Luca Giuzzi

• DPHIL
• Professor (Associate) at University of Brescia

The long root geometry $A_{n,\{1,n\}}(\mathbb{K})$ for the special linear group $\mathrm{SL}(n+1,\mathbb{K})$ admits an embedding in the (projective space of) the vector space of the traceless square matrices of order $n+1$ with entries in the field $\mathbb{K}$, usually regarded as the {\em natural} embedding of $A_{n,\{1,n\}}(\mathbb{K})$. S. Smi...

Consider the point line-geometry ${\mathcal P}_t(n,k)$ having as points all the $[n,k]$-linear codes having minimum dual distance at least $t+1$ and where two points $X$ and $Y$ are collinear whenever $X\cap Y$ is a $[n,k-1]$-linear code having minimum dual distance at least $t+1$. We are interested in the collinearity graph $\Lambda_t(n,k)$ of ${\...

Let Γ(n,k) be the Grassmann graph formed by the k-dimensional subspaces of a vector space of dimension n over a field F and, for t∈N∖{0}, let Δt(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 |F|≥(nt) then Δt(n,k) is connected and it is isometrically embedded in Γ(n,...

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...

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 > 2 n . We also study generating sets for the 2-Grassmannians arising from qu...

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 $d=n-k+1$.

Let Γ be an embeddable non-degenerate polar space of finite rank n≥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 ε:Γ→PG(V) be the universal embedding of Γ. Let S be a subspace of Γ and...

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 num...

We provide a geometric construction of [n,9,n-9]q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[n,9,n-9]_q$$\end{document} near-MDS codes arising from elliptic curves...

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 univ...

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.

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.

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...

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...

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 $\Gamma(K) = Gr_J(X_n(K))$ be the $J$-Grassmannian of $X_n(K)$. We prove that $\Gamma(K)$ cannot be generated over any proper sub-division ring $K_0$...

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 f...

For any admissible value of the parameters there exist Maximum Rank distance (shortly MRD) $\mathbb{F}_{q^n}$-linear codes of $\mathbb{F}_q^{n\times n}$. It has been shown in \cite{H-TNRR} (see also \cite{ByrneRavagnani}) that, if field extensions large enough are considered, then \emph{almost all} (rectangular) rank distance codes are MRD. On the...

For any admissible value of the parameters n and k there exist [n,k]-Maximum Rank Distance F q -linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank metric codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present paper we study some...

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...

For any admissible value of the parameters $n$ and $k$ there exist $[n,k]$-Maximum Rank distance ${\mathbb F}_q$-linear codes. Indeed, it can be shown that if field extensions large enough are considered, almost all rank distance codes are MRD. On the other hand, very few families up to equivalence of such codes are currently known. In the present...

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 application...

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.

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 $\varphi$ 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 $k$-Grassman...

In this article we construct new minimal intersection sets in AG(r,q2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {AG}(r,q^2)$$\end{document} sporting three...

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 {\...

We introduce the class of transparent embeddings for a point-line geometry $\Gamma = ({\mathcal P},{\mathcal L})$ as the class of full projective embeddings $\varepsilon$ of $\Gamma$ such that the preimage of any projective line fully contained in $\varepsilon({\mathcal P})$ is a line of $\Gamma$. We will then investigate the transparency of Pl\"uc...

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-Gr...

We present a concise description of Orthogonal Polar Grassmann Codes and motivate their relevance. We also describe efficient encoding and decoding algorithms for the case of Line Grassmannians and introduce some open problems.

Polar Grassmann codes of orthogonal type have been introduced in [1]. They are sub-codes of the Grassmann code arising from the projective system defined by the Plücker embedding of a polar Grassmannian of orthogonal type. In the present paper we fully determine the minimum distance of line polar Grassmann Codes of orthogonal type for q odd. [Warni...

In this article we construct new minimal intersection sets in $AG(r,q^2)$ with respect to hyperplanes, of size $q^{2r-1}$ and multiplicity $t$, where $t\in \{ q^{2r-3}-q^{(3r-5)/2}, q^{2r-3}+q^{(3r-5)/2}-q^{(3r-3)/2}\}$, for $r$ odd or $t \in \{ q^{2r-3}-q^{(3r-4)/2}, q^{2r-3}-q^{r-2}\}$, for $r$ even. As a byproduct, for any odd $q$ we get a new f...

