Research

Topics

My research interests include:

Preprints and papers

A full pdf list can be found here.

Accepted papers

  1. I. Cardinali, L. Giuzzi, Enumerative Coding for Line Polar Grassmannians with applications to codes, Finite Fields Appl. 46 (2017) 107-138, doi:10.1016/j.ffa.2017.03.005 (arXiv:1412.5466)
    Abstract: 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 \(\mu\) 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 \(\mu\) 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.
  2. I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating \(k\)-linear forms, to appear on J. Algebraic Combin., doi:10.1007/s10801-016-0730-6 (arXiv:1601.08115)
    Abstract: Given an \(n\)-dimensional vector space \(V\) over a field \({\mathbb K}\), let \(2\leq k < n\). A natural one-to-one correspondence exists between the alternating \(k\)-linear forms of \(V\) and the linear functionals of \(\bigwedge^kV\), an alternating \(k\)-linear form \(\varphi\) and a linear functional \(f\) being matched in this correspondence precisely when \(\varphi(x_1,\ldots, x_k) = f(x_1\wedge\cdots\wedge x_k)\) for all \(x_1,\ldots, x_k \in V\). Let \(\varepsilon_k:{\mathcal G}_k(V)\rightarrow {\mathrm{PG}}(\bigwedge^kV)\) be the Pl\"{u}cker 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 \({\mathcal G}_k(V)\). It is well known that all hyperplanes of \({\mathcal G}_k(V)\) can be obtained in this way, namely every hyperplane of \({\mathcal G}_k(V)\) is the family of \(k\)-subspaces of \(V\) where a given alternating \(k\)-linear form identically vanishes. For a hyperplane \(H\) of \({\mathcal G}_k(V)\), let \(R^\uparrow(H)\) be the subset (in fact a 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 \(\varphi\) 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 \(\varphi(a_1,\ldots, a_{k-1},x) = 0\) for all \(x\in V\). In principle, when \(n-k\) is even it might happen 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 \(R^\uparrow(H)\), but it can happen that every \((k-2)\)-subspace of \(V\) is contained in precisely one member of \(R^\uparrow(H)\). If this is the case, we say that \(R^\uparrow(H)\) is \emph{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 \(R^\uparrow(H) \not= \emptyset\) 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.
  3. A. Aguglia, L. Giuzzi, Intersection sets, three-character multisets and associated codes, Des. Codes Cryptogr. 83:269-282 (2017),doi:10.1007/s10623-016-0302-8 (arXiv:1504.00503)
    Abstract: In this article we construct new minimal intersection sets in \({\mathrm{AG}}(r,q^2)\) sporting three intersection numbers with hyperplanes; we then use these sets to obtain linear error correcting codes with few weights, whose weight enumerator we also determine. Furthermore, we provide a new family of three-character multisets in \({\mathrm{PG}}(r,q^2)\) with \(r\) even and we also compute their weight distribution.
  4. A. Aguglia, L. Giuzzi, Intersections of the Hermitian Surface with irreducible Quadrics in even Characteristic, Electron. J. of Combin. 23 (4): P4.13 (2016),(arXiv:1407.8498)
    Abstract: 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.
  5. I. Cardinali, L. Giuzzi, K.V. Kaipa, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, J. Pure Appl. Algebra 220 (5): 1924-1934 (2016), doi:10.1016/j.jpaa.2015.10.007
    Abstract: Polar Grassmann codes of orthogonal type have been introduced in [19]. They are subcodes 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.
  6. I. Cardinali, L. Giuzzi, Minimum distance of Symplectic Grassmann codes, Linear Algebra Appl. 488: 124-134 (2016), doi:10.1016/j.laa.2015.09.031 (arXiv:1503.05456)
    Abstract:In this paper we introduce symplectic Grassmann codes, in analogy to ordinary Grassmann codes and orthogonal Grassmann codes, as projective codes defined by symplectic Grassmannians. Lagrangian-Grassmannian codes are a special class of symplectic Grassmann codes. We describe all the parameters of line symplectic Grassmann codes and we provide the full weight enumerator for the Lagrangian-Grassmannian codes of rank \(2\) and \(3\).
  7. L. Giuzzi, V. Pepe, On some subvarieties of the Grassmann variety, Linear Multilinear Algebra 63 (11): 2121-2134 (2015), ISSN 0308-1087, doi:10.1080/03081087.2014.983449 (arXiv:1405.6926)
    Abstract: Let \(\mathcal S\) be a Desarguesian \((t-1)\)-spread of \(\mathrm{PG}(rt-1,q)\), \(\Pi\) a \(m\)--dimensional subspace of \(\mathrm{PG}(rt-1,q)\) and \(\Lambda\) the linear set consisting of the elements of \(\mathcal S\) with non-empty intersection with \(\Pi\). It is known that the Plücker embedding of the elements of \(\mathcal S\) is a variety of \(\mathrm{PG}(r^t-1,q)\), say \({\mathcal V}_{rt}\). In this paper, we describe the image under the Plücker embedding of the elements of \(\Lambda\) and we show that it is an \(m\)-dimensional algebraic variety, projection of a Veronese variety of dimension \(m\) and degree \(t\), and it is a suitable linear section of \({\mathcal V}_{rt}\).
