A primitive non-symmetric 3-class association scheme on 36 elements with \(p_{11}^ 1=0\) exists and is unique
As is shown in an other paper by the same authors [Feasibility conditions for non-symmetric 3-class association schemes, to appear in Discrete Math.], the existence of a primitive non-symmetric 3-class association scheme with the parameters \(v= 36\), \(v_ 1= 7\), \(p^ 1_{11}= 0\) and \(p^ 2_{11}= 4\) is feasible. Assuming that such a scheme should exist, we show that the digraph of one of its anti-symmetric relations can be constructed, only using the intersection numbers. In fact this digraph on 36 vertices is build on the 7 points, the 7 lines and the 21 flags of the Fano-plane and one additional vertex, extending in a natural way the point-line incidence relations of this projective plane. Finally its uniqueness follows directly from its construction.
scientific article; zbMATH DE number 700926