We introduce the Symplectic Grassmann codes as projective codes defined by symplectic Grassmannians, in analogy with the orthogonal Grassmann codes introduced in [4]. Note that the Lagrangian-Grassmannian codes are a special class of Symplectic Grassmann codes. We describe the weight enumerator of the Lagrangian--Grassmannian codes of rank $2$ and...

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 symplect...

We determine the possible intersection sizes of a Hermitian surface $\mathcal H$ with an irreducible quadric of ${\mathrm PG}(3,q^2)$ sharing at least a tangent plane at a common non-singular point when $q$ is even.

Let $\mathcal S$ be a Desarguesian $(t-1)$--spread of $PG(rt-1,q)$, $\Pi$ a $m$-dimensional subspace of $PG(rt-1,q)$ and $\Lambda$ the linear set consisting of the elements of $\mathcal S$ with non-empty intersection with $\Pi$. It is known that the Pl\"{u}cker embedding of the elements of $\mathcal S$ is a variety of $PG(r^t-1,q)$, say ${\mathcal...

In PG(3,q^2), with q odd, we determine the possible intersection sizes of a Hermitian surface H and an irreducible quadric Q having the same tangent plane at a common point P.

In this note we offer a short summary of some recent results, to be contained in a forthcoming paper [Cardinali, I., and L. Giuzzi, Caps and codes from orthogonal Grassmannians, submitted], on projective caps and linear error correcting codes arising from the Grassmann embedding εkgr of an orthogonal Grassmannian ΔkΔk. More precisely, we consider t...

In this paper we investigate linear error correcting codes and projective caps related to the Grassmann embedding $\varepsilon_k^{gr}$ of an orthogonal Grassmannian $\Delta_k$. In particular, we determine some of the parameters of the codes arising from the projective system determined by $\varepsilon_k^{gr}(\Delta_k)$. We also study special sets o...

Let Γ ' be a subgraph of a graph Γ. We define a down-link from a (K v ,Γ)-design ℬ to a (K n ,Γ ' )-design ℬ ' as a map f:ℬ→ℬ ' mapping any block of ℬ into one of its subgraphs. This is a new concept, closely related with both the notion of metamorphosis and that of embedding. In the present paper we study down-links in general and prove at any (K...

Let Gamma' be a subgraph of a graph Gamma. We define a down-link from a (K-nu, Gamma)-design B to a (K-n, Gamma')-design B' as a map f : B -> B' mapping any block of B into one of its subgraphs. This is a new concept, closely related with both the notion of metamorphosis and that of embedding. In the present paper we study down-links in general and...

In this note we introduce a computational approach to the construction of ovoids of the Hermitian surface and present some related experimental results.

Let UU be a unital embedded in the Desarguesian projective plane PG(2,q2). Write MM for the subgroup of PGL(3,q2) which preserves UU. We show that UU is classical if and only if UU has two distinct points P,QP,Q for which the stabiliser G=MP,QG=MP,Q has order q2−1q2−1.

Suppose Gamma' to be a subgraph of a graph Gamma. We define a sampling of a Gamma-design B = (V, B) into a Gamma'-design B' = (V. B') as a surjective map xi : B -> B' mapping each block of B into one of its subgraphs. A sampling will be called regular when the number of preimages of each block of B' under xi is a constant. This new concept is close...

Using geometric properties of the variety Vr,t, the image under the Grassmannian map of a Desarguesian (t − 1)-spread of PG (rt − 1, q), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We exactly determine the parameters of these codes and characterise the wor...

In (arXiv:1004.4127) the concept of down-link from a (K_v, G)-design B to a (K_n, G')-design B' has been introduced. In the present paper the spectrum problems for G'= P4 are studied. General results on the existence of path-decompositions and embeddings between path- decompositions playing a fundamental role for the construction of down-links are...

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(...

Let $\cU$ be a unital embedded in the Desarguesian projective plane $\PG(2,q^2)$. Write $M$ for the subgroup of $\PGL(3,q^2)$ which preserves $\cU$. We show that $\cU$ is classical if and only if $\cU$ has two distinct points $P,Q$ for which the stabiliser $G=M_{P,Q}$ has order $q^2-1$.

Let G' be a subgraph of a graph G. We define a down-link from a (K_v,G)-design B to a (K_n,G')-design B' as a map f:B->B' mapping any block of B into one of its subgraphs. This is a new concept, closely related with both the notion of metamorphosis and that of embedding. In the present paper we study down-links in general and prove that any (K_v,G)...

