Regulus-free spreads of PG\((3,q)\)
type: Item
instance of: scholarly article
title: Regulus-free spreads of PG\((3,q)\)
zbMATH Open document ID: 0872.51003
DOI: 10.1007/BF00130569
published in: Designs, Codes and Cryptography
publication date: 1997-05-29T00:00:00Z
review text: A spread in \(PG(3,q)\) is a set of mutually disjoint lines which cover the space. This amounts exactly to what we call a spread on a 4-dimensional vector space over \(F= GF(q)\): A set of 2-dimensional subspaces which are disjoint except for the zero vector which cover the vector space. Then there is a translation plane whose lines are the translates of the components of this vector space spread. In a similar way a regulus corresponds to a derivable set defined on the vector space. An affine plane (of order \(q^2)\) which contains a derivable set can be derived to form a new plane. A translation plane is non-derivable if it contains no derivable sets -- i.e. the spread in \(PG(3,q)\) contains no reguli. There are many known examples of regulus-free spreads (non-derivable planes), particularly if \(q\) is a prime power but not a prime. The authors show that such non-derivable planes exist if \(q\) is an odd prime \(p\) such that \(p\equiv 1\) mod 3. The planes in question are non-Desarguesian flag transitive planes. A flag transitive translation plane is transitive on the points of \(l_\infty\).
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 51A45
zbMATH DE Number: 994655
zbMATH Keywords: derivation
zbMATH Keywords: spread
zbMATH Keywords: translation plane
author: R. D. Baker
author: Gary L. Ebert
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 994655