arXiv subjects

Luca Giuzzi

Publications and source records attributed to Luca Giuzzi.

At least 19 recordsLinked to original sources

arXiv2025

On minimal codes arising from projective embeddings of point-line geometries

Let ${\mathcal C}(\Omega)$ be the linear code arising from a projective system $\Omega$ of $\mathrm{PG}(V).$ Consider the point-line geometry $\Gamma=({\mathcal P},{\mathcal L})$ and a projective embedding $\varepsilon\colon \Gamma\rightarrow \mathrm{PG}(V)$ of $\Gamma.$ We show that the projective code obtained by taking as projective system $\Omega:=\varepsilon(\mathcal{P})$ is minimal if the graph induced on the set $\Gamma\setminus\varepsilon^{-1}(H)$ by the collinearity graph of $\Gamma$ is connected for any hyperplane $H$ of $\mathrm{PG}(V)$. As an application, we prove that Grassmann codes, Segre codes, line polar Grassmann codes of orthogonal, symplectic, hermitian type, codes arising from dual polar spaces of orthogonal and symplectic type and codes arising from the point-hyperplane geometry of a projective space are minimal codes.

math.CO↗

arXiv2025

Linear codes arising from the point-hyperplane geometry -- Part II: the twisted embedding

Let $\bar{\Gamma}$ be the point-hyperplane geometry of a projective space $\mathrm{PG(V)},$ where $V$ is a $(n+1)$-dimensional vector space over a finite field $\mathbb{F}_q$ of order $q.$ Suppose that $\sigma$ is an automorphism of $\mathbb{F}_q$ and consider the projective embedding $\varepsilon_{\sigma}$ of $\bar{\Gamma}$ into the projective space $\mathrm{PG}(V\otimes V^*)$ mapping the point $([x],[\xi])\in \bar{\Gamma}$ to the projective point represented by the pure tensor $x^{\sigma}\otimes \xi$, with $\xi(x)=0.$ In [I. Cardinali, L. Giuzzi, Linear codes arising from the point-hyperplane geometry -- part I: the Segre embedding (Jun. 2025). arXiv:2506.21309, doi:10.48550/ARXIV.2506.21309] we focused on the case $\sigma=1$ and we studied the projective code arising from the projective system $\Lambda_1=\varepsilon_{1}(\bar{\Gamma}).$ Here we focus on the case $\sigma\not=1$ and we investigate the linear code ${\mathcal C}(\Lambda_{\sigma})$ arising from the projective system $\Lambda_{\sigma}=\varepsilon_{\sigma}(\bar{\Gamma}).$ In particular, after having verified that $\mathcal{C}( \Lambda_{\sigma})$ is a minimal code, we determine its parameters, its minimum distance as well as its automorphism group. We also give a (geometrical) characterization of its minimum and second lowest weight codewords and determine its maximum weight when $q$ and $n$ are both odd.

math.CO↗

arXiv2025

Linear codes arising from the point-hyperplane geometry-Part I: the Segre embedding

Let $V$ be a vector space over the finite field $\mathbb{F}_q$ with $q$ elements and $\Lambda$ be the image of the Segre geometry $\mathrm{PG}(V)\otimes\mathrm{PG}(V^*)$ in $\mathrm{PG}(V\otimes V^*)$. Consider the subvariety $\Lambda_{1}$ of $\Lambda$ represented by the pure tensors $x\otimes \xi$ with $x\in V$ and $\xi\in V^*$ such that $\xi(x)=0$. Regarding $\Lambda_1$ as a projective system of $\mathrm{PG}(V\otimes V^*)$, we study the linear code $\mathcal{C}(\Lambda_1)$ arising from it. The code $\mathcal{C}(\Lambda_1)$ is minimal code and we determine its basic parameters, itsfull weight list and its linear automorphism group. We also give a geometrical characterization of its minimum and second lowest weight codewords as well as of some of the words of maximum weight.

math.CO↗

arXiv2025

On the $1$-cohomology of $\mathrm{SL}(n,{\mathbb K})$ on the dual of its adjoint module

Given a field $\mathbb K$, for any $n\geq 3$ the first cohomology group $H^1(G_n,A^*_n)$ of the special linear group $G_n = \mathrm{SL}(n,{\mathbb K})$ over the dual $A^*_n$ of its adjoint module $A_n$ is isomorphic to the space $\mathrm{Der}({\mathbb K})$ of the derivations of $\mathbb K$, except possibly when $|{\mathbb K}| \in \{2, 4\}$ and $n$ is even. This fact is stated by S. Smith and H. V\"{o}lklein in their paper "A geometric presentation for the adjont module of $\mathrm{SL}_3(k)$" (J. Algebra 127 (1989), 127--138). They claim that when $|{\mathbb K}| > 9$ this fact follows from the main result of V\"{o}lklein's paper "The 1-cohomology of the adjoint module of a Chevalley group" (Forum Math. 1 (1989), 1--13), but say nothing that can help the reader to deduce it from that result. When $|{\mathbb K}| \leq 9$ they obtain the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ by means of other results from homological algebra, which however miss the case $|{\mathbb K}| \in\{2, 4\}$ with $n $ even. In the present paper we shall provide a straightforward proof of the isomorphism $H^1(G_n,A^*_n) \cong \mathrm{Der}({\mathbb K})$ under the hypothesis $n > 3$. Our proof also covers the above mentioned missing case.