The main goal of coding theory is to devise efficient systems to exploit the full capacity of a communication channel, thus achieving an arbitrarily small error probability. Low Density Parity Check (LDPC) codes are a family of block codes–characterised by admitting a sparse parity check matrix–with good correction capabilities. In the present pape...

No regular hyperoval of the Desarguesian affine plane AG(2,2^(2h)), with h> 1, is inherited by a dual André plane of order 2^(2h) and dimension 2 over its kernel.

We present a new construction of non-classical unitals from a classical unital $U$ in $PG(2,q^2)$. The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model $\Pi$ of $PG(2,q^2)$ with the following three properties: 1. points of $\Pi$ are those of $PG(2,q^2)$; 2. lines of $\Pi$ are certain lines and conics of $PG(...

No regular hyperoval of the Desarguesian affine plane $AG(2,2^{2h})$, with $h>1$, is inherited by a dual Andr\'e plane of order $2^{2h}$ with dimension 2 over its centre.

In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order q 2. The construction relied upon the existence of a regular spread of tangent lines to an ovoid in PG(3, q) and, in particular, it does apply to the Suzuki–Tits ovoid. In this paper, we describe an algorithm for obtaining a possible representation of such...

An orthogonal array OA(q^{2n-1},q^{2n-2}, q,2) is constructed from the action of a subset of PGL(n+1,q^2) on some non--degenerate Hermitian varieties in PG(n,q^2). It is also shown that the rows of this orthogonal array correspond to some blocks of an affine design, which for q> 2 is a non--classical model of the affine space AG(2n-1,q). Comment: C...

A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set \(\mathcal{K}\) of the Desarguesian plane PG(2,q) is obtained. The case where \(\mathcal{K}\) is a maximal (k,n)-arc is considered in greater depth.

In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$. Comment: 8 Pages Minor typesetting changes; dedication

In this paper, we describe a procedure for constructing $q$--ary $[N,3,N-2]$--MDS codes, of length $N\leq q+1$ (for $q$ odd) or $N\leq q+2$ (for $q$ even), using a set of non--degenerate Hermitian forms in $PG(2,q^2)$.

Multiple derivation of the classical ovoid of the Hermitian surface H(3,q2)\mathcal{H}(3,q^2) of PG(3,q2)PG(3,q^2) is a well known, powerful method for constructing large families of non classical ovoids of H(3,q2)\mathcal{H}(3,q^2) . In this paper, we shall provide a geometric costruction of a family of ovoids amenable to multiple derivation.

We provide a description of the configuration arising from intersection of two Hermitian surfaces in PG(3, q), provided that the linear system they generate contains at least a degenerate variety.

Richiamiamo la definizione di grado per un polinomio in più variabili.

in questo capitolo introdurremo una famiglia di codici che generalizza, in modo diverso rispetto i codici di Reed Müller, la costruzione de Reed Solomon: i codici Algebrico Geometrici di Goppa. Tali lineari rivestono un ruolo importante per svariati motivi: 1. possono essere descritti in modo compatto, senza dover esplicitamente fornire una matrice...

Un [n, k] codice ciclico q] ario, come visto nel Capitolo 5, é univocamente individuato dal proprio polinomio geeratore g(x) ∈ Fqq [x] di grado n—k

A partire da questo paragrafo, per ragioni che saranno chiare in seguito, le componenti dei vettori vengono sempre enumerate a partire da 0.

In questo paragrafo introdurremo il codice binario di Golay esteso e mostreremo come esso sia legato ad un sistema di Steiner particolarmente interessante.

Nel capitolo 15, si è investigato il comportamento dei codici lineari al crescere della dei blocchi. Nel paragrafo 15.3, in particolare, si è visto che “quasi tutti” i codici lineari, purché sufficientemente lunghi, abbiano, a livello teorico, una capacità correttiva non inferiore a quella descritta nella limitazione di Gilbert-Varshamov. D’altro c...

Ci sono, essenzialmente, due approcci per presentare i codici di Reed-Solomon: uno è basato sulla nozione di valutazione di polinomi; l’altro, sfrutta la costruzione BCH. In questo paragrafo seguiremo il primo approccio; nel successivo Paragrafo 9.5 vedremo l’equivalenza con il secondo.

Definizione 17.1. Una matrice H si dice dv, dc)-sparsa se gode delie seguenti due proprietà strutturali: 1. ogni riga contiene al più dc entrate diverse da 0; 2. ogni colonna contiene al più dv entrate diverse da 0.

