Minimal Binary Linear Codes from Vectorial Boolean Functions

TL;DR

This paper introduces a novel method for constructing minimal binary linear codes using vectorial Boolean functions, resulting in higher-dimensional codes and new families that violate existing conditions.

Contribution

It presents the first construction of minimal linear codes from vectorial Boolean functions, including new classes and infinite families with higher dimensions.

Findings

Derived necessary and sufficient conditions for minimality of codes from vectorial Boolean functions.

Constructed new three-weight minimal linear codes with known weight distributions.

Identified infinite families of minimal codes that violate the AB condition.

Abstract

Recently, much progress has been made to construct minimal linear codes due to their preference in secret sharing schemes and secure two-party computation. In this paper, we put forward a new method to construct minimal linear codes by using vectorial Boolean functions. Firstly, we give a necessary and sufficient condition for a generic class of linear codes from vectorial Boolean functions to be minimal. Based on that, we derive some new three-weight minimal linear codes and determine their weight distributions. Secondly, we obtain a necessary and sufficient condition for another generic class of linear codes from vectorial Boolean functions to be minimal and to be violated the AB condition. As a result, we get three infinite families of minimal linear codes violating the AB condition. To the best of our knowledge, this is the first time that minimal liner codes are constructed from vectorial Boolean functions. Compared with other known ones, in general the minimal liner codes obtained in this paper have higher dimensions.