Non-symmetric 3-class association schemes

There are 24 feasible parameter sets for a primitive non-symmetric association schemes with 3 classes and at most 100 vertices. Using computer search, we prove non-existence for three feasible parameter sets. Eleven cases are still open. In the imprimitive case, we survey the known results including some constructions of infinite families of schemes. In the smallest case that has been open up to now we construct a new scheme. This scheme is equivalent to a “skew” Bush-type Hadamard matrix of order 36. We also consider directed graphs that satisfy only some of the conditions required for a non-symmetric association scheme with 3 classes. Let X be finite set (|X| = v) and let {R0, R1, . . . , Rd} be a partition of X×X. Then we say that X = (X, {R0, R1, . . . , Rd}) is an association scheme with d classes if the following conditions are satisfied • R0 = {(x, x) | x ∈ X}. • for each i, R i := {(x, y) | (y, x) ∈ Ri} = Ri′, for some i . • for each triple (i, j, h), i, j, h ∈ {0, . . . , d} there exists a so-called intersection number phij such that for all x, y ∈ X with (x, y) ∈ Rh there are exactly phij elements z ∈ X so that (x, z) ∈ Ri and (z, y) ∈ Rj. If i = i for all i then X is said to be symmetric, otherwise it is nonsymmetric. If the graphs R1, . . . , Rd all are connected then we say that X is primitive, otherwise it is imprimitive.