Linear codes with few weights from non-weakly regular plateaued functions

TL;DR

This paper constructs new linear codes with few weights from non-weakly regular plateaued functions using two generic methods, determines their weight distributions, and explores their applications in secret sharing schemes.

Contribution

It addresses open problems by constructing and analyzing linear codes from non-weakly regular plateaued functions via two generic constructions, expanding the class of codes with known weight distributions.

Findings

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

Determined the weight distributions of the constructed codes.

Obtained some optimal and almost optimal linear codes and secret sharing schemes.

Abstract

Linear codes with few weights have significant applications in secret sharing schemes, authentication codes, association schemes, and strongly regular graphs. There are a number of methods to construct linear codes, one of which is based on functions. Furthermore, two generic constructions of linear codes from functions called the first and the second generic constructions, have aroused the research interest of scholars. Recently, in \cite{Nian}, Li and Mesnager proposed two open problems: Based on the first and the second generic constructions, respectively, construct linear codes from non-weakly regular plateaued functions and determine their weight distributions. Motivated by these two open problems, in this paper, firstly, based on the first generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions and determine the weight distributions of the constructed codes. Next, based on the second generic construction, we construct some three-weight and at most five-weight linear codes from non-weakly regular plateaued functions belonging to $\mathcal{NWRF}$ (defined in this paper) and determine the weight distributions of the constructed codes. We also give the punctured codes of these codes obtained based on the second generic construction and determine their weight distributions. Meanwhile, we obtain some optimal and almost optimal linear codes. Besides, by the Ashikhmin-Barg condition, we have that the constructed codes are minimal for almost all cases and obtain some secret sharing schemes with nice access structures based on their dual codes.