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Alessandro Montinaro

Publications and source records attributed to Alessandro Montinaro.

18 recordsLinked to original sources
arXiv2026

$2$-designs admitting a flag-transitive automorphism group with socle $PSL(2,q)$

$2$-designs admitting a flag-transitive automorphism group $G$ with socle $PSL(2,q)$, where $q=p^{f}\geq 4$, are investigated in both the point-primitive and point-imprimitive cases. In the latter case, a complete classification is achieved, and three known examples occur, namely: the complementary designs of $PG(3,2)$ and $PG(3,4)$, and the $2$-$(36,8,4)$ design constructed by Devillers and Praeger in [14]. In the point-primitive case, apart from the Witt-Bose-Shrikhande linear spaces of even order $q$, $48$ sporadic examples are classified. Surprisingly, one of these numerical examples is the linear space with $v=496$ and $k=4$ admitting $PΓL(2,2^{5})$ as a flag-transitive automorphism group, which was missing in the 1990 classification by Buekenhout et al. [7,36,12].

math.CO↗
arXiv2025

$\mathrm{ EA}(q)$-additive Steiner 2-designs

A design is $G$-additive with $G$ an abelian group, if its points are in $G$ and each block is zero-sum in $G$. All the few known ``manageable" additive Steiner 2-designs are $\mathrm{EA}(q)$-additive for a suitable $q$, where $\mathrm{EA}(q)$ is the elementary abelian group of order $q$. We present some general constructions for $\mathrm{EA}(q)$-additive Steiner 2-designs which unify the known ones and allow to find a few new ones: an additive $\mathrm{EA}(2^8)$-additive 2-$(52,4,1)$ design which is also resolvable, and three pairwise non-isomorphic $\mathrm{EA}(3^5)$-additive 2-$(121,4,1)$ designs, none of which is the point-line design of $\mathrm{PG}(4,3)$. In the attempt to find also an $\mathrm{EA}(2^9)$-additive 2-$(511,7,1)$ design, we prove that a putative 2-analog of a 2-$(9,3,1)$ design cannot be cyclic.

math.CO↗
arXiv2025

The Flag-Transitive and Point-Imprimitive Symmetric $(v,k,λ)$ Designs with $v<100$

A complete classification of the flag-transitive point-imprimitive symmetric $2$-$(v,k,λ)$ designs with $v<100$ is provided. Apart from the known examples with $λ\leq 10$, the complementary design of $PG_{5}(2)$, and the $2$-design $\mathcal{S}^{-}(3)$ constructed by Kantor in \cite{Ka75}, we found two non isomorphic $2$-$(64,28,12)$ designs. They were constructed via computer as developments of $(64,28,12)$-difference sets by AbuGhneim in \cite{OAG}. In the present paper, independently from \cite{OAG}, we construct the aforementioned two $2$-designs and we prove that their full automorhpism group is flag-transitive and point-imprimitive. The construction is theoretical and relies on the the absolutely irreducible $8$-dimensional $\mathbb{F}_{2}$-representation of $PSL_{2}(7)$. Our result, together with that about the flag-transitive point-primitive symmetric $2$-designs with $v<2500$ by Braić-Golemac-Mandić-Vučičić \cite{BGMV}, provides a complete classification of the flag-transitive $2$-designs with $v<100$.

math.GR↗
arXiv2025

On a class of quasi-Hermitian surfaces in even characteristic

In [1], a new quasi-Hermitian variety $\mathcal{H}_\varepsilon^r$ in $\mathrm{PG}(r, q^2)$, with $q = 2^e$ and $e \geq 3$ an odd integer, was constructed. The variety depends on a primitive element $\varepsilon$ of the underlying field $\mathrm{GF}(q^2)$.11 In the present paper, we first provide a classification of such varieties up to projective equivalence in finite projective spaces of arbitrary dimension. Then, we focus on the case $r = 3$ and study the structure of the lines contained in $\mathcal{H}_\varepsilon^3$; as a consequence, we determine the full automorphism group of $\mathcal{H}_\varepsilon^3$ . Finally, as a byproduct, we prove the equivalence of certain minimal codes introduced in [3].

