Search

In this article, we investigate symmetric designs admitting a flag-transitive and point-primitive affine automorphism group. We prove that if an automorphism group $G$ of a symmetric $(v,k,\lambda)$ design with $\lambda$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,\lambda)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $\lambda$ prime, which says that such an incidence structure is a projective space $\mathrm{PG}(n,q)$, it has parameter set $(15,7,3)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 2)$, $(16,6,2)$ or $(45, 12, 3)$, or $v=p^d$ where $p$ is an odd prime and the automorphism group is a subgroup of $\mathrm{A\Gamma L}_{1}(q)$.

The classification of the $2$-designs with $\lambda=2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in \cite{ABDT}. The result is achieved by using conics and hyperovals of $PG(2,q)$.

In this paper, we provide a complete classification of the $2$-$(v,3,\lambda )$ designs with $v\equiv 1,3\pmod{6}$ and $% v \equiv 6 \pmod{\lambda}$ admitting a flag-transitive automorphism group non-isomorphic to a subgroup of $A\Gamma L_{1}(v)$.

In this paper, we provide a complete classification of $2$-$(v,k,2)$ design admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear $1$-dimensional group. Alongside this analysis we provide a construction of seven new families of such flag-transitive $2$-designs, two of them infinite, and some of them involve remarkable objects such as $t$-spreads, translation planes, quadrics and Segre varieties. Our result together with those Alavi et al. [1,2], Praeger et al. [15], Zhou and the first author [37,38] provides a complete classification of $2$-$(v,k,2)$ design admitting a flag-transitive automorphism group with the only exception of the semilinear $1$-dimensional case.

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchm\`aros, On quasi-Hermitian Varieties, J. Combin. Des. 20 (2012) 433-447.] in characteristc $2$ and dimension $3$. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the classification project started in that paper. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays $O(q^5,q^4,q,2)$, with entries in $\mathrm{GF}{q}$, where $q$ is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed in order to investigate how variables in testing interact with each other., Comment: Work supported by INDAM. A. Aguglia and V. Siconolfi have been partially supported by the EU under the Italian NRRP of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program RESTART, CUP: D93C22000910001) and by the Italian Ministry of University and Research under the Programme "Department of Excellence" Legge 232/2016 (Grant No. CUP - D93C23000100001)

The symmetric $2$-$(v,k,\lambda )$ designs, with $k>\lambda \left(\lambda-3 \right)/2$, admitting a flag-transitive, point-imprimitive automorphism group are completely classified: they are the known $2$-designs with parameters $(16,6,2),(45,12,3),(15,8,4)$ or $(96,20,4)$.

The pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a non-trivial $2$-$(k^{2},k,\lambda )$ design, with $\lambda \mid k$, and $G$ is a flag-transitive automorphism group of $\mathcal{D}$ of affine type such that $G \nleq A \Gamma L_{1}(k^{2})$, are classified.

In this paper, we present a classification of $2$-designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,\lambda)=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q = p^{d}$ points with $p$ prime and $G$ is a subgroup of $A\Gamma L_{1}(q)$.

Non-trivial $2$-$(k^{2},k,\lambda )$ designs, with $\lambda \mid k$, admitting a flag-transitive almost simple automorphism group are classified.

Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a symmetric $2$-$(v,k,\lambda )$ design admitting a flag-transitive, point-imprimitive automorphism group $G$ that leaves invariant a non-trivial partition $\Sigma$ of $\mathcal{P}$. Praeger and Zhou \cite{PZ} have shown that, there is a constant $k_{0}$ such that, for each $B \in \mathcal{B}$ and $\Delta \in \Sigma$, the size of $\left\vert B \cap \Delta \right \vert$ is either $0$ or $k_{0}$. In the present paper we show that, if $k>\lambda \left(\lambda-3 \right)/2$ and $k_{0} \geq 3$, $\mathcal{D}$ is isomorphic to one of the known flag-transitive, point-imprimitive symmetric $2$-designs with parameters $(45,12,3)$ or $(96,20,4)$.

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, \lambda)$ design D, with $\lambda \mid k$, is either an affine group or an almost simple classical group. Moreover, when $G$ is the smallest Ree group, $\mathcal{D}$ is isomorphic either to the $2$-$(62, 6, 2)$ design or to one of the three $2$- $(62, 6, 6)$ designs constructed in this paper. All the four $2$-designs have the $36$ secants of a nondegenerate conic $\mathcal{C}$ of $PG_{2}(8)$ as a point set and 6-sets of secants in a remarkable configuration as a block set.