Implications of vanishing Krein parameters on Delsarte designs, with applications in finite geometry

In this paper we show that if $\theta$ is a $T$-design of an association scheme $(\Omega, \mathcal{R})$, and the Krein parameters $q_{i,j}^h$ vanish for some $h \not \in T$ and all $i, j \not \in T$ ($i, j, h \neq 0$), then $\theta$ consists of precisely half of the vertices of $(\Omega, \mathcal{R})$ or it is a $T'$-design, where $|T'|>|T|$. We then apply this result to various problems in finite geometry. In particular, we show for the first time that nontrivial $m$-ovoids of generalised octagons of order $(s, s^2)$ do not exist. We give short proofs of similar results for (i) partial geometries with certain order conditions; (ii) thick generalised quadrangles of order $(s,s^2)$; (iii) the dual polar spaces $\mathsf{DQ}(2d, q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$, for $d \ge 3$; (iv) the Penttila-Williford scheme. In the process of (iv), we also consider a natural generalisation of the Penttila-Williford scheme in $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.Comment: This paper builds on part of the doctoral work of the second author under the supervision of the first. The second author acknowledges the support of an Australian Government Research Training Program Scholarship and Australian Research Council Discovery Project DP20010195

On Bruen chains

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\le q\leq 49$. We develop a model, based on finite fields, which allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs

Separating rank 3 graphs

We classify, up to some notoriously hard cases, the rank 3 graphs which fail to meet either the Delsarte or the Hoffman bound. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give new examples of synchronising, but not $\mathbb{Q}\mathrm{I}$, groups of affine type

Designs in finite geometry

This thesis is concerned with the study of Delsarte designs in symmetric association schemes, particularly in the context of finite geometry. We prove that m-ovoids of regular near polygons satisfying certain conditions must be hemisystems, and as a consequence, that for d ≥ 3 m-ovoids of DH(2d−1, q^2), DW(2d−1, q), and DQ(2d, q) are hemisystems. We also construct an infinite family of hemisystems of Q(2d, q), for q an odd prime power and d ≥ 2, the first known family for d ≥ 4. We generalise the AB-Lemma to constructions of m-covers other than just hemisystems. In the context of general Delsarte designs, we show that either the size of a design, or the strata in which it lies, may be constrained when certain Krein parameters vanish, and explore various consequences of this result. We also study the concept of a “witness” to the non-existence of a design, in particular by considering projection and inclusion of association schemes, and the implications this has on the existence of designs when the strata of a projected design is constrained. We furthermore introduce strong semi-canonicity and use it in a black-box pruned orderly algorithm for effective generation of designs and combinatorial objects. We use these techniques to find new computational results on various m-ovoids, partial ovoids, and hemisystems

The non-existence of block-transitive subspace designs

Let $q$ be a prime power and $V\cong\mathbb{F}_q^d$. A $t$-$(d,k,\lambda)_q$ design, or simply a subspace design, is a pair $\mathcal{D}=(V,\mathcal{B})$, where $\mathcal{B}$ is a subset of the set of all $k$-dimensional subspaces of $V$, with the property that each $t$-dimensional subspace of $V$ is contained in precisely $\lambda$ elements of $\mathcal{B}$. Subspace designs are the $q$-analogues of balanced incomplete block designs. Such a design is called block-transitive if its automorphism group $\mathrm{Aut}(\mathcal{D})$ acts transitively on $\mathcal{B}$. It is shown here that if $t\geq 2$ and $\mathcal{D}$ is a block-transitive $t$-$(d,k,\lambda)_q$ design then $\mathcal{D}$ is trivial, that is, $\mathcal{B}$ is the set of all $k$-dimensional subspaces of $V$.Mathematics Subject Classifications: 05E18, 05B9