Polarities, quasi-symmetric designs, and Hamada’s conjecture. (English) Zbl 1247.05032
Summary: We prove that every polarity of \(\text{PG}(2k-1,q)\), where \(k\geq 2\), gives rise to a design with the same parameters and the same intersection numbers as, but not isomorphic to, \(\text{PG}_k(2k,q)\). 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.
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 51E20 | Combinatorial structures in finite projective spaces |
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
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