Abstract: We describe the spreading property for finite transitive permutation groups in terms of properties of their associated coherent configurations, in much the same way that separating and synchronising groups can be described via properties of their orbital graphs. We also show how the other properties in the synchronisation hierarchy naturally fit inside this framework. This combinatorial description allows for more efficient computational tools, and we deduce that every spreading permutation group of degree at most $8191$ is a $\mathbb{Q}$I-group. We also consider design-orthogonality more generally for noncommutative homogeneous coherent configurations.

Abstract: Delsarte theory, more specifically the study of codes and designs in association schemes, has proved invaluable in studying an increasing assortment of association schemes in recent years. Tools motivated by the study of error-correcting codes in the Hamming scheme and combinatorial $t$-designs in the Johnson scheme apply equally well in association schemes with irrational eigenvalues. We assume here that we have a commutative association scheme with irrational eigenvalues and wish to study its Delsarte $T$-designs. We explore when a $T$-design is also a $T'$-design where $T'\supseteq T$ is controlled by the orbits of a Galois group related to the splitting field of the association scheme. We then study Delsarte designs in the association schemes of finite groups, with a detailed exploration of the dicyclic groups.

Abstract: A chamber of the vector space $\mathbb{F}_q^n$ is a set $\{S_1,\dots,S_{n-1}\}$ of subspaces of $\mathbb{F}_q^n$ where $S_1\subset S_2\subset\dots\subset S_{n-1}$ and $\dim(S_i)=i$ for $i=1,\dots,n-1$. By $\Gamma_n(q)$ we denote the graph whose vertices are the chambers of $\mathbb{F}_q^n$ with two chambers $C_1=\{S_1,\dots,S_{n-1}\}$ and $C_2=\{T_1,\dots,T_{n-1}\}$ adjacent in $\Gamma_n(q)$, if $S_i\cap T_{n-i}=\{0\}$ for $i=1,\dots,n-1$. The Erdős-Ko-Rado problem on chambers is equivalent to determining the structure of independent sets of $\Gamma_n(q)$. The independence number of this graph was determined in De Beule, Mattheus, Metsch (2022) for $n$ even and given a subspace $P$ of dimension one, the set of all chambers whose subspaces of dimension $\frac n2$ contain $P$ attains the bound. The dual example of course also attains the bound. It remained open in De Beule, Mattheus, Metsch (2022) whether or not these are all maximum independent sets. Using a description from De Beule, Mattheus, Metsch, (2024+) of the eigenspace for the smallest eigenvalue of this graph, we prove an Erdős-Ko-Rado theorem on chambers of $\mathbb{F}_q^n$ for sufficiently large $q$, giving an affirmative answer for n even.

Abstract: For finite classical groups acting naturally on the set of points of their ambient polar spaces, the symmetry properties of synchronising and separating are equivalent to natural and well-studied problems on the existence of certain configurations in finite geometry. The more general class of spreading permutation groups is harder to describe, and it is the purpose of this paper to explore this property for finite classical groups. In particular, we show that for most finite classical groups, their natural action on the points of its polar space is non-spreading. We develop and use a result on tactical decompositions (an AB-Lemma) that provides a useful technique for finding witnesses for non-spreading permutation groups. We also consider some of the other primitive actions of the classical groups.

Abstract: 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 $41\leqslant q\leqslant 49$. 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.

Abstract: Using the classification of transitive groups of degree $n$, for $2 \leqslant n \leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 \leqslant n \leqslant 48$. The results of this classification are made available as a supplementary database.

Abstract: We classify, up to some notoriously hard cases, the vertex-primitive strongly regular graphs which meet both the Delsarte and Hoffman bounds. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give the first known examples of synchronising but not $\mathbb{Q}\mathrm{I}$ groups of affine type.

Abstract: 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 $\mathsf{Q}^-(2n-1, q)$, $n\geqslant 3$.

Abstract: Every synchronising permutation group is primitive and of one of three types: affine, almost simple, or diagonal. We exhibit the first known example of a synchronising diagonal type group. More precisely, we show that $\mathrm{PSL}(2,q)\times \mathrm{PSL}(2,q)$ acting in its diagonal action on $\mathrm{PSL}(2,q)$ is separating, and hence synchronising, for $q=13$ and $q=17$. Furthermore, we show that such groups are non-spreading for all prime powers $q$.

Abstract: Let $q$ be a prime power and $V\cong\mathbb{F}_q^d$. A \emph{$t$-$(d,k,\lambda)_q$ 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 \emph{$q$-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 $t\geqslant 2$ 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 $V$.

Abstract: We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \geqslant 2$ and all odd prime powers $q$, that admit $\Omega_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \geqslant 4$.

Abstract: We generalise the work of Segre (1965), Cameron – Goethals – Seidel (1978), and Vanhove (2011) by showing that nontrivial $m$-ovoids of the dual polar spaces $\mathsf{DQ}(2d,q)$, $\mathsf{DW}(2d-1,q)$ and $\mathsf{DH}(2d-1,q^2)$ $(d>3)$ are hemisystems. We also provide a more general result that holds for regular near polygons.

Abstract: In Bachmann’s Aufbau der Geometrie aus dem Spiegelungsbegriff (1959), it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines, and conversely. Sherk (1967) generalised this result to charac- terise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow non-collinear points.