Geometric construction of association schemes from non-degenerate quadrics (Q1594864)

Consider the projective space \({\mathcal S}=\text{PG}(2\nu+ \delta, q)\) over the finite field \(\text{GF}(q)\). Here \(\nu\geq 1\) and \(\delta=0\), 1, or 2. The geometry can be partitioned into the affine geometry \({\mathcal A}=\text{AG} (2\nu+ \delta,q)\) and a hyperplane \({\mathcal H}=\text{PG}(2\mu+ \delta-1,q)\) of dimension \(2\nu+ \delta-1\) at infinity. The authors consider a quadric \(Q\) in \({\mathcal H}\) given by a symmetric \(2\nu+\delta\) by \(2\nu+\delta\) matrix \(S\) whose definition depends on the value of the parameter \(\delta\). A point \(x\) in \({\mathcal H}\) not on \(Q\) is of square type or of nonsquare type if \(xSx^t\) is a square or a nonsquare, respectively, in \(\text{GF}(q)\). The authors then define a three-class association scheme on the points of the affine space \({\mathcal A}\) as follows. Let \(a\) and \(b\) be two points of \({\mathcal A}\). The points are first associates if the line \(ab\) intersects the hyperplane \({\mathcal H}\) in a point of the quadric \(Q\), they are second associates if the line \(ab\) intersects \({\mathcal H}\) in a point of square type, and, finally, they are third associates if the line \(ab\) intersects \({\mathcal H}\) in a point of nonsquare type. The main result of the article is a determination of the parameters of this association scheme.