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We provide classification results for translation generalized quadrangles of order less or equal to $64$, and hence, for all incidence geometries related to them. The results consist of the classification of all pseudo-ovals in $PG(3n-1,2)$, for $n=3,4$, and that of the pseudo-ovals in $PG(3n-1,q)$, for $n=5,6$, such that one of the associated projective planes is Desarguesian.

Brown et al. provide a representation of a spread of the Tits quadrangle $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,q)$, $q$ even, in terms of a certain family of $q$ ovals of $\mathrm PG(2,q)$. By combining this representation with the Vandendriessche classification of hyperovals in $\mathrm PG(2,64)$ and the classification of flocks of the quadratic cone in $\mathrm PG(3,64)$, recently given by the authors, in this paper, we classify all the spreads of $T_2(\mathcal O)$, $\mathcal O$ an oval of $\mathrm PG(2,64)$, up to equivalence. These complete the classification of spreads of $T_2(\mathcal O)$ for $q\le 64$.

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the Andr\'e/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of PG(4m-1,q), m>=1, are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order 3^{10}, which is represented in PG(20,3) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in PG(19,3). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.

H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger's axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala's formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.

Flocks are an important topic in the field of finite geometry, with many relations with other objects of interest. This paper is a contribution to the difficult problem of classifying flocks up to projective equivalence. We complete the classification of flocks of the quadratic cone in PG(3,q) for q <= 71, by showing by computer that there are exactly three flocks of the quadratic cone in PG(3,64), up to equivalence. The three flocks had previously been discovered, and they are the linear flock, the Subiaco flock and the Adelaide flock. The classification proceeds via the connection between flocks and herds of ovals in PG(2,q), q even, and uses the prior classification of hyperovals in PG(2,64).

We construct a new infinite family of Cameron-Liebler line classes in $PG(3,q)$ with parameter $x=\frac{q^2+1}{2}$ for all odd $q$.

In this article we construct a series of new infinite families of strongly regular graphs with the same parameters as the point-graphs of non-singular quadrics in PG(n,2).

A {\em pseudo-arc} in $\mathrm{PG}(3n-1,q)$ is a set of $(n-1)$-spaces such that any three of them span the whole space. A pseudo-arc of size $q^n+1$ is a {\em pseudo-oval}. If a pseudo-oval $\mathcal{O}$ is obtained by applying field reduction to a conic in $\mathrm{PG}(2,q^n)$, then $\mathcal{O}$ is called a {\em pseudo-conic}. We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens-Szekeres GQ is obtained from a $q$-arc in $\mathrm{PG}(2,q)$ and we extend this construction to that of a GQ of order $(q^n-1,q^n+1)$ from a pseudo-arc of $\mathrm{PG}(3n-1,q)$ of size $q^n$. The main theorem of this paper shows that if $\mathcal{K}$ is a pseudo-arc in $\mathrm{PG}(3n-1,q)$, $q$ odd, of size larger than the size of the second largest complete arc in $\mathrm{PG}(2,q^n)$, where for one element $K_i$ of $\mathcal{K}$, the partial spread $\mathcal{S}=\{K_1,\ldots,K_{i-1},K_{i+1},\ldots,K_{s}\}/K_i$ extends to a Desarguesian spread of $\mathrm{PG}(2n-1,q)$, then $\mathcal{K}$ is contained in a pseudo-conic. The main result of \cite{Casse} also follows from this theorem.

We show that an elation generalised quadrangle which has p+1 lines on each point, for some prime p, is classical or arises from a flock of a quadratic cone (i.e., is a flock quadrangle)., Comment: 14 pages