# 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 *s*_{min}(*q*) is the minimal size of a stopping set in a finite projective plane of odd order *q*, then either *s*_{min}(*q*) ≥ *q* + 5 if $3\not\vert q$ or *s*_{min}(*q*) ≥ *q* + 3 if 3|*q*. In this note, we prove that *s*_{min}(*q*) ≥ *q* + 5 for any odd *q* ≠ 3. If *q* = 3, then *s*_{min}(3) = 6 and a stopping set of minimal size 6 in *P**G*(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

2021-08-23

Issue

Section

Bled 2019 – Polytopes, Configurations, and Symmetries (Grünbaum Issue)