  8. A. Aguglia, L. Giuzzi, Intersections of the Hermitian surface with irreducible quadrics in \(PG(3,q^2)\), \(q\) odd, Finite Fields Appl. 30: 1-13 (2014), ISSN: 1071-5797, doi:10.1016/j.ffa.2014.05.005 (arXiv:1307.8386)
    Abstract: In \(\mathrm{PG}(3,q^2)\), with \(q\) odd, we determine the possible intersection sizes of a Hermitian surface \(\mathcal H\) and an irreducible quadric \(\mathcal Q\) having the same tangent plane \(\pi\) at a common point \( P\in{\mathcal Q}\cap{\mathcal H}\).
  9. I. Cardinali, L. Giuzzi, Codes and caps from orthogonal Grassmannians, Finite Fields Appl. 24: 148-169 (2013), ISSN: 1071-5797, doi:10.1016/j.ffa.2013.07.003 (arXiv:1303.5636)
    Abstract: 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 of points of \(\Delta_k\) which are met by any line of \(\Delta_k\) in at most \(2\) points and we show that their image under the Grassmann embedding \(\varepsilon_k^{gr}\) is a projective cap.
  10. L. Giuzzi, V. Pepe, Families of twisted tensor product codes, Des. Codes Cryptogr. 67: 375-384 (2013), doi:10.1007/s10623-012-9613-6 (arXiv:1107.1066)
    Abstract: Using geometric properties of the variety \({\mathcal V}_{r,t}\), the image under the Grassmannian map of a Desarguesian \((t-1)\)-spread of \(\mathrm{PG}(rt-1,q)\), we introduce error correcting codes related to the twisted tensor product construction, producing several families of constacyclic codes. We determine the precise parameters of these codes and characterise the words of minimum weight.
  11. A. Benini, L. Giuzzi, A. Pasotti, Down-linking \((K_v,\Gamma)\)-designs to \(P_3\)-designs, Util. Math., 90: 3-21 (2013), ISSN: 0315-3681 (arXiv:1004.4127)
    Abstract: Let \(\Gamma'\) be a subgraph of a graph \(\Gamma\). We define a down-link from a \((K_v,\Gamma)\)-design \(\mathcal B\) to a \((K_n,\Gamma')\)-design \({\mathcal B}'\) as a map \(f:{\mathcal B}\to {\mathcal B}'\) mapping any block of \(\mathcal B\) 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 \((K_v,\Gamma)\)-design might be down-linked to a \((K_n,\Gamma')\)-design, provided that \(n\) is admissible and large enough. We also show that if \(\Gamma'=P_3\), it is always possible to find a down-link to a design of order at most \(v+3\). This bound is then improved for several classes of graphs \(\Gamma\), by providing explicit constructions.
  12. L. Giuzzi, G. Korchmáros, Unitals in \(PG(2,q^2)\) with a large 2-point stabiliser, Discrete Math. 312 (3): 532-535 (2012), ISSN: 0012-365X, doi:10.1016/j.disc.2011.03.017 (arXiv:1009.6109)
    Abstract: Let \(\mathcal U\) be a unital embedded in the Desarguesian projective plane \(\mathrm{PG}(2,q^2)\). Write \(M\) for the subgroup of \(\mathrm{PGL}(3,q^2)\) which preserves \(\mathcal U\). We show that \(\mathcal U\) is classical if and only if \(\mathcal U\) has two distinct points \(P,Q\) for which the stabiliser \(G=M_{P,Q}\) has order \(q^2-1\).
  13. L. Giuzzi, A. Pasotti, Sampling complete graphs, Discrete Math. 312 (3): 488-497 (2012), ISSN: 0012-365X, doi:10.1016/j.disc.2011.02.034 (arXiv:0907.3199)
    Abstract: In the present paper, complete designs of graphs are considered. The notion of (regular) sampling is introduced and analyzed in detail, showing that the trivial necessary condition for its existence is actually sufficient. Some examples are also provided.
  14. A. Aguglia, L. Giuzzi, G. Korchmáros, Construction of unitals in Desarguesian planes, Discrete Math. 310 (22): 3162-3167 (2010), ISSN: 0012-365X, doi:10.1016/j.disc.2009.06.023 (arXiv:0810.2233)
    Abstract: We present a new construction of non-classical unitals from a classical unital \(\mathcal U\) in \(\mathrm{PG}(2,q^2)\). The resulting non-classical unitals are B--M unitals. The idea is to find a non-standard model \(\Pi\) of \(\mathrm{PG}(2,q^2)\) with the following three properties:
    1. points of \(\Pi\) are those of \(\mathrm{PG}(2,q^2)\);
    2. lines of \(\Pi\) are certain lines and conics of \(\mathrm{PG}(2,q^2)\);
    3. the points in \(\mathcal U\) form a non-classical B-M unital in \(\Pi\).
    Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.
  15. L. Giuzzi, A. Sonnino, LDPC codes from Singer cycles, Discrete Appl. Math. 157: 1723-1728 (2009), ISSN: 0166-218X, doi:10.1016/j.dam.2009.01.013 (arXiv:0709.2813)
    Abstract: 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 paper the orbits of subspaces of a finite projective space under the action of a Singer cycle are investigated. The incidence matrix associated to each of these structures yields an LDPC code in a natural manner.
