Research

Topics

My research interests include:

Preprints and papers

A full pdf list can be found here.

Accepted papers

  1. I. Cardinali, L. Giuzzi, Grassmannians of codes, Finite Fields Appl. 94 (2024), 102342 doi:j.ffa.2023.102342 (arXiv:2304.08397)
    Abstract:Consider the point line-geometry Pt(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 XY is a [n,k1]-linear code having minimum dual distance at least t+1. We are interested in the collinearity graph Λt(n,k) of Pt(n,k). The graph Λt(n,k) is a subgraph of the Grassmann graph and also a subgraph of the graph Δt(n,k) of the linear codes having minimum dual distance at least t+1 introduced in [M. Kwiatkowski, M. Pankov, On the distance between linear codes, Finite Fields Appl. 39 (2016), 251--263]. We shall study the structure of Λt(n,k) in relation to that of Δt(n,k) and we will characterize the set of its isolated vertices. We will then focus on Λ1(n,k) and Λ2(n,k) providing necessary and sufficient conditions for them to be connected.
  2. I. Cardinali, L. Giuzzi, On orthogonal polar spaces, Linar Algebra Appl. 674 (2023), 493-518 doi:10.1016/j.laa.2023.06.013 (arXiv:2301.05876)
    Abstract:Let 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 P, called the anisotropic gap, defined as the least upper bound of the lengths of the well-ordered chains of subspaces of P containing a frame; when P is orthogonal, we also defined two other parameters of P, called the elliptic and parabolic gap, related to the universal embedding of P.
    In this paper, assuming P is an orthogonal polar space, we prove that the elliptic and parabolic gaps can be described as intrinsic invariants of P without making recourse to the embedding.
  3. A. Aguglia, L. Giuzzi, On the equivalence of certain quasi-Hermitian varieties, J. Combin. Des. (2022), 1-15 doi:10.1002/jcd.21870 (arXiv:2108.04813)
    Abstract: In [A. Aguglia, A. Cossidente, G. Korchmaros, On quasi-Hermitian varieties, J. Comb. Des. 20 (2012), 433-447] new quasi-Hermitian varieties Mα,β in PG(r,q2) depending on a pair of parameters α,β from the underlying field GF(q2) have been constructed. In the present paper we determine the projective equivalence classes of such varieties for r=3 and q odd.
  4. I. Cardinali, L. Giuzzi, A. Pasini, On the generation of some Lie-type geometries, J. Combin. Theory A 193 (2023) 105673 doi:10.1016/j.jcta.2022.105673 (arXiv:1912.03484)
    Abstract: Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-Grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 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 Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)21 or (n+1)2.
  5. I. Cardinali, H. Cuypers, L. Giuzzi, A. Pasini, Characterizations of symplectic polar spaces, to appear on Adv. Geom. doi:10.1515/advgeom-2023-0006 (arXiv:2205.14426)
    Abstract: A polar space S is said to be symplectic if it admits an embedding ε in a projective geometry PG(V) such that the ε-image ε(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.
  6. A. Aguglia, M. Ceria, L. Giuzzi, Some hypersurfaces over finite fields, minimal codes and secret sharing schemes, Des. Codes. Cryptogr. 90 (2022), 1503-1519 doi:10.1007/s10623-022-01051-1 (arXiv:2105.14508)
    Abstract: 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, we prove that the secret sharing schemes thus obtained are democratic that is, each participant belongs to the same number of minimal access sets.
  7. I. Cardinali, L. Giuzzi, A. Pasini, Nearly all subspaces of a classical polar space arise from its universal embedding, Linear Algebra Appl. 627 (2021), 287-307 doi:10.1016/j.laa.2021.06.013 (arXiv:2010.07640)
    Abstract:Let Γ be an embeddable non-degenerate polar space of finite rank n2. 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:ΓPG(V) be the universal embedding of Γ. Let S be a subspace of Γ and suppose that S, regarded as a polar space, has non-degenerate rank at least 2. We shall prove that S is the ε-preimage of a projective subspace of PG(V).
