Two-Weight Codes, Partial Geometries and Steiner Systems

Abstract

Two-weight codes and projective sets having two intersection sizes with hyperplanes 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 investigate these sets; we prove that if q=2 the corresponding strongly regular graphs are switching equivalent and that they contain subconstituents that 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).