Classification of minimal 1-saturating sets in $PG(2,q)$, $q\leq 23$

TL;DR

This paper classifies all minimal 1-saturating sets in projective planes PG(2,q) for q ≤ 23, using exhaustive computer search, with implications for coding theory and combinatorics.

Contribution

It provides the first complete classifications for PG(2,9) and PG(2,11), and minimal sets of smallest size for PG(2,q) with 16 ≤ q ≤ 23, advancing understanding of these structures.

Findings

Complete classification for PG(2,9) and PG(2,11)

Minimal 1-saturating sets identified for 16 ≤ q ≤ 23

Computer-based exhaustive search exploiting projective equivalence

Abstract

Minimal 1-saturating sets in the projective plane $PG(2,q)$ are considered. They correspond to covering codes which can be applied to many branches of combinatorics and information theory, as data compression, compression with distortion, broadcasting in interconnection network, write-once memory or steganography (see \cite{Coh} and \cite{BF2008}). The full classification of all the minimal 1-saturating sets in PG(2,9) and PG(2,11) and the classification of minimal 1-saturating sets of smallest size in PG(2,q), $16\leq q\leq 23$ are given. These results have been found using a computer-based exhaustive search that exploits projective equivalence properties.