  8. I. Cardinali, L. Giuzzi, M. Kwiatkowski, On the Grassmann Graph of Linear Codes, Finite Fields Appl. 75 (2021), 101895 doi:10.1016/j.ffa.2021.101895 (arXiv:2005.04402)
    Abstract: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 tN{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,k).
  9. 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)
    Abstract: 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,Fq) for q=4,8,9 and 2nN2n+2. We prove that for N>6 they can be generated over the prime subfield, thus determining their generating rank.
  10. A. Aguglia, L. Giuzzi, M. Homma, On Hermitian varieties in PG(6,q2), Ars Mathematica Contemporanea (2021) doi:10.26493/1855-3974.2358.3c9 (arXiv:2006.04099)
    Abstract: In this paper we characterize the non-singular Hermitian variety H(6,q2) of PG(6,q2), q2 among the irreducible hypersurfaces of degree q+1 in PG(6,q2) not containing solids by the number of its points and the existence of a solid S meeting it in q4+q2+1 points.
  11. A. Aguglia, L. Giuzzi, A. Sonnino, Near-MDS codes from elliptic curves, Des. Codes. Cryptogr. 89 (2021), 965-972 doi:10.1007/s10623-021-00852-0 (arXiv:2009.05623)
    Abstract:We provide a new construction of [n,9,n9]q near-MDS codes arising from elliptic curves with n Fq-rational points. Furthermore we show that in some cases these codes cannot be extended to longer near-MDS codes.
  12. I. Cardinali, L. Giuzzi, A. Pasini, Grassmann embeddings of polar Grassmannians, J. Combin. Theory Series A 170 (2020) 105133 doi:10.1016/j.jcta.2019.105133 (arXiv:1810.12811)
    Abstract: 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.
  13. I. Cardinali, L. Giuzzi, Implementing Line-Hermitian Grassmann codes, Linear Algebra Appl. 580 (2019), 96-120 doi:10.1016/j.laa.2019.06.020 (arXiv:1804.03024)
    Abstract: 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.
  14. I. Cardinali, L. Giuzzi, Geometries arising from trilinear forms on low-dimensional vector spaces, Adv. Geom. 19:2 (2019), 269-290, doi:10.1515/advgeom-2018-0027 (arXiv:1703.06821)
    Abstract:Let Gk(V) be the k-Grassmannian of a vector space V with dimV=n. Given a hyperplane H of Gk(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 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 n7 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 PG(5,K) arising from hyperplanes of G3(V).
  15. L. Giuzzi, F. Zullo, Identifiers for MRD codes, Linar Algebra Appl. 575 (2019), 66-86, doi:10.1016/j.laa.2019.03.030 (arXiv:1807.09476)
    Abstract: For any admissible value of the parameters n and k there exist [n,k]-Maximum Rank distance Fq-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 paper we study some invariants of MRD codes and evaluate their value for the known families, providing a new characterization of generalized twisted Gabidulin codes.
  16. I. Cardinali, L. Giuzzi, Line Hermitian Grassmann Codes and their Parameters, Finite Fields Appl. 51 (2018), 407-432 doi:10.1016/j.ffa.2018.02.006 (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.
  17. I. Cardinali, L. Giuzzi, Minimum distance of Line Orthogonal Grassmann Codes in even characteristic, J. Pure Applied Algebra 222:10 (2018), 2975-2988 doi:10.1016/j.jpaa.2017.11.009 (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.
  18. I. Cardinali, L. Giuzzi, A. Pasini, On transparent embeddings of point-line geometries, J. Combin. Theory Series A 155 (2018), 190-224, doi:10.1016/j.jcta.2017.11.001 (arXiv:1611.07877)
    Abstract:We introduce the class of transparent embeddings for a point-line geometry Γ=(P,L) as the class of full projective embeddings ε of Γ such that the preimage of any projective line fully contained in ε(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.