  16. A. Aguglia, L. Giuzzi, An algorithm for constructing some maximal arcs in \(\mathrm{PG}(2,q^2)\), Results Math. 52 no. 1-2: 17-33 (2008), ISSN: 1422-6383, doi:10.1007/s00025-007-0268-y (arXiv:math.CO/0611466)
    Abstract: 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 \(\mathrm{PG}(3,q)\) 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 arcs in \(\mathrm{PG}(2,q^2)\).
  17. A. Aguglia, L. Giuzzi, On the non-existence of some inherited ovals in Moulton planes of even order, Electron. J. Combin. 15(1): N37 (2008) (arXiv:0803.1597)
    Abstract: No oval contained in a regular hyperoval of the Desarguesian plane \(\mathrm{PG}(2,q^2)\), \(q\) even, is inherited by a Moulton plane of order \(q^2\).
  18. A. Aguglia, L. Giuzzi, G. Korchmáros, Algebraic curves and maximal arcs, J. Algebraic Combin. 28: 531-544 (2008), ISSN: 0925-9899, doi:10.1007/s10801-008-0122-7 (arXiv:math/0702770)
    Abstract: 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 \(\mathrm{PG}(2,q)\) is obtained. The case where \(\mathcal K\) is a maximal \((k,n)\)-arc is considered to greater extent.
  19. A. Aguglia, L. Giuzzi, Construction of a 3-Dimensional MDS code, Contrib. Discrete Math. 3 no. 1: 39-46 (2007), ISSN: 1715-0868 (arXiv:0708.1558)
    Abstract: 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 \(\mathrm{PG}(2,q^2)\).
  20. A. Aguglia, L. Giuzzi, Orthogonal arrays from Hermitian varieties, Innov. Incidence Geom. 5: 129-144 (2007), ISSN: 1781-6475, (arXiv:0705.3590)
    Abstract: A simple orthogonal array \(\mathrm{OA}(q^{2n-1},q^{2n-2}, q,2)\) is constructed by using the action of a large subgroup of \(\mathrm{PGL}(n+1,q^2)\) on a set of non--degenerate Hermitian varieties in \(\mathrm{PG}(n,q^2)\).
  21. L. Giuzzi, On the intersection of Hermitian surfaces, J. Geom. 85: 49-60 (2006), ISSN: 0047-2468, doi:10.1007/s00022-006-0042-4
    Abstract: We provide a description of the configuration arising from intersection of two Hermitian surfaces in \(\mathrm{PG}(3,q)\), provided that the linear system they generate contains at least a degenerate variety.
  22. L. Giuzzi, A geometric construction for some ovoids of the Hermitian Surface, Results Math. 49: 81-88 (2006), ISSN: 1422-6383, doi:10.1007/s00025-006-0210-8
    Abstract: Multiple derivation of the classical ovoid of the Hermitian surface \({\mathcal H}(3,q^2)\) of \(\mathrm{PG}(3,q^2)\) is a well known, powerful method for constructing large families of non classical ovoids of \({\mathcal H}(3,q^2)\). In this paper, multiple derivation is generalised and applied to non-classical ovoids. A resulting new family of ovoids is investigated.
  23. L. Giuzzi, G. Korchmáros, Ovoids of the Hermitian Surface in Odd Characteristic, Adv. Geom Special Issue: S49-S58 (2003), ISSN: 1615-715X.
    Abstract: We construct a new ovoid of the polar space arising from the Hermitian surface of \(\mathrm{PG}(3,q^2)\) with \(q\geq 5\) odd. The automorphism group \(\Gamma\) of such an ovoid has a normal cyclic subgroup \(\Phi\) of order \(\frac{1}{2}(q+1)\) such that \(\Gamma/\Phi\cong\mathrm{PGL}(2,q)\). Furthermore, \(\Gamma\) has three orbits on the ovoid, one of size \(q+1\) and two of size \(\frac{1}{2}q(q-1)(q+1)\).
  24. L. Giuzzi, H. Karzel, Co-Minkowski spaces, their reflection structure and K-loops, Discrete Math. 255: 161-179 (2002), ISSN: 0012-365X, doi:10.1016/S0012-365X(01)00396-X
    Abstract: In this work an infinite family of K-loops is constructed from the reflection structure of co-Minkowski planes and their properties are analysed.
  25. L. Giuzzi, Collineation groups of the intersection of two classical unitals, J. Combin. Des. 9: 445-459 (2001), ISSN: 1063-8539, doi:10.1002/jcd.1023
    Abstract: Kestenband proved that there are only seven pairwise non-isomorphic Hermitian intersections in the Desarguesian projective plane \(\mathrm{PG}(2,q^2)\) of square order \(q^2\). 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 Hermitian intersections from the same class might not be projectively equivalent in \(\mathrm{PG}(2,q^2)\) and might have different collineation groups. The projective classification of Hermitian intersections in \(\mathrm{PG}(2,q^2)\) is the main goal in this paper. It turns out that each of Kestenband's classes consists of projectively equivalent Hermitian intersections. A complete classification of the linear collineation groups preserving a Hermitian intersection is also given.
  26. L. Giuzzi, A characterisation of classical unitals, J. Geom. 74: 86-89 (2002), ISSN: 0047-2468, doi:10.1007/PL00012541
    Abstract: A short proof is given of the following result: A unital in \(\mathrm{PG}(2,q)\) is classical if and only if it is preserved by a cyclic linear collineation group of order \(q-\sqrt{q}+1\).

