Dina Ghinelli

Flag-transitive extensions ofC n geometries

We consider locally Cn-geometries where all planes are circular spaces (i.e., complete graphs). We call them extensions of Cn-geometries or (c.Cn) geometries, for short. We give a classification of finite flag-transitive (c.Cn) geometries) geometries when n≥3. A classification is given also in the case of n=2 under the hypothesis that residues of points are thick classical generalized quadrangles.

Regular groups on generalized quadrangles and nonabelian difference sets with multiplier -1

This is a first approach to the study of regular generalized quadrangles (i.e. generalized quadrangles with an automorphism group sharply 1-transitive on points). In this paper we point out how the problem is connected to the theory of difference sets with multiplier-1. First, some of the results in [3] on difference sets with multiplier-1 are extended to the nonabelian case; then, these new results on difference sets are used to prove nonexistence theorems for regular GQs of even order s=t.

(0,n)-Sets in a generalized quadrangle

Abstract We study subsets of points in a generalized quadrangle (GQ), in particular, those meeting every line in just n points or in none at all, (i.e: sets of class (0, n )), simply called here (0, n )-sets.

Arcs and Ovals from Abelian Groups

We use large abelian collineation groups of finite projective planes (via the associated representation by some sort of difference set) to construct interesting families of (hyper)ovals. This approach is of particular interest for groups of type (b) in the Dembowski-Piper classification, i.e., for abelian relative (n, n, n, 1)-difference sets. Here we obtain the first series of ovals in planes of Lenz-Barlotti class II.1, namely in the Coulter-Matthews planes, and a partition of the affine part of any commutative semifield plane of even order into translation ovals; we also provide a somewhat surprising embedding of the dual affine translation plane into the original projective plane and an explicit description of a maximal arc of degree q/2 which leads to the embedding of a certain Hadamard design as a family of maximal arcs. We also survey previous results for groups of type (a) and (d), i.e., for planar and affine difference sets, respectively. Finally, we study the case of groups of type (f) (which correspond to direct product difference sets); here we also use the resulting ovals to give considerably simpler proofs for some known restrictions concerning planes admitting such a group.

Diameter bounds for locally partial geometries

We obtain a new upper bound d 1 =[ s /2] − α + 3 for the diameter Δ of a locally partial geometry LpG α ( r, s, t ) of order ( r, s, t ) with α ⩾ 2 (see [10] for α= r = 1). This is attained by an infinite family of LpG 2 (1, s , 1), say D s +2 (see example 1.1). Furthermore, other upper bounds for Δ are obtained, which are often better than the previous one.

Codes from incidence matrices and line graphs of Paley graphs

We examine the

Generalized quadrangles with a regular point and association schemes

p

Tubes of even order and flat π.C2 geometries

-ary codes from incidence matrices of Paley graphs

Nets and generalized quadrangles

P(q)

Extended partial geometries with minimal μ

where