Searcharxiv⌕ Search

arXiv subjects

Valentino Smaldore

Publications and source records attributed to Valentino Smaldore.

At least 19 recordsLinked to original sources
arXiv2026

Regular fat linear sets

In this work, we introduce $(r,i)$-regular fat linear sets, which are defined as linear sets containing exactly $r$ points of weight $i$ and all other points of weight one. This notion generalizes and unifies existing constructions; scattered linear sets, clubs, and other previously studied families are special cases. We present new classes of regular fat linear sets in PG$(k-1,q^n)$ for composite $n$ and study their equivalence classes. Finally, we show that regular fat linear sets naturally yield three-weight rank-metric codes, which we use to obtain bounds on their parameters.

math.CO↗
arXiv2026

Vertex-transitive strongly regular graphs in the switching class of doubly transitive two-graphs

Let $G$ be a permutation group that acts $2$-transitively on the finite set $V$ and let $\mathcal{T}=(V,T)$ be a two-graph whose automorphism group contains $G$. In this paper, we classify those strongly regular graphs $Γ$ with vertex set $V$ whose automorphism group is a transitive maximal subgroup of $G$ and whose associated two-graph is $\mathcal{T}$. In doing so, we obtain a new family of vertex-transitive strongly regular graphs whose associated two-graph arises from $PΣL(2,q)$.

math.CO↗
arXiv2026

Classification of Deza graphs from anisotropic association schemes of quadrics

Let $Q^\varepsilon(3,q)$, where $\varepsilon\in\{+,-\}$ and $q>3$ is odd, be a non-degenerate hyperbolic or elliptic quadric of $PG(3,q)$. Fix one of the two quadratic classes of anisotropic points. Since the line joining two distinct points of this class is tangent, secant, or external to the quadric, one obtains a $3$-class association scheme. We classify all non-trivial unions of its relations which define Deza graphs. In addition to the previously known tangency family, exactly four exceptional strictly Deza graphs occur, with parameters $(360,135,54,45)$, $(369,108,36,27)$, $(65,34,18,15)$ and $(168,111,75,70)$. We determine their spectra and Deza children and give geometric or group-theoretic descriptions of all four exceptional graphs.

math.CO↗
arXiv2026

Strongly regular graphs from hyperbolic quadrics and their maximal cliques

Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $Π$ of the quadric. Define $\cG_n$ as the graph with as vertex set the points of $Q^+(2n+1,q)\setminus Π$ and two vertices adjacent if they either span a secant to $Q^+(2n+1,q)$ or a line contained in $Q^+(2n+1,q)$ meeting $Π$ non-trivially. Then such a construction defines a strongly regular graph, which is the complement of a (non-induced) subgraph of the collinearity graph of $Q^+(2n+1,q)$. In this paper, we directly compute the parameters of $\cG_n$, which is cospectral, when $q=2$, to the tangent graph $NO^+(2n+2,2)$, but it is non-isomorphic for $n\geq3$. We also classify the maximal cliques of $\cG_3$ for $q=2$, proving as a by-product the non-isomorphism with the graph $NO^+(8,2)$.

math.CO↗
arXiv2026

Ramanujan polar graphs

Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such graphs Ramanujan, since all known non-trivial constructions imply the Ramanujan conjecture on arithmetical functions. In this paper, we prove that some families of tangent graphs of finite classical polar spaces satisfy Ramanujan's condition. If the polarity is unitary, or it is orthogonal and the quadric is over the binary field, the tangent graphs are strongly regular, and we know their spectrum. By direct computation, it is possible to show which families of tangent graphs are Ramanujan.

math.CO↗
arXiv2026

Hermitian-Singer Functional and Differential Codes

Algebraic geometry codes on the Hermitian curve have been the subject of several papers, since they happen to have good performances and large automorphism groups. Here, those arising from the Singer cycle of the Hermitian curve are investigated.

math.AG↗
arXiv2025

Switching equivalence of strongly regular polar graphs

We prove the switching equivalence of the strongly regular polar graphs $NO^\pm(4m,2)$, $NO^\mp(2m+1,4)$, and $Γ(O^\mp(4m,2))$ plus an isolated vertex by giving an analytic description for them and their associated two-graphs.

math.CO↗
arXiv2025

On Geometry and Combinatorics of Finite Classical Polar Spaces

Polar spaces over finite fields are fundamental in combinatorial geometry. The concept of polar space was firstly introduced by F. Veldkamp who gave a system of 10 axioms in the spirit of Universal Algebra. Later the axioms were simplified by J. Tits, who introduced the concept of subspaces. Later on, from the point of view of incidence geometry, axioms of polar spaces were also given by F. Buekenhout and E. Shult in 1974. The reader can find the three systems of axioms of polar spaces in Appendix A. Examples of polar spaces are the so called Finite classical polar spaces, i.e. incidence structures arising from quadrics, symplectic spaces and Hermitian varieties, which are in correspondance with reflexive sesquilinear forms. It is still an open problem to show whether or not classical polar spaces are the only example of finite polar spaces. Nowadays, some research problems related to finite classical polar space are: existence of spreads and ovoids; existence of regular systems and $m$-ovoids; upper or lower bounds on partial spreads and partial ovoids. Moreover, polar spaces are in relation with combinatorial objects as regular graphs, block designs and association schemes. In this Ph.D. Thesis we investigate the geometry of finite classical polar spaces, giving contributions to the above problems. The thesis is organized as follows. Part I is more focused on the geometric aspects of polar spaces, while in Part II some combinatorial objects are introduced such as regular graphs, association schemes and combinatorial designs. Finally Appendix B, C and D are dedicated to give more details on, respectively, maximal curves, linear codes and combinatorial designs, giving useful results and definitions.