  19. 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 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.
  20. I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating k-linear forms, J. Algebraic Combin. 45 (2017) 931-963, doi:10.1007/s10801-016-0730-6 (arXiv:1601.08115)
    Abstract: Given an n-dimensional vector space V over a field K, let 2k<n. A natural one-to-one correspondence exists between the alternating k-linear forms of V and the linear functionals of kV, an alternating k-linear form φ and a linear functional f being matched in this correspondence precisely when φ(x1,,xk)=f(x1xk) for all x1,,xkV. Let εk:Gk(V)PG(kV) be the Pl\"{u}cker embedding of the k-Grassmannian Gk(V) of V. Then εk1(ker(f)εk(Gk(V))) is a hyperplane of the point-line geometry Gk(V). It is well known that all hyperplanes of Gk(V) can be obtained in this way, namely every hyperplane of Gk(V) is the family of k-subspaces of V where a given alternating k-linear form identically vanishes. For a hyperplane H of Gk(V), let R(H) be the subset (in fact a subspace) of Gk1(V) formed by the (k1)-subspaces AV 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(H) are the (k1)-subspaces A=a1,,ak1 of V such that φ(a1,,ak1,x)=0 for all xV. In principle, when nk is even it might happen that R(H)=. When nk is odd then R(H), since every (k2)-subspace of V is contained in at least one member of R(H), but it can happen that every (k2)-subspace of V is contained in precisely one member of R(H). If this is the case, we say that R(H) is \emph{spread-like}. In this paper we obtain some results on R(H) which answer some open questions from the literature and suggest the conjecture that, if nk is even and at least 4, then R(H) but for one exception with KR and (n,k)=(7,3), while if nk is odd and at least 5 then R(H) is never spread-like.
  21. 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 AG(r,q2) 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 PG(r,q2) with r even and we also compute their weight distribution.
  22. 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 H with an irreducible quadric of PG(3,q2) sharing at least a tangent plane at a common non-singular point when q is even.
  23. 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.
  24. 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.
  25. 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 S be a Desarguesian (t1)-spread of PG(rt1,q), Π a m--dimensional subspace of PG(rt1,q) and Λ the linear set consisting of the elements of S with non-empty intersection with Π. It is known that the Plücker embedding of the elements of S is a variety of PG(rt1,q), say Vrt. In this paper, we describe the image under the Plücker embedding of the elements of Λ 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 Vrt.
  26. A. Aguglia, L. Giuzzi, Intersections of the Hermitian surface with irreducible quadrics in PG(3,q2), 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 PG(3,q2), with q odd, we determine the possible intersection sizes of a Hermitian surface H and an irreducible quadric Q having the same tangent plane π at a common point PQH.
  27. 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 εkgr of an orthogonal Grassmannian Δk. In particular, we determine some of the parameters of the codes arising from the projective system determined by εkgr(Δk). We also study special sets of points of Δk which are met by any line of Δk in at most 2 points and we show that their image under the Grassmann embedding εkgr is a projective cap.
  28. 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 Vr,t, the image under the Grassmannian map of a Desarguesian (t1)-spread of PG(rt1,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.
  29. A. Benini, L. Giuzzi, A. Pasotti, Down-linking (Kv,Γ)-designs to P3-designs, Util. Math., 90: 3-21 (2013), ISSN: 0315-3681 (arXiv:1004.4127)
    Abstract: Let Γ be a subgraph of a graph Γ. We define a down-link from a (Kv,Γ)-design B to a (Kn,Γ)-design B as a map f:BB 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 (Kv,Γ)-design might be down-linked to a (Kn,Γ)-design, provided that n is admissible and large enough. We also show that if Γ=P3, 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 Γ, by providing explicit constructions.
