Abstract
We prove that every polarity of PG(2k − 1,q), where k≥ 2, gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, PG k (2k,q). In particular, the case k = 2 yields a new family of quasi-symmetric designs. We also show that our construction provides an infinite family of counterexamples to Hamada’s conjecture, for any field of prime order p. Previously, only a handful of counterexamples were known.
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Jungnickel, D., Tonchev, V.D. Polarities, quasi-symmetric designs, and Hamada’s conjecture. Des. Codes Cryptogr. 51, 131–140 (2009). https://doi.org/10.1007/s10623-008-9249-8
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DOI: https://doi.org/10.1007/s10623-008-9249-8