Conference matrices from projective planes of order 9

There exist at least 4 nonisomorphic projective planes of order 9. They determine 1+ 2 + 2 + 2 = 7 nonisomorphic affine planes of order 9, that is, 7 nonisomorphic transversal designs T(10, 9). These yield 7 x (10/5) = 1764 transversal designs T(5,9), each of which defines a strongly regular graph on 81 vertices, and a conference matrix of order 82. We show that these constructions reduce to 26 nonequivalent conference matrices of order 82, which give rise to 175 nonisomorphic strongly regular graphs (81, 40, 19, 20). Our tools are considerations of symmetry, use of a computer, and general results such as the following. For any strongly regular graph (4µ+ 1,2µ, µ - 1, µ) the number of 4-cliques equals the number of 4-cocliques. Section 2 recalls the definitions of a transversal design T(k, n), the corresponding strongly regular graph and, in the case 2k = n + I, the conference matrix of order n2+ 1. Section 3 adds some new results to the general theory of conference matrices (Theorems 3.3 and 3.4). Section 4 explains the tables, which lead to the results mentioned above. Table 1 copies the 7 alline planes of order 9 from [4], Table 2 mentions some useful permutations, and Table 3 lists the 26 nonisomorphic graphs (81, 40, 19, 20). We gratefully acknowledge the collaboration by M. J. van Althuis and the advice by H. A. Wilbrink