The existence and construction of a family of block-transitive 2- designs

Let G be a block-transitive automorphism group of a 2- ( v , k , 1 ) design D . It has been shown that the pairs ( G , D ) fall into three classes: those where G is unsolvable and is flag-transitive, those where G is a subgroup of A Γ L ( 1 , q ) , and those where G is solvable and is of small order. Not much is known about the latter two classes. s paper, we investigate the existence of 2- ( v , 6 , 1 ) designs admitting a block-transitive automorphism group G < AGL ( 1 , q ) . Using Weilʹs theorem on character sums, the following theorem is proved: if a prime power q is large enough and q ≡ 31 ( mod 60 ) then there is a 2- ( v , 6 , 1 ) design which has a block-transitive, but nonflag-transitive automorphism group G. Moreover, using computers, some concrete examples are given when q is small.