  30. L. Giuzzi, G. Korchmáros, Unitals in PG(2,q2) 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 U be a unital embedded in the Desarguesian projective plane PG(2,q2). Write M for the subgroup of PGL(3,q2) which preserves U. We show that U is classical if and only if U has two distinct points P,Q for which the stabiliser G=MP,Q has order q21.
  31. 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.
  32. 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 U in PG(2,q2). The resulting non-classical unitals are B--M unitals. The idea is to find a non-standard model Π of PG(2,q2) with the following three properties:
    1. points of Π are those of PG(2,q2);
    2. lines of Π are certain lines and conics of PG(2,q2);
    3. the points in U form a non-classical B-M unital in Π.
    Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.
  33. 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.
  34. A. Aguglia, L. Giuzzi, An algorithm for constructing some maximal arcs in PG(2,q2), Results Math. 52 no. 1-2: 17-33 (2008), ISSN: 1422-6383, doi:10.1007/s00025-007-0268-y (arXiv:0611466)
    Abstract: In 1974, J. Thas constructed a new class of maximal arcs for the Desarguesian plane of order q2. 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 arcs in PG(2,q2).
  35. 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 PG(2,q2), q even, is inherited by a Moulton plane of order q2.
  36. 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:0702770)
    Abstract: A lower bound on the minimum degree of the plane algebraic curves containing every point in a large point-set K of the Desarguesian plane PG(2,q) is obtained. The case where K is a maximal (k,n)-arc is considered to greater extent.
  37. A. Aguglia, L. Giuzzi, Construction of a 3-Dimensional MDS code, Contrib. Discrete Math. 3 no. 1: 39-46 (2008), ISSN: 1715-0868 (arXiv:0708.1558)
    Abstract: In this paper, we describe a procedure for constructing q-ary [N,3,N2]-MDS codes, of length Nq+1 (for q odd) or Nq+2 (for q even), using a set of non-degenerate Hermitian forms in PG(2,q2).
  38. 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 OA(q2n1,q2n2,q,2) is constructed by using the action of a large subgroup of PGL(n+1,q2) on a set of non--degenerate Hermitian varieties in PG(n,q2).
  39. 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 PG(3,q), provided that the linear system they generate contains at least a degenerate variety.
  40. 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 H(3,q2) of PG(3,q2) is a well known, powerful method for constructing large families of non classical ovoids of H(3,q2). In this paper, multiple derivation is generalised and applied to non-classical ovoids. A resulting new family of ovoids is investigated.
  41. 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 PG(3,q2) with q5 odd. The automorphism group Γ of such an ovoid has a normal cyclic subgroup Φ of order 12(q+1) such that Γ/ΦPGL(2,q). Furthermore, Γ has three orbits on the ovoid, one of size q+1 and two of size 12q(q1)(q+1).
  42. 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.
  43. 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 PG(2,q2) of square order q2. 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 PG(2,q2) and might have different collineation groups. The projective classification of Hermitian intersections in PG(2,q2) 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.
  44. 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 PG(2,q) is classical if and only if it is preserved by a cyclic linear collineation group of order qq+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. A. Aguglia, L. Giuzzi, V. Siconolfi, On mutually μ-intersecting quasi-Hermitian varieties with some applications, (arXiv:2406.15589)
    Abstract:Mutually intersecting varieties of a finite Desarguesian projective space are varieties with the same size, such that the intersection of two of them contains the same number of points. Here, we show the existence of a new family of mutually intersecting varieties by using certain quasi-Hermitian varieties of PG(n,q2), 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 Fq as well as an infinite family of simple orthogonal arrays OA(q2n1,q2n2,q,2) of index μ=q2n3.
  2. L. Giuzzi, G. Lunardon, A remark on Fqn-Linear MRD codes, (arXiv:2106.15708)
    Abstract: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 nk is obtained, as well as parametric equations for MRD-codes of distance d=nk+1.
  3. 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