On complete caps in the projective geometries over \(\mathbb F_3\)
type: Item
instance of: scholarly article
title: On complete caps in the projective geometries over \(\mathbb F_3\)
zbMATH Open document ID: 0971.51005
DOI: 10.1007/BF01220305
author: Yanyan Li
published in: Journal of Geometry
publication date: 2001-05-01T00:00:00Z
review text: Let \(\text{PG}(n,q)\) be the \(n\)-dimensional projective space over \(\text{GF}(q)\). A \(K\)-cap in \(\text{PG}(n,q)\) is a set of \(K\) points such that no three of them are collinear. A \(K\)-cap is said to be complete if it is maximal with respect to set-theoretic inclusion. The largest size of a cap in \(\text{PG}(5,3)\) is \(56\), but very little is known about smaller caps. In the paper under review, using both geometric and coding theoretic techniques, the authors construct complete \(48\)-caps and show that any \(53\)-cap is extendable to a \(56\)-cap. From the construction they derive new upper bounds on the largest size of a cap in \(\text{PG}(r,3)\), \(r\geq 6\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51E22
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 1443430
zbMATH Keywords: complete cap
zbMATH Keywords: linear code
reviewed by: Antonio Cossidente
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01220305
OpenAlex ID: W1986145264
description: scientific article; zbMATH DE number 1443430