-
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 \({\cal P}_t(n,k)\) 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 \({\cal P}_t(n,k)\). The graph \(\Lambda_t(n,k)\) is a subgraph of the Grassmann graph and also a subgraph of the graph \(\Delta_t(n,k)\) 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 \(\Lambda_t(n,k)\) in relation to that of \(\Delta_t(n,k)\) and we will characterize the set of its isolated vertices. We will then focus on \(\Lambda_1(n,k)\) and \(\Lambda_2(n,k)\) providing necessary and sufficient conditions for them to be connected.
-
A. Aguglia, B. Csajbók, Luca Giuzzi, On regular sets of affine type in finite Desarguesian planes and related codes, Discrete Math. 347 (2024), 113835 doi:10.1016/j.disc.2023.113835
(arXiv:2305.17103)
Abstract:In this paper, we consider point sets of finite Desarguesian planes whose multisets of intersection numbers with lines is the same for all but one exceptional parallel class of lines. We call such sets regular of affine type. When the lines of the exceptional parallel class have the same intersection numbers, then we call these sets regular of pointed type. Classical examples are e.g. unitals; a detailed study and constructions of such sets with few intersection numbers is due to Hirschfeld and Szőnyi from 1991. We here provide some general construction methods for regular sets and describe a few infinite families. The members of one of these families have the size of a unital and meet affine lines of \(\mathrm{PG}(2, q^2)\) in one of \(4\) possible intersection numbers, each of them congruent to \(1\) modulo \(\sqrt{q}