A Four-Class Association Scheme

We show the existence of a four-class association scheme defined on the unordered pairs of distinct points from PG(1, q2), for q⩾4 a power of 2, thereby proving a conjecture of D. de Caen and E. van Dam (Fissioned triangular schemes via the cross-ratio, European J. Combin.22 (2001), 297–301). This is a fusion of certain relations in the fission scheme FT(q2+1) obtained from the triangular association scheme. Combining three relations in the above four-class association scheme yields a strongly regular graph, which we show is isomorphic to one constructed by Brouwer and Wilbrink using hyperbolic solid sections of the parabolic quadric in PG(4, q).