Minimal linear codes from weakly regular plateaued balanced functions

Abstract

Minimal linear codes have diverse applications in many areas such as secret sharing schemes and secure two-party computation. There are several construction methods for these codes, one of which is based on functions over finite fields. In this paper, to construct minimal codes with few weights, we make use of weakly regular plateaued balanced functions over Fp, where p is an odd prime, in the second generic construction method. We obtain several three-weight and four-weight minimal codes with desirable parameters from these functions. The weight distributions of the obtained codes are completely determined with the help of the Walsh distributions of these functions. We then derive projective three-weight punctured codes from some obtained codes, by deleting some coordinates of the defining sets. It is worth noting that they may include the (almost) optimal codes. We finally analyze the minimum Hamming distances of the dual codes of our minimal codes for secret sharing schemes and association schemes.