# Minimal linear codes from characteristic functions

## TL;DR

This paper introduces a new method using characteristic functions to construct and analyze minimal linear codes, expanding existing classes and providing a characterization of such codes with applications in cryptography.

## Contribution

It presents a novel approach using characteristic functions to generate and characterize minimal linear codes, generalizing previous results and introducing new classes.

## Key findings

- More minimal binary linear codes derived from known codes.
- Construction of many minimal linear codes from subspaces of _q.
- A new characterization of minimal linear codes from the defining set method.

## Abstract

Minimal linear codes have interesting applications in secret sharing schemes and secure two-party computation. This paper uses characteristic functions of some subsets of $\mathbb{F}_q$ to construct minimal linear codes. By properties of characteristic functions, we can obtain more minimal binary linear codes from known minimal binary linear codes, which generalizes results of Ding et al. [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018]. By characteristic functions corresponding to some subspaces of $\mathbb{F}_q$, we obtain many minimal linear codes, which generalizes results of [IEEE Trans. Inf. Theory, vol. 64, no. 10, pp. 6536-6545, 2018] and [IEEE Trans. Inf. Theory, vol. 65, no. 11, pp. 7067-7078, 2019]. Finally, we use characteristic functions to present a characterization of minimal linear codes from the defining set method and present a class of minimal linear codes.