One factors and the existence of affine designs
type: Item
instance of: scholarly article
title: One factors and the existence of affine designs
zbMATH Open document ID: 0783.05010
DOI: 10.1016/0012-365X(93)90562-8
author: Neal Brand
author: Somporn Sutinuntopas
published in: Discrete Mathematics
publication date: 1993-12-22T00:00:00Z
review text: In his paper ``Quadruple systems over \(\mathbb{Z}_ p\) admitting the affine group'' [Combinatorial theory, Lect. Notes Math. 969, 212-228 (1982; Zbl 0496.05010)], \textit{Egmont Köhler} obtained examples of quadruple systems (with various \(\lambda\)-values) admitting the affine group \(\text{AGL}(1,p)\) as an automorphism group. His method consists of reducing the construction of the desired design to finding a 1-factor of a certain graph; this method of ``Köhler graphs'' has recently been applied for \(\lambda=1\) and \(p \equiv 5 \pmod{12}\) with spectacular success by H. Siemon in a series of papers. The present authors generalize Köhler's graph theoretic approach to the case of the affine group \(\text{AGL}(1,q)\) over an arbitrary finite field \(\text{GF}(q)\), but do not address the much harder problem of actually finding the required 1-factors. Since they do not even include examples, the value of their generalization is not quite clear. One final remark regarding terminology: The author's use of the term ``affine design'' to denote a \(t\)-design invariant under an affine group is quite unfortunate, since usually this term denotes a design with a certain type of parallelism.
reviewed by: Dieter Jungnickel
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05E20
Mathematics Subject Classification ID: 05C70
zbMATH DE Number: 429132
zbMATH Keywords: \(t\)-design
zbMATH Keywords: quadruple systems
zbMATH Keywords: affine group
zbMATH Keywords: automorphism group
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 429132