Polarities, quasi-symmetric designs, and Hamada's conjecture

Authors:
Dieter Jungnickel;Vladimir D. Tonchev
Affiliations:
Lehrstuhl für Diskrete Mathematik, Optimierung, und Operations Research, Universität Augsburg, Augsburg, Germany 86135;Department of Mathematical Sciences, Michigan Technological University, Houghton, USA 49931
Venue:
Designs, Codes and Cryptography
Year:
2009

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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.