Several classes of minimal linear codes with few weights from weakly regular plateaued functions

TL;DR

This paper constructs new minimal linear codes with few weights from weakly regular plateaued functions over finite fields of odd characteristic, expanding the methods for secure secret sharing applications.

Contribution

It generalizes existing construction methods to weakly regular plateaued functions, producing minimal codes with specific weight distributions and parameters.

Findings

Constructed three-weight linear codes from weakly regular plateaued functions.

Identified subcodes with two or three weights and their parameters.

Proved the minimality of the constructed codes, enabling secure secret sharing schemes.

Abstract

Minimal linear codes have significant applications in secret sharing schemes and secure two-party computation. There are several methods to construct linear codes, one of which is based on functions over finite fields. Recently, many construction methods of linear codes based on functions have been proposed in the literature. In this paper, we generalize the recent construction methods given by Tang et al. in [IEEE Transactions on Information Theory, 62(3), 1166-1176, 2016] to weakly regular plateaued functions over finite fields of odd characteristic. We first construct three weight linear codes from weakly regular plateaued functions based on the second generic construction and determine their weight distributions. We next give a subcode with two or three weights of each constructed code as well as its parameter. We finally show that the constructed codes in this paper are minimal, which confirms that the secret sharing schemes based on their dual codes have the nice access structures.