Books

  1. L. Giuzzi, Codici correttori, Unitext Springer Verlag n. 27 (2006), ISBN: 88-470-0539-6.

Proceedings/Extended Abstracts

  1. I. Cardinali, L. Giuzzi, Polar Grassmannians and their Codes, extended abstract accepted for the MEGA2015 conference (2015) (arXiv:1509.07686)

Theses

  1. L. Giuzzi, Gruppi di Frobenius e strutture geometriche associate, Tesi di Laurea (Advisor: Prof. S. Pianta) 1996.
  2. L. Giuzzi, Hermitian varieties over finite fields, DPhil thesis (Supervisor: Prof. J.W.P. Hirschfeld) 2000.

Preprints

Most of my old preprints might be found here; some other notes on mathematics are available on this page.

The following are current preprints:

  1. I. Cardinali, L. Giuzzi, Line Hermitian Grassmann Codes and their Parameters, (arXiv:1706.10255)
    Abstract: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.
  2. I. Cardinali, L. Giuzzi, Geometries arising from trilinear forms on low-dimensional vector spaces, (arXiv:1703.06821)
    Abstract: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 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 \({\mathcal G}_3(V).\)
  3. I. Cardinali, L. Giuzzi, A. Pasini, On transparent embeddings of point-line geometries, (arXiv:1611.07877)
    Abstract: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 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.
  4. I. Cardinali, L. Giuzzi, Minimum distance of Line Orthogonal Grassmann Codes in even characteristic, (arXiv:1605.09333)
    Abstract: 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 (doi:10.1016/j.jpaa.2015.10.007 )" 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.
  5. I. Cardinali, L. Giuzzi, A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, (arXiv:1407.6149)
    Abstract: Polar Grassmann codes of orthogonal type have been introduced in [19]. They are subcodes 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.

Lecture notes

  1. L.Giuzzi, lecture notes of the course Algebraic geometry over a field of positive characteristic, given by J.W.P. Hirschfeld (1998).
  2. L.Giuzzi, A. Sonnino, Alcune note introduttive sulla crittografia (2005).

Patents

  1. L. Giuzzi, G. Korchmáros, A. Sonnino, Perfezionamenti nella crittografia a chiave pubblica basata su curve ellittiche, patent no. 0001379714, 30 August 2010