Block-transitive, point-imprimitive designs with λ = 1

AbstractLet D be a 2-(v,k,1) design with a group G of automorphisms which is transitive on the blocks of D and transitive but imprimitive on the points of D. Delandtsheer and Doyen (1989) proved that v is bounded above by (k−2)2(k+1)14. Carrying on from the work of Cameron and Praeger (1989), we show that if v is equal to this upper bound then v=729 and k=8. Further work of Nickel et al. (1992) has shown that, up to an isomorphism, there are 467 block-transitive, point-imprimitive 2-(729,8,1) designs