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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机译：我们证明PG（2k − 1，q）的每个极性（其中k≥2）都会导致设计具有与PG k 相同但不同构的参数和相交数（2k，q）。特别是在k = 2的情况下，产生了一个新的准对称设计族。我们还表明，对于任何阶数p的字段，我们的构造都为Hamada的猜想提供了无穷无尽的反例。以前，只有少数反例是已知的。
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