent functions is presented. This technique is then used to obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.     

A class of binary cyclic codes with generalized Niho exponents

In this paper, a class of binary cyclic codes with three generalized Niho-type nonzeros is introduced. Based on some techniques in solving certain equations over finite fields, the proposed cyclic codes are shown to have six nonzero weights and the weight distribution is also completely determined.     

Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

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.