A construction of minimal linear codes from partial difference sets

Abstract

In this paper, we study a class of linear codes defined by characteristic function of some subsets of a finite field. We derive a sufficient and necessary condition for such code to be minimal, and apply partial difference sets to obtain some new minimal linear codes with three or four weights that do not satisfy the Ashikhmin-Barg condition. Based on the constructions of this paper, we show some sporadic examples of minimal linear code not arising from cutting vectorial blocking sets. Our approach is charactertheoretical. The interesting things about these codes are their connections with strongly regular graphs and association schemes, and their applications in secret schemes. In addition, they usually have large automorphism groups, which makes it possible to have a fast decoding algorithm.