Pseudocyclic association schemes and strongly regular graphs

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated symmetric design is trivial. We present several non-amorphous examples, which are either cyclotomic association schemes, or their fusion schemes. Special properties of symmetric designs guarantee the existence of further fusions, and the two known non-amorphous association schemes of class 4 discovered by van Dam and by the authors, are recovered in this way. We also give another pseudocyclic non-amorphous association scheme of class 7 on GF(221), and a new pseudocyclic amorphous association scheme of class 5 on GF(212).