# Minimal linear codes arising from blocking sets

## TL;DR

This paper introduces new constructions of minimal linear codes using combinatorial and geometric methods, specifically from blocking sets, expanding beyond the Ashikhmin-Barg condition and providing explicit examples.

## Contribution

It generalizes recent minimal linear code constructions by employing blocking sets and geometric techniques, including the concept of cutting blocking sets, to produce codes not satisfying Ashikhmin-Barg's condition.

## Key findings

- Presented a family of codes from blocking sets
- Proved relations between blocking sets and minimal codes
- Provided explicit examples of codes not meeting Ashikhmin-Barg's condition

## Abstract

Minimal linear codes are algebraic objects which gained interest in the last twenty years, due to their link with Massey's secret sharing schemes. In this context, Ashikhmin and Barg provided a useful and a quite easy to handle sufficient condition for a linear code to be minimal, which has been applied in the construction of many minimal linear codes. In this paper, we generalize some recent constructions of minimal linear codes which are not based on Ashikhmin-Barg's condition. More combinatorial and geometric methods are involved in our proofs. In particular, we present a family of codes arising from particular blocking sets, which are well-studied combinatorial objects. In this context, we will need to define cutting blocking sets and to prove some of their relations with other notions in blocking sets' theory. At the end of the paper, we provide one explicit family of cutting blocking sets and related minimal linear codes, showing that infinitely many of its members do not satisfy the Ashikhmin-Barg's condition.