Quasi-symmetric 2-(31, 7, 7) designs and a revision of Hamada's conjecture
Journal of Combinatorial Theory Series A
Designs and their codes
Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
Handbook of Combinatorial Designs, Second Edition (Discrete Mathematics and Its Applications)
Polarities, quasi-symmetric designs, and Hamada's conjecture
Designs, Codes and Cryptography
A class of majority logic decodable codes (Corresp.)
IEEE Transactions on Information Theory
Majority logic decoding using combinatorial designs (Corresp.)
IEEE Transactions on Information Theory
Characterizing geometric designs, II
Journal of Combinatorial Theory Series A
A Hamada type characterization of the classical geometric designs
Designs, Codes and Cryptography
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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 PG"d(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 AG"d"+"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.