One factors and the existence of affine designs
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.
scientific article; zbMATH DE number 429132
scientific article