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

We prove that every polarity of PG(2k - 1,q), where ka parts per thousand yen 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.