Small stopping sets in finite projective planes of order q

Keywords: Low density parity check codes, projective planes, KM--arcs, stopping sets, linear spaces

Abstract

A configuration C in a (finite) incidence structure is a subset C of blocks. If every point on a block of C belongs to at least one other block of C, then C is called stopping set (or equivalently full configuration). If smin(q) is the minimal size of a stopping set in a finite projective plane of odd order q, then either smin(q) ≥ q + 5 if $3\not\vert q$ or smin(q) ≥ q + 3 if 3|q. In this note, we prove that smin(q) ≥ q + 5 for any odd q ≠ 3. If q = 3, then smin(3) = 6 and a stopping set of minimal size 6 in PG(2, 3) is the dual set of the symmetric difference of two lines. Also, we study stopping sets of size q + 4 in a finite projective plane of order q.

Published
2020-07-16
Section
Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)