Generalized quadrangles, flocks, and BLT sets

Generalized quadrangles, flocks, and BLT sets

Approximately five years ago Thas observed a remarkable coincidence relating certain generalized quadrangles constructed in [Ka3] to flocks of cones and translation planes [Th] (cf. [Ka2]). While there is a rapidly growing literature concerning this connection, there as yet has been no explanation for it. This note contains an observation providing at least some kind of explanation. It also con...

Flocks, ovoids and generalized quadrangles

Let (∞, π∞) be an incident point-plane pair of PG(3, q). A tetrad with respect to (∞, π∞) is a set {X,Y, Z,W} of points of PG(3, q) \ π∞ such that {∞, X, Y, Z,W} is a cap of PG(3, q), ∞ ∈ 〈X,Y, Z〉 and W 6∈ 〈X,Y, Z〉. A set Θ of ovoids of PG(3, q) is tetradic with respect to (∞, π∞) if each ovoid contains ∞, has tangent plane π∞ and is such that each tetrad with respect to (∞, π∞) is contained in...

BLT-sets over small fields

A BLT-set is a set X of q+l points of the generalized quadrangle Q(4, q), q odd, such that no point of Q(4, q) is collinear with more than 2 points of X. BLT-sets are closely related to flocks of the quadratic cone, elation generalised quadrangles and certain translation planes. In this paper we report on the results of computer searches for BLT-sets for odd q ::; 25. We complete the classifica...

Finite Generalized Quadrangles

Monomial graphs and generalized quadrangles

Let Fq be a finite field, where q = p for some odd prime p and integer e ≥ 1. Let f, g ∈ Fq[x, y] be monomials. The monomial graph Gq(f, g) is a bipartite graph with vertex partition P ∪L, P = Fq = L, and (x1, x2, x3) ∈ P is adjacent to [y1, y2, y3] ∈ L if and only if x2 + y2 = f(x1, y1) and x3 + y3 = g(x1, y1). Dmytrenko, Lazebnik, and Williford proved in [6] that if p ≥ 5 and e = 23 for integ...