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

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.

Recent results on designs with classical parameters

“…A recent joint paper of the author with Tonchev [29] provides the first infinite families of counterexamples to Conjecture 4.2:…”

“…This method was introduced by Tonchev and the present author in [29,30], but the idea of distorting a classical design in some appropriate manner is by no means new and has been used successfully before, see e.g. [13,26,32].…”

“…In this section, we outline the proof of Theorem 4.3 given in [29] and discuss some properties of the "pseudo-geometric" designs constructed there. To this purpose, we recall that a polarity of a projective space Π = P G(k, q) is an involutory isomorphism between P G(k, q) and its dual space; in other words, a polarity is an incidence preserving bijection interchanging points and hyperplanes.…”

“…Then the hyperplane H is isomorphic to P G(2d−1, q), and thus we may use a polarity α of P G(2d − 1, q) to distort D. As shown in [29], the resulting design α(D) then has the following two important properties, which do not hold for arbitrary permutations α.…”

“…The following lemma was proved in [29]; a crucial part of the argument involves proving that the incidence vectors of the blocks of the complementary design α(D) span a linear self-orthogonal code of length (q 2d+1 − 1)/(q − 1) over GF (p); this follows from the fact that α(D) has the same intersection numbers as the classical design P G d (2d, q).…”