Further results on multiple coverings of the farthest-off points

TL;DR

This paper introduces new algebraic methods to construct small multiple covering codes and saturating sets in projective spaces, achieving near-optimal density and classifying minimal sets for small fields.

Contribution

It develops novel algebraic constructions and bounds for multiple covering codes and saturating sets, improving their size and density properties.

Findings

New small $(1,)$-saturating sets in projective spaces

Short $(2,)$-MCF codes with near-optimal density

Classification of minimal and optimal saturating sets in $PG(2,q)$ for small $q$

Abstract

Multiple coverings of the farthest-off points ($(R,\mu)$-MCF codes) and the corresponding $(\rho,\mu)$-saturating sets in projective spaces $PG(N,q)$ are considered. We propose and develop some methods which allow us to obtain new small $(1,\mu)$-saturating sets and short $(2,\mu)$-MCF codes with $\mu$-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly $1+1/cq$, $c\ge1$). In particular, we provide new algebraic constructions and some bounds. Also, we classify minimal and optimal $(1,\mu)$-saturating sets in $PG(2,q)$, $q$ small.