Quasi-quadrics and related structures

In a projective space PG(n, q) a quasi-quadric is a set of points that has the same intersection numbers with respect to hyperplanes as a non­ degenerate quadric in that space. Of course, non-degenerate quadrics themselves are examples of quasi-quadrics, but many other examples ex­ ist. In the case that n is odd, quasi-quadrics have two sizes of inter­ sections with hyperplanes and so are two-character sets. These sets are known to give rise to strongly regular graphs, two-weight codes, differ­ ence sets, SDP-designs, Reed-Muller codes and bent functions. When n is even, quasi-quadrics have three sizes of intersection with respect to hyperplanes. Certain of these may be used to construct antipodal dis­ tance regular covers of complete graphs. The aim of this paper is to draw together many of the known results about quasi-quadrics, as well as to provide some new geometric construction methods and theorems.