1. Short minimal codes and covering codes via strong blocking sets in projective spaces
- Author
-
Héger, Tamás and Nagy, Zoltán Lóránt
- Subjects
Mathematics - Combinatorics ,Computer Science - Information Theory - 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., Comment: Minor improvement for higgledy-piggledy line sets in the even order case. The main proof is slightly simplified. Some smaller mistakes are corrected
- Published
- 2021