Short minimal codes and covering codes via strong blocking sets in projective spaces

TL;DR

This paper establishes tighter bounds on the minimal length of linear minimal codes over finite fields by leveraging properties of strong blocking sets in projective spaces, advancing understanding of code and covering set sizes.

Contribution

It introduces improved bounds on minimal code length and covering codes using geometric and probabilistic methods, refining previous superlinear estimates.

Findings

Proves an upper bound on minimal code length linear in q and k.

Improves bounds on higgledy-piggledy line sets in projective spaces.

Provides enhanced bounds on covering codes and saturating sets.

Abstract

Minimal linear codes are in one-to-one correspondence with special types of blocking sets of projective spaces over a finite field, which are called strong or cutting blocking sets. In this paper we prove an upper bound on the minimal length of minimal codes of dimension $k$ over the $q$-element Galois field which is linear in both $q$ and $k$, hence improve the previous superlinear bounds. This result determines the minimal length up to a small constant factor. We also improve the lower and upper bounds on the size of so called higgledy-piggledy line sets in projective spaces and apply these results to present improved bounds on the size of covering codes and saturating sets in projective spaces as well. The contributions rely on geometric and probabilistic arguments.