Quasi-quadrics and related structures

Abstract

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 nondegenerate quadric in that space. Of course, non-degenerate quadrics themselves are examples of quasi-quadrics, but many other examples exist. In the case that n is odd, quasi-quadrics have two sizes of intersections with hyperplanes and so are two-character sets. These sets are known to give rise to strongly regular graphs, two-weight codes, difference 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 distance 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.