math.GR↗

arXiv2025

Minimal codes from hypersurfaces in even characteristic

The setting of projective systems can be used to study the parameters of a projective linear code $\mathcal{C}$. This can be done by considering the intersections of the point set $\Omega$ defined by the columns of a generating matrix for $\mathcal{C}$ with the hyperplanes of a projective space. In particular, $\mathcal{C}$ is minimal if $\Omega$ is cutting, i.e., every hyperplane is spanned by its intersection with $\Omega$. Minimal linear codes have important applications for secret sharing schemes and secure two-party computation. In this article we first investigate the properties of some algebraic hypersurfaces $\mathcal{V}_{\varepsilon}^r$ related to certain quasi-Hermitian varieties of $\mathrm{PG}(r,q^2)$, with $q=2^e$, $e>1$ odd. These varieties give rise to a new infinite family of linear codes which are minimal except for $r=3$ and $e\equiv 1 \pmod 4$. In the case $r \in \{3,4\}$, we exhibit codes having at most 6 non-zero weights whose we provide the complete list. As a byproduct, we obtain $(r+1)$-dimensional codes with just $3$ non-zero weights. We point out that linear codes with few weights are also important in authentication codes and association schemes. In the last part of the paper we consider an extension of the notion of being cutting with respect to subspaces other than hyperplanes and introduce the definition of cutting gap in order to characterize and measure what happens when this property is not satisfied. Finally, we then apply these notions to Hermitian codes and to the codes related to $\mathcal{V}_\varepsilon^r$ discussed before.

math.CO↗

arXiv2024

Nash Equilibria in Traffic Networks with Multiple Populations and Origins-Destinations

Different populations of vehicles travel along a network. Each population has its origin, destination and travel costs - which may well be unbounded. Under the only requirement of the continuity of the travel costs, we prove the existence of a Nash equilibrium for all populations. Conditions for its uniqueness are also provided. A few cases are treated in detail to show specific situations of interest.

math.OC↗

arXiv2024

On mutually $\mu$-intersecting quasi-Hermitian varieties with some applications

Let $\mathcal W$ be a non-empty set of points of a finite Desarguesian projective space $\mathrm{PG}(n,q)$. A collection of varieties of $\mathrm{PG}(n,q)$ is \emph{mutually $\mu$-intersecting (relatively to $\mathcal W$)} if its elements meet all $\mathcal W$ in the same number of points and pairwise intersect in $\mathcal W$ in exactly $\mu$-points. Here we construct a new family of mutually $\mu$-intersecting algebraic varieties by using certain quasi-Hermitian varieties of $\mathrm{PG}(n,q^2)$ where $q$ is any prime power. With the help of these quasi-Hermitian varieties we provide a new construction of $5$-dimensional MDS codes over ${\mathbb F}_q$ as well as an infinite family of simple orthogonal arrays $OA(q^{2n-1},q^{2n-2},q,2)$ of index $\mu=q^{2n-3}$.

math.CO↗

arXiv2023

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. 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.

math.CO↗

arXiv2023

On regular sets of affine type in finite Desarguesian planes and related codes

In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Sz\H{o}nyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of $\mathrm{PG}(2, q^2)$ in one of $4$ possible intersection numbers, each of them congruent to $1$ modulo $\sqrt{q}$. As a byproduct, we determine the intersection sizes of the Hermitian curve defined over $\mathrm{GF}(q^2)$ with suitable rational curves of degree $\sqrt{q}$ and we obtain $\sqrt{q}$-divisible codes with $5$ non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some $q$-powers.

math.CO↗

arXiv2023

On orthogonal polar spaces

Let $\cal P$ be a non-degenerate polar space. In [I. Cardinali, L. Giuzzi, A. Pasini, "The generating rank of a polar grassmannian", Adv. Geom. 21:4 (2021), 515-539 doi:10.1515/advgeom-2021-0022 (arXiv:1906.10560)] we introduced an intrinsic parameter of $\cal P$, called the anisotropic gap, defined as the least upper bound of the lengths of the well-ordered chains of subspaces of $\cal P$ containing a frame; when $\cal P$ is orthogonal, we also defined two other parameters of $\cal P$, called the elliptic and parabolic gap, related to the universal embedding of $\cal P$. In this paper, assuming $\cal P$ is an orthogonal polar space, we prove that the elliptic and parabolic gaps can be described as intrinsic invariants of $\cal P$ without making recourse to the embedding.

