Overlarge sets and partial geometries
The starting point of this very interesting paper is the construction of two partial geometries with parameters (7,8,4) (i.e., 7+1 points per line, 8+1 lines per point, 4 lines per point meet a given opposite line) using a certain type of overlarge sets of Fano planes (an overlarge set of Fano planes is a set of 8 Fano planes, each one based on a different 7-set of a fixed 8-set, and such that every triple of the 8-set is a block of a unique Fano plane). One of these partial geometries was known before (it is related to the alternating group \(Alt_9)\), but the second is a new one. Analysing the relation between these two partial geometries, the authors find a general construction method of new partial geometries out of old ones using the notion of a regular spread. Let us, for simplicity, explain this procedure for partial geometries with parameters (7,8,4) (but it works as well in general for parameters \((2t-1,2t,t))\). Let \({\mathcal S}\) be a spread of such a partial geometry \(\Gamma\). It contains 15 lines, and it is said to be regular if the 120 remaining lines come in 15 classes of 8 non-intersecting elements each, such that all the lines of each class meet the same 8 elements of the spread. If we delete the 15 lines of \({\mathcal S}\) and add 15 new points, one for each class of lines (and we declare such a new point to be incident only with each line of the corresponding class), then the main result of the paper under review says that this new geometry is a partial geometry with parameters (8,7,4). Taking the dual, one can try to repeat the procedure. Starting with the known partial geometry mentioned above, the authors construct in this way seven new partial geometries (and nine new strongly regular graphs). This construction outclasses the other result of the paper: in the last section the authors explain a new construction of an eightfold cover of the complete graph on sixteen vertices, also using overlarge sets.
scientific article; zbMATH DE number 1106605
scientific article