math.CO↗
arXiv2024

New scattered linearized quadrinomials

Let $1 8$ only three families of scattered polynomials in $\mathbb F_{q^n}[X]$ are known: $(i)$~monomials of pseudoregulus type, $(ii)$~binomials of Lunardon-Polverino type, and $(iii)$~a family of quadrinomials defined in [1,10] and extended in [8,13]. In this paper we prove that the polynomial $φ_{m,q^J}=X^{q^{J(t-1)}}+X^{q^{J(2t-1)}}+m(X^{q^J}-X^{q^{J(t+1)}})\in\mathbb F_{q^{2t}}[X]$, $q$ odd, $t\ge3$ is R-$q^t$-partially scattered for every value of $m\in\mathbb F_{q^t}^*$ and $J$ coprime with $2t$. Moreover, for every $t>4$ and $q>5$ there exist values of $m$ for which $φ_{m,q}$ is scattered and new with respect to the polynomials mentioned in $(i)$, $(ii)$ and $(iii)$ above. The related linear sets are of $ΓL$-class at least two.

math.CO↗
arXiv2024

Bent functions and strongly regular graphs

The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on $\mathbb{Z}_{2}^{n}$ by the support of a bent function is a strongly regular graph $srg(v,kλ,μ)$, with $λ=μ$. In this note we list the parameters of such Cayley graphs. Moreover, it is given a condition on $(n,m)$-bent functions $F=(f_1,\ldots,f_m)$, involving the support of their components $f_i$, and their $n$-ary symmetric differences.

cs.IT↗
arXiv2024

Some non-existence results on $m$-ovoids in classical polar spaces

In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$ for $r>2$. In [4] a lower bound on $m$ for the existence of $m$-ovoids of $H(4,q^2)$ is found by using the connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approach is generalized in [3] for the polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$, $r>2$. In [1] an improvement for the particular case $H(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by the results from [10], to improve the bounds from [3].

math.CO↗
arXiv2024

On the stabilizer of the graph of linear functions over finite fields

In this paper we will study the action of $\mathbb{F}_{q^n}^{2 \times 2}$ on the graph of an $\mathbb{F}_q$-linear function of $\mathbb{F}_{q^n}$ into itself. In particular we will see that, under certain combinatorial assumptions, its stabilizer (together with the sum and product of matrices) is a field. We will also see some examples for which this does not happen. Moreover, we will establish a connection between such a stabilizer and the right idealizer of the rank-metric code defined by the linear function and give some structural results in the case in which the polynomials are partially scattered.

math.CO↗
arXiv2023

On a graph isomorphic to $NO^{+}(6,2)$

Let $Q^{+}(2n-1,2)$ be a non-degenerate hyperbolic quadric of $PG(2n-1,2)$. Let $NO^{+}(2n,2)$ be the tangent graph, whose vertices are the points of $PG(2n-1,2) \setminus Q^{+}(2n-1,2)$ and two vertices $u,~v$ are adjacent if the line joining $u$ and $v$ is tangent to $Q^{+}(2n-1,2)$. Then $NO^{+}(2n-1,q)$ is a strongly regular graph. Let $\mathcal{V}^{4}_{2}$ be the \textit{Veronese surface} in $PG(5,q)$, and $\mathcal{M}^{3}_{4}$ its \textit{secant variety}. When $q=2$, $|Q^{+}(5,2)|=|\mathcal{M}^{3}_{4}|=35$. In this paper we define the graph $N\mathcal{M}^{3}_{4}$, with 28 vertices in $PG(5,2)\setminus\mathcal{M}^{3}_{4}$ and with the analogue incidence rule of the tangent graph. Such graph is isomorphic to $NO^{+}(6,2)$.

math.CO↗
arXiv2022

All minimal $[9,4]_{2}$-codes are hyperbolic quadrics

Minimal codes are being intensively studied in last years. $[n,k]_{q}$-minimal linear codes are in bijection with strong blocking sets of size $n$ in $PG(k-1,q)$ and a lower bound for the size of strong blocking sets is given by $(k-1)(q+1)\leq n$. In this note we show that all strong blocking sets of length 9 in $PG(3,2)$ are the hyperbolic quadrics $Q^{+}(3,2)$.

math.CO↗
arXiv2022

On large partial ovoids of symplectic and Hermitian polar spaces

In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces ${\cal W}(3, q)$, $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal W}(5, q)$ and of the Hermitian polar spaces ${\cal H}(4, q^2)$, $q$ even or $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal H}(6, q^2)$, ${\cal H}(8, q^2)$.

math.CO↗
arXiv2022

The Automorphism Group of $NU(3,q^2)$

Let $H(n, q^2)$ be a non-degenerate Hermitian variety of $PG(n,q^2)$, $n \geq 2$. Let $NU(n+1,q^2)$ be the graph whose vertices are the points of $PG(n,q^2) \setminus H(n,q^2)$ and two vertices $u,~v$ are adjacent if the line joining $u$ and $v$ is tangent to $H(n, q^2 )$. Then $NU(n + 1, q^2)$ is a strongly regular graph. In this paper we show that the automorphism group of the graph $NU(3,q^2)$ is isomorphic either to $PΓU(3,q)$, the automorphism group of the projective unitary group $PGU(3,q)$, or to $S_{3} \wr S_{4}$, according as $q \neq 2$, or $q=2$.

math.CO↗
arXiv2021

New hemisystems of the Hermitian surface

Finding Hemisystems is a challenging problem and just few examples arising from the Hermitian surface are known. A recent method to obtain Hemisystems is based on using maximal curves. Along this side of research, we provide new examples of Hemisystems in $PG(3,p^2)$, for each prime of the form $p=1+4a^2$, with $ a $ integer. Last, we use these results to obtain two weight linear codes and strongly regular graphs.

math.CO↗