math.RT↗

arXiv2022

Characterizations of symplectic polar spaces

A polar space S is said to be symplectic if it admits an embedding e in a projective geometry PG(V) such that the e-image e(S) of S is defined by an alternating form of V. In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when S admits different (non isomorphic) embeddings, as it is the case (precisely) when S is defined over a field of characteristic 2.

math.SG↗

arXiv2021

On the equivalence of certain quasi-Hermitian varieties

In [A. Aguglia, A. Cossidente, G. Korchmaros, "On quasi-Hermitian varieties", J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties ${\mathcal M}_{\alpha,\beta}$ in $\mathrm{PG}(r,q^2)$ depending on a pair of parameters $\alpha,\beta$ from the underlying field $\mathrm{GF}(q^2)$ have been constructed. In the present paper we determine the projective equivalence classes of such varieties for $r=3$ and $q$ odd.

math.AG↗

arXiv2021

A remark on ${\mathbb F}_{q^n}$-Linear MRD codes

In this note, we provide a description of the elements of minimum rank of a generalized Gabidulin code in terms of Grassmann coordinates. As a consequence, a characterization of linearized polynomials of rank at most $n-k$ is obtained, as well as parametric equations for MRD-codes of distance $d=n-k+1$.

cs.IT↗

arXiv2021

Some hypersurfaces over finite fields, minimal codes and secret sharing schemes

Linear error-correcting codes can be used for constructing secret sharing schemes; however finding in general the access structures of these secret sharing schemes and, in particular, determining efficient access structures is difficult. Here we investigate the properties of certain algebraic hypersurfaces over finite fields, whose intersection numbers with any hyperplane only takes a few values; these varieties give rise to $q$-divisible linear codes with at most $5$ weights. Furthermore, for $q$ odd these codes turn out to be minimal and we characterize the access structures of the secret sharing schemes based on their dual codes. Indeed, the secret sharing schemes thus obtained are democratic, that is each participant belongs to the same number of minimal access sets and can easily be described.

cs.IT↗

arXiv2020

Nearly all subspaces of a classical polar space arise from its universal embedding

Let $\Gamma$ be an embeddable non-degenerate polar space of finite rank $n \geq 2$. Assuming that $\Gamma$ admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least $5$ and certain generalized quadrangles defined over quaternion division rings), let $\varepsilon:\Gamma\to\mathrm{PG}(V)$ be the universal embedding of $\Gamma$. Let $\cal S$ be a subspace of $\Gamma$ and suppose that $\cal S$, regarded as a polar space, has non-degenerate rank at least $2$. We shall prove that $\cal S$ is the $\varepsilon$-preimage of a projective subspace of $\mathrm{PG}(V)$.

math.RT↗

arXiv2020

Near-MDS codes from elliptic curves

We provide a new construction of $[n,9,n-9]_q$ near-MDS codes arising from elliptic curves with $n$ ${\mathbb F}_q$-rational points. Furthermore we show that in some cases these codes cannot be extended to longer near-MDS codes.

math.CO↗

arXiv2020

On Hermitian varieties in $\mathrm{PG}(6,q^2)$

In this paper we characterize the non-singular Hermitian variety ${\mathcal H}(6,q^2)$ of $\mathrm{PG}(6, q^2)$, $q\neq2$ among the irreducible hypersurfaces of degree $q+1$ in $\mathrm{PG}(6, q^2)$ not containing solids by the number of its points and the existence of a solid $S$ meeting it in $q^4+q^2+1$ points.

math.CO↗

arXiv2020

On the Grassmann Graph of Linear Codes

Let $\Gamma(n,k)$ be the Grassmann graph formed by the $k$-dimensional subspaces of a vector space of dimension $n$ over a field $\mathbb F$ and, for $t\in \mathbb{N}\setminus \{0\}$, let $\Delta_t(n,k)$ be the subgraph of $\Gamma(n,k)$ formed by the set of linear $[n,k]$-codes having minimum dual distance at least $t+1$. We show that if $|{\mathbb F}|\geq{n\choose t}$ then $\Delta_t(n,k)$ is connected and it is isometrically embedded in $\Gamma(n,k)$. This generalizes some results of [M. Kwiatkowski, M. Pankov, "On the distance between linear codes", Finite Fields Appl. 39 (2016), 251--263] and [M. Kwiatkowski, M. Pankov, A. Pasini, "The graphs of projective codes" Finite Fields Appl. 54 (2018), 15--29].

math.CO↗