Two-weight codes, partial geometries and Steiner systems

Two-weight codes and projective sets having two intersection sizes with hyp erplanes are equivalent objects and they define strongly regular graphs. We construct projective sets in PG(2m - 1, q) that have the same intersection numbers with hyperplanes as the hyperbolic quadric Q(+)(2m - 1, q). We inv estigate these sets; we prove that if q = 2 the corresponding strongly regu lar graphs are switching equivalent and that they contain subconstituents t hat are point graphs of partial geometries. If m = 4 the partial geometries have parameters s = 7, t = 8, alpha = 4 and some of them are embeddable in Steiner systems S(2, 8, 120).