arXiv subjects

Ilaria Cardinali

Publications and source records attributed to Ilaria Cardinali.

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↗

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

On the generation of some Lie-type geometries

Let $X_n(K)$ be a building of Coxeter type $X_n = A_n$ or $X_n = D_n$ defined over a given division ring $K$ (a field when $X_n = D_n$). For a non-connected set $J$ of nodes of the diagram $X_n$, let $Γ(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $Γ(K)$ cannot be generated over any proper sub-division ring $K_0$ of $K$. As a consequence, the generating rank of $Γ(K)$ is infinite when $K$ is not finitely generated. In particular, if $K$ is the algebraic closure of a finite field of prime order then the generating rank of $Gr_{1,n}(A_n(K))$ is infinite, although its embedding rank is either $(n+1)^2-1$ or $(n+1)^2$.

math.CO↗

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 Grassmann Graph of Linear Codes

Let $Γ(n,k)$ 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 $Δ_t(n,k)$ be the subgraph of $Γ(n,k)$ 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 $Δ_t(n,k)$ is connected and it is isometrically embedded in $Γ(n,k)$. 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↗

arXiv2020

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

Let $Γ$ be an embeddable non-degenerate polar space of finite rank $n \geq 2$. Assuming that $Γ$ 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 $\varepsilon:Γ\to\mathrm{PG}(V)$ be the universal embedding of $Γ$. Let $\cal S$ be a subspace of $Γ$ 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↗

arXiv2019

Grassmann embeddings of polar Grassmannians

In this paper we compute the dimension of the Grassmann embeddings of the polar Grassmannians associated to a possibly degenerate Hermitian, alternating or quadratic form with possibly non-maximal Witt index. Moreover, in the characteristic $2$ case, when the form is quadratic and non-degenerate with bilinearization of minimal Witt index, we define a generalization of the so-called Weyl embedding (see [I. Cardinali and A. Pasini, Grassmann and Weyl embeddings of orthogonal Grassmannians. J. Algebr. Combin. 38 (2013), 863-888]) and prove that the Grassmann embedding is a quotient of this generalized "Weyl-like" embedding. We also estimate the dimension of the latter.

math.AG↗

arXiv2019

The generating rank of a polar Grassmannian

In this paper we compute the generating rank of $k$-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of $k$-Grassmannians arising from Hermitian forms of Witt index $n$ defined over vector spaces of dimension $N > 2n$. We also study generating sets for the $2$-Grassmannians arising from quadratic forms of Witt index $n$ defined over $V(N,{\mathbb F}_q)$ for $q=4,8,9$ and $2n \leq N \leq 2n+2$. We prove that for $N >6$ they can be generated over the prime subfield, thus determining their generating rank.

math.RT↗

arXiv2018

Enumerative Coding for Line Polar Grassmannians with applications to codes

A $k$-polar Grassmannian is the geometry having as pointset the set of all $k$-dimensional subspaces of a vector space $V$ which are totally isotropic for a given non-degenerate bilinear form $μ$ defined on $V.$ Hence it can be regarded as a subgeometry of the ordinary $k$-Grassmannian. In this paper we deal with orthogonal line Grassmannians and with symplectic line Grassmannians, i.e. we assume $k=2$ and $μ$ a non-degenerate symmetric or alternating form. We will provide a method to efficiently enumerate the pointsets of both orthogonal and symplectic line Grassmannians. This has several nice applications; among them, we shall discuss an efficient encoding/decoding/error correction strategy for line polar Grassmann codes of both types.

cs.IT↗

arXiv2018

On transparent embeddings of point-line geometries

We introduce the class of transparent embeddings for a point-line geometry $Γ= ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $Γ$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $Γ$. We will then investigate the transparency of Plücker embeddings of projective and polar grassmannians and spin embeddings of half-spin geometries and dual polar spaces of orthogonal type. As an application of our results on transparency, we will derive several Chow-like theorems for polar grassmannians and half-spin geometries.

math.AG↗

arXiv2018

Minimum distance of Line Orthogonal Grassmann Codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for $q$ even. The case $q$ odd was solved in "I. Cardinali, L. Giuzzi, K. Kaipa, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, J. Pure Applied Algebra." We also show that for $q$ even all minimum weight codewords are equivalent and that symplectic line-Grassmann codes are proper subcodes of codimension $2n$ of the orthogonal ones.

math.CO↗

arXiv2018

Implementing Line-Hermitian Grassmann codes

In [I. Cardinali and L. Giuzzi. Line Hermitian Grassmann codes and their parameters. Finite Fields Appl., 51: 407-432, 2018] we introduced line Hermitian Grassmann codes and determined their parameters. The aim of this paper is to present (in the spirit of [I. Cardinali and L. Giuzzi. Enumerative coding for line polar Grassmannians with applications to codes. Finite Fields Appl., 46:107-138, 2017]) an algorithm for the point enumerator of a line Hermitian Grassmannian which can be usefully applied to get efficient encoders, decoders and error correction algorithms for the aforementioned codes.

math.CO↗

arXiv2017

Line Hermitian Grassmann Codes and their Parameters

In this paper we introduce and study line Hermitian Grassmann codes as those subcodes of the Grassmann codes associated to the $2$-Grassmannian of a Hermitian polar space defined over a finite field of square order. In particular, we determine their parameters and characterize the words of minimum weight for $m\neq5$.

math.CO↗

arXiv2017

Geometries arising from trilinear forms on low-dimensional vector spaces

Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J. Algebraic Combin. doi:10.1007/s10801-016-0730-6] a point-line subgeometry of ${\mathrm{PG}}(V)$ called the {\it geometry of poles of $H$}. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for $k=3$ and $n\leq 7$ and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi:10.1016/j.laa.2010.03.040] regarding the existence of line spreads of ${\mathrm{PG}}(5,{\mathbb K})$ arising from hyperplanes of ${\mathcal G}_3(V).$

math.AG↗

arXiv2016

A geometric approach to alternating $k$-linear forms

Given an $n$-dimensional vector space $V$ over a field $\mathbb K$, let $2\leq k < n$. There is a natural correspondence between the alternating $k$-linear forms $φ$ of $V$ and the linear functionals $f$ of $\bigwedge^kV$. Let $\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)$ be the Plucker embedding of the $k$-Grassmannian ${\mathcal G}_k(V)$ of $V$. Then $\varepsilon_k^{-1}(\ker(f)\cap\varepsilon_k(\mathcal{G}_k(V)))$ is a hyperplane of the point-line geometry ${\mathcal G}_k(V)$. All hyperplanes of ${\mathcal G}_k(V)$ can be obtained in this way. For a hyperplane $H$ of ${\mathcal G}_k(V)$, let $R^\uparrow(H)$ be the subspace of ${\mathcal G}_{k-1}(V)$ formed by the $(k-1)$-subspaces $A\subset V$ such that $H$ contains all $k$-subspaces that contain $A$. In other words, if $φ$ is the (unique modulo a scalar) alternating $k$-linear form defining $H$, then the elements of $R^\uparrow(H)$ are the $(k-1)$-subspaces $A = \langle a_1,\ldots, a_{k-1}\rangle$ of $V$ such that $φ(a_1,\ldots, a_{k-1},x) = 0$ for all $x\in V$. When $n-k$ is even it might be that $R^\uparrow(H) = \emptyset$. When $n-k$ is odd, then $R^\uparrow(H) \neq \emptyset$, since every $(k-2)$-subspace of $V$ is contained in at least one member of $R^\uparrow(H)$. If every $(k-2)$-subspace of $V$ is contained in precisely one member of $R^\uparrow(H)$ we say that $R^\uparrow(H)$ is spread-like. In this paper we obtain some results on $R^\uparrow(H)$ which answer some open questions from the literature and suggest the conjecture that, if $n-k$ is even and at least $4$, then $R^\uparrow(H) \neq \emptyset$ but for one exception with ${\mathbb K}\leq{\mathbb R}$ and $(n,k) = (7,3)$, while if $n-k$ is odd and at least $5$ then $R^\uparrow(H)$ is never spread-like.

math.AG↗