New partial geometries constructed from old ones
In [J. Geom. 60, No. 1-2, 85-104 (1997; Zbl 0897.51003)], \textit{R. Mathon} and \textit{A. P. Street} construct 7 new partial geometries with \((s,t, \alpha)=(7,8,4)\), out of one such partial geometry \({\mathcal S}\). The method was to look for a certain substructure (in partial geometries with \(s+1=t= 2\alpha)\) allowing one to define a new partial geometry with the same parameters. These substructures were then found by computer. In the paper under review, the author generalizes the construction by introducing the notion of replaceable spread in finite partial geometries with arbitrary parameters. By considering the model \(PQ^+(7,2)\) of \({\mathcal S}\) using the quadric \(Q^+(7,2)\), he constructs geometrically 3 replaceable spreads (and hence 3 of the 7 new partial geometries). Moreover, he similarly founds 3 replaceable spreads in \(PQ^+ (7,3)\), yielding 3 new partial geometries with parameters \((26, 27, 18)\).
scientific article; zbMATH DE number 1266812