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

Abstract

Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of two-weight and three-weight linear codes are presented and their weight distributions are determined using Weil sums. Some of the linear codes obtained are optimal or almost optimal with respect to the Griesmer bound.

Complete Weight Enumerators of a Class of Linear Codes From Weil Sums

“…If c = 1, by Theorem 4, the code C D1 is a [24,4,12] linear code. Its weight enumerator is 1 + 24z 12 + 56z 18 , and its complete weight enumerator is…”

“…This is an extension for the work in [12], [18], [20], [23]. When α = t = 1, the weight distribution of C D0 is determined in [12]. In this paper, by employing Weil sums, we shall illustrate explicitly the complete weight enumerators of C Dc for all c ∈ F p .…”

“…Let D = {d 1 The set D is called the defining set. Many classes of linear codes were produced in this way, such as [1], [12], [15], [18], [19], [23], [27], [28]. Particularly, Kong et al [18] constructed some linear codes with two or three weights and presented their complete weight enumerators, by choosing D = {x ∈ F * q : Tr(ax p u +1 ) = c}, where F * q = F q \{0}, u is a positive integer, a ∈ F * q and c ∈ F p .…”

“…If we take a = 1 and c = 0, it is the case considered in [23]. In [12], [20], the authors investigated several classes of linear codes, which can be viewed as a generalization of cyclic codes whose duals have two zeros.…”

“…The punctured code C D0 has parameters [26,4,20] and weight enumerator 1 + 520z 20 + 104z 25 [20,4,12] linear code with weight enumerator 1+60z 12 + 20z 18 , and complete weight enumerator…”