Affine geometry designs, polarities, and Hamada's conjecture

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In a recent paper, two of the authors used polarities in PG(2d − 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 PGd(2d, p) having as blocks the d-subspaces in the projective space PG(2d, 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 Hamada’s 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 AGd+1(2d + 1, 2) of the (d + 1)-subspaces in the binary affine geometry AG(2d + 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.

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Copyright © 2010 Elsevier Inc. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jcta.2010.06.007

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Journal of Combinatorial Theory, Series A