Affine geometry designs, polarities, and Hamadaʹs conjecture

In a recent paper, two of the authors used polarities in PG ( 2 d − 1 , p ) ( p ⩾ 2 prime, d ⩾ 2 ) to construct non-geometric designs having the same parameters and the same p-rank as the geometric design PG d ( 2 d , p ) having as blocks the d-subspaces in the projective space PG ( 2 d , p ) , hence providing the first known infinite family of examples where projective geometry designs are not characterized by their p-rank, as it is the case in all known proven cases of Hamadaʹs conjecture. In this paper, the construction based on polarities is extended to produce designs having the same parameters, intersection numbers, and 2-rank as the geometric design AG d + 1 ( 2 d + 1 , 2 ) of the ( d + 1 ) -subspaces in the binary affine geometry AG ( 2 d + 1 , 2 ) . These designs generalize one of the four non-geometric self-orthogonal 3 - ( 32 , 8 , 7 ) designs of 2-rank 16 (V.D. Tonchev, 1986 [12]), and provide the only known infinite family of examples where affine geometry designs are not characterized by their rank.