Quasimultiples of geometric designs
type: Item
instance of: scholarly article
title: Quasimultiples of geometric designs
zbMATH Open document ID: 1059.51006
published in: Discrete Mathematics
publication date: 2004-08-20T00:00:00Z
review text: A block design \(D= (P,B)\) is a collection \(B\) of \(k\)-subsets (called blocks) of a \(v\)-set \(P\) such that every \(t\)-subset of \(P\) is contained in exactly \(\lambda\) blocks, for some positive constant \(\lambda\). Such a design is often referred to as a \(t\)-\((v, k,\lambda)\) design. With this notation a \(t\)-\((v, k,s\lambda)\) design is called an \(s\)-fold quasimultiple of a \(t\)-\((v,k,\lambda)\) design, for any integer \(s\geq 2\). Quasimultiple projective and affine planes (that is \(2\)-\((n^2+ n+ 1,n +1,\lambda)\) and \(2\)-\((n^2,n,\lambda)\) designs, respectively) have received considerable attention in recent years and it is conjectured that quasidouble affine planes exist for all values of \(n\). In this paper, quasimultiples of affine planes, inversive planes, and unitals from geometrical configurations in various finite geometries are constructed. All designs have \(k= \lambda.\) These designs are simple (no repeated blocks), and all appear to be irreducible in the sense that a quasimultiple of type \(P\) will have no subdesigns isomorphic to \(P\). This is verified for small orders by computer computations.
Mathematics Subject Classification ID: 51E05
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 2093773
zbMATH Keywords: affine planes
zbMATH Keywords: quasimultiples
zbMATH Keywords: inversive planes
zbMATH Keywords: unitals
author: Gary L. Ebert
reviewed by: Giorgio Faina
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: j.disc.2003.11.028
OpenAlex ID: W2119726200
cites work: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe
cites work: Regulus-free spreads of PG\((3,q)\)
cites work: Two-dimensional flag-transitive planes revisited
cites work: Singer line orbits in \(\text{PG} (3,q)\)
cites work: Unitals and codes
cites work: Some unitals on 28 points and their embeddings in projective planes of order 9
cites work: A unital in the Hughes plane of order nine
cites work: The construction of translation planes from projective spaces
cites work: Existence of unitals in finite translation planes of order \(q^2\) with a kernel of order \(q\)
cites work: Q3692865
cites work: Q5545191
cites work: A method for constructing quasimultiple affine planes of arbitrary order
cites work: Q4221862
cites work: Complete systems of lines on a Hermitian surface over a finite field
cites work: Q3694152
cites work: Q3998727
cites work: Quasimultiples of projective and affine planes
cites work: On the existence of small quasimultiples of affine and projective planes of arbitrary order
cites work: On a class of unitals
cites work: Sets of type \((m,n)\) in the affine and projective planes of order nine
cites work: Unitäre Polaritäten endlicher projektiver Ebenen
cites work: Unital designs in planes of order 16
cites work: A class of non-embeddable designs
DOI: 10.1016/J.DISC.2003.11.028
description: scientific article; zbMATH DE number 2093773