On block-transitive designs with affine automorphism group

Detailed necessary and sufficient conditions for a k-subset of AG(d, 3) to generate the block set of a block-transitive t-design with automorphism group AGL(d, 3) are derived for t = 3, 4, 5. Similar necessary conditions are found for the existence of a block-transitive design with automorphism group AGL(d, p) when p is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with k or v − k not exceeding 1000 were for t = 4 and (d, p) = (7, 3), (8, 3), and (3, 7). Examples of block-transitive 4-designs admitting AGL(7, 3) are found for each of the values k = 115, 116, 230, 437, and 552.