  In this paper, we study a class of linear codes defined by characteristic
functions of certain subsets of a finite field. We derive a sufficient and
necessary condition for such a code to be a minimal linear code by a
character-theoretical approach. We obtain new three-weight or four-weight
minimal linear codes that do not satisfy the Ashikhmin-Barg condition by using
partial difference sets. We show that our construction yields minimal linear
codes that do not arise from cutting vectorial blocking sets, and also discuss
their applications in secret sharing schemes.