math.CO↗
arXiv2025

On flag-transitive automorphism groups of $2$-designs with $λ$ prime

In this article, we study $2$-$(v,k,λ)$ designs $\mathcal{D}$ with $λ$ prime admitting flag-transitive and point-primitive almost simple automorphism groups $G$ with socle $T$ a finite exceptional simple group or a sporadic simple groups. If the socle of $G$ is a finite exceptional simple group, then we prove that $\mathcal{D}$ is isomorphic to one of two infinite families of $2$-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set $(v,b,r,k,λ)=(q^{2}+1,q^{2}(q^{2}+1)/(q-1),q^{2},q,q-1)$ design, where $q-1$ is a Mersenne prime, and the other is newly constructed in this paper and has parameter set $(v,b,r,k,λ)=(q^{3}(q^{3}-1)/2,(q+1)(q^{6}-1),(q+1)(q^{3}+1),q^{3}/2,q+1)$, where $q+1$ a Fermat prime. If $T$ is a sporadic simple group, then we show that $\mathcal{D}$ is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set $(v,b,r,k,λ)=(176,1100,50,2)$, $(12,22,11,6,5)$ or $(22,77,21,6,5)$.

math.GR↗
arXiv2025

The Higman-M\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $λ$ prime

A famous result of Higman and McLaughlin \cite{HM} in 1961 asserts that any flag-transitive automorphism group $G$ of a $2$-design $\mathcal{D}$ with $λ=1$ acts point-primitively on $\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $λ$ is a prime and $\mathcal{D}$ is not isomorphic to one of the two $2$-$(16,6,2)$ designs as in [42, Section 1.2], or the $2$-$(45,12,3)$ design as in [44, Construction 4.2], or, when $2^{2^{j}}+1$ is a Fermat prime, a possible $2$-$(2^{2^{j+1}}(2^{2^{j}}+2),2^{2^{j}}(2^{2^{j}}+1),2^{2^{j}}+1)$ design having very specific features.

math.CO↗
arXiv2024

On quasi-Hermitian varieties in even characteristic and related orthogonal arrays

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchmàros, 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.

math.CO↗
arXiv2024

Affine groups as flag-transitive and point-primitive automorphism groups of symmetric designs

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 $(v,k,λ)$ design with $λ$ prime is point-primitive of affine type, then $G=2^{6}{:}\mathrm{S}_{6}$ and $(v,k,λ)=(16,6,2)$, or $G$ is a subgroup of $\mathrm{AΓL}_{1}(q)$ for some odd prime power $q$. In conclusion, we present a classification of flag-transitive and point-primitive symmetric designs with $λ$ 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 $\mathrm{AΓL}_{1}(q)$.

math.GR↗
arXiv2024

A Classification of the flag-transitive $2$-$(v,k,2)$ designs

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.

math.CO↗
arXiv2022

Flag-transitive, point-imprimitive symmetric $2$-$(v,k,λ)$ designs with $k>λ\left(λ-3 \right)/2$

Let $\mathcal{D}=\left(\mathcal{P},\mathcal{B} \right)$ be a symmetric $2$-$(v,k,λ)$ design admitting a flag-transitive, point-imprimitive automorphism group $G$ that leaves invariant a non-trivial partition $Σ$ 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 $Δ\in Σ$, the size of $\left\vert B \cap Δ\right \vert$ is either $0$ or $k_{0}$. In the present paper we show that, if $k>λ\left(λ-3 \right)/2$ 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 $(96,20,4)$.

math.CO↗
arXiv2022

Block designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups

In this paper, we present a classification of $2$-designs with $\gcd(r,λ)=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd(r,λ)=1$, 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 $AΓL_{1}(q)$.

math.GR↗
arXiv2022

On Flag-Transitive $2$-$(k^{2}, k, λ)$ Designs with $λ\mid k$

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group $G$ of a $2$-$(k^{2}, k, λ)$ design D, with $λ\mid k$, 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.

math.CO↗