Two-Weight Codes, Partial Geometries and Steiner Systems

Two-weight codes and projectivesets having two intersection sizes with hyperplanes are equivalentobjects and they define strongly regular graphs. We construct projective sets in PG(2m − 1,q) that have the sameintersection numbers with hyperplanes as the hyperbolic quadricQ+(2m − 1,q). We investigate these sets; we provethat if q = 2 the corresponding strongly regular graphsare switching equivalent and that they contain subconstituentsthat are point graphs of partial geometries. If m = 4the partial geometries have parameters s = 7, t = 8,α = 4 and some of them are embeddable in Steinersystems S(2,8,120).