A geometric characterization of minimal codes and their asymptotic performance

TL;DR

This paper provides a geometric framework for understanding minimal linear codes, establishing bounds on their parameters, demonstrating their asymptotic goodness, and offering new geometric constructions.

Contribution

It introduces a geometric characterization of minimal codes, relates them to cutting blocking sets, and proves their asymptotic goodness with new construction methods.

Findings

Minimal codes are related to cutting blocking sets.

Bounds on length and distance of minimal codes are derived.

Minimal codes are shown to be asymptotically good.

Abstract

In this paper, we give a geometric characterization of minimal linear codes. In particular, we relate minimal linear codes to cutting blocking sets, introduced in a recent paper by Bonini and Borello. Using this characterization, we derive some bounds on the length and the distance of minimal codes, according to their dimension and the underlying field size. Furthermore, we show that the family of minimal codes is asymptotically good. Finally, we provide some geometrical constructions of minimal codes.