On the (22, 4)-arcs in PG(2, 7) and related codes
type: Item
instance of: scholarly article
title: On the (22, 4)-arcs in PG(2, 7) and related codes
zbMATH Open document ID: 1027.51009
DOI: 10.1016/S0012-365X(02)00812-9
author: Chris P. Love
author: Raymond Hill
published in: Discrete Mathematics
publication date: 2003-06-09T00:00:00Z
review text: An \((n,t)_q\)-arc \(K\) in \(PG(2,q)\) is a set of \(n\) points such that every line of \(PG(2,q)\) meets \(K\) in at most \(t\) points. The largest size \(n\) for an \((n,4)_7\)-arc is 22. The authors classify all \((22,4)_7\)-arcs. They prove that there are exactly three inequivalent \((22,4)_7\)-arcs. These three \((22,4)_7\)-arcs are members of three infinite classes of \(((q^2-q+2)/2,(q+1)/2)_q\)-arcs for \(q\) odd. The infinite classes are constructed by using properties of the internal points of a conic in \(PG(2,q)\), \(q\) odd. The classification of the \((22,4)_7\)-arcs is then used to prove the non-existence of \([23,4,18]_7\)-codes, showing that 24 is the shortest length for an \([n,4,18]_7\)-code.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 1924770
zbMATH Keywords: arcs in projective geometries
zbMATH Keywords: optimal linear codes
reviewed by: Leo Storme
MaRDI profile type: Publication
cites work: Multiple Blocking Sets and Arcs in Finite Planes
cites work: Q4370894
cites work: Q5683869
cites work: Q5340480
cites work: Classification of the \((n,3)\)-arcs in PG\((2,7)\)
description: scientific article; zbMATH DE number 1924770