In questo capitolo si consideraranna codici lineari costruiti a partire da strutture di incidenza dotate di una certa regolarità, quali i desegni. I codici cosi costruiti riffelettono bene le proprietà della struttura originarial pertano, risulta possibile studiarli sfruttando anche metodi geometrici. Come referenza per gli argomenti qui trattati,...

L’operazione più semplice di modifica di un codice è l’accorciamento: esso consiste nel cancellare delle posizioni di informazione in ogni parola di un codice.

In un codice a blocchi, il flusso di dati da trasmettere viene codificato in unità discrete, ognuna contenente n caratteri, ove n è preassegnato: i blocchi. La proprietà fondamentale di questo tipo di codici è che ogni blocco è codificato e decodificato in modo indipendente e autonomo sia da quelli che lo precedono che da quelli che lo seguono. In...

In un sistema di commuinicazione sono spesso disponibili informazioni sulla quallità de; segnale in ricezione. In practicolare, può risultare possibile sapere a priori, prima di iniziare la decodifica vera e propria dei messaggi, che alcuni carattri ricevuti sono sicuramente non affidabili. Qeuesto corrisponde concretamente ad avere informazioni su...

Il modello di sistema di communicaziobe di Shannon è stato originariamente ispirato dal funzionemento del telegrafo, ma prescinde da come i segnali vengono effettivamente trasmessi su di un canale di communicazione. In particolare, per poter concretamente comunicare, si rivela indispensabile difinire preventivamente in che modo i segnali trasmessi...

L'atto di nascita ufficiale della teoria della commnnicazione è l'articolo [89]. Nelle parole di Shannon “il problema fondamentale della communicazione consiste den riprodurre in un punto esattamente (o cobe [buona]approssimazione) un messaggio preparato in un altro punto” Il reallizare un sistema che raggiunga questo obiettivo è il fine ultimo del...

L’alfabeto A su cui un codice correttore è definito è, a priori, solamente un insieme finito di simboli, senza alcuna ipotesi su sue ulteriori proprietà. Casi interessanti si verificano quando su A sono definite delle operazioni algebriche; in particolare, si rivela molto utile investigare il caso in cui A sia quantomeno un gruppo. In questo capito...

La distanza minima di u codice fornisce, in prima istanza, un'indicaziobe delle sue possibili capacità correttive. Pertanto, assegnato un problema concreto e formulate delle ipotesi sul numero di errori che è lecito attendersi, si rivela fondamentale poster costruire un codice abbia distanza minima preassegnata. La costruzione BCH, dovutta a R. C....

La nozione di errore concentrato, o burst di errore, è strettamente legata a quella di catena ciclica, già introdotta nella Definizione 5.9.

Le carratteristiche principali di un codice correttore sono, quantomeo in prima approssimazione, descritte da due fondamcentali quantià: la ridondanza r e l'effizcienza R. Come visto nel Captolo 3, la prima fornisce una limitazione superiore alla massina distanza minima (e dunque alla massima capacità correttiva garantita); la seconda indica quanto...

We construct a new ovoid of the polar space arising from the Hermitian surface of PGð3; q2Þ with qd 5 odd. The automorphism group G of such an ovoid has a normal cyclic subgroup F of order 1 2 ðq þ 1Þ such that G=F G PGLð2; qÞ. Furthermore, G has three orbits on the ovoid, one of size q þ 1 and two of size 1 2 qðq � 1Þðq þ 1Þ.

In this work an infinite family of K-loops is constructed from the reflection structure of co-Minkowski planes and their properties are analysed.

A short proof is given for the main result of [1]:¶¶THEOREM 1. A unital in PG(2,q) is classical if and only if it is preserved by a cyclic linear collineation group of order \( q - \sqrt{q} +1 \).

Kestenband proved in (12) that there are only seven pairwise non-isomorphic Hermitian intersections in the desarguesian projective plane PGÖ2; qÜ of square order q. His classification is based on the study of the minimal polynomials of the matrices associated with the curves and leads to results of purely combinatorial nature: in fact, two Hermitia...

Curves over finite fields not only are interesting structures in themselves, but they are also remarkable for their application to coding theory and to the study of the geometry of arcs in a finite plane. In this note, the basic properties of curves and the number of their points are recounted. Preface These notes were inspired by the lectures give...