利用射影平面中的二次曲线构作结合方案

GEOMETRIC CONSTRUCTION OF ASSOCIATION SCHEMES FROM CONICS IN A PROJECTIVE PLANE

设Ｃ是射影平面Ｈ＝ＰＧ（２，Ｆｑ）中的一条二次曲线．把Ｈ看作射影空间ＰＧ（３，Ｆｑ） 的无穷远超平面，那么Ｈ在ＰＧ（３，Ｆｑ）中的补空间是仿射空间Ｘ＝ＡＧ（３，Ｆｑ）．我们把 Ｈ上的点集划分为２个或３个子集的并．设ａ≠ｂ∈Ｘ．若线ａｂ与Ｈ的交点属于第ｉ个 集合，定义ａ和ｂ属于第ｉ个结合类．我们证明上述构作是结合方案．最后，把Ｈ的某一 点集作为处理集，构作出结合方案．

Let C be a conic in a projective plane H = PG(2,Fq). Considering H as the hyperplane at infinity in a projective space PG(3, Fq), then the affine space complementary to H in PG(3, Fq) is X = AG(3, Fq). The points of H are then partitioned into 2 or 3 subset. Let a ≠ b ∈ X. Define a and b to belong to the i-th associates if the line ab intersects the subset i of H. We prove that the above constructions are association schemes. Finally, taking the set of some points on H as the set of treatment, some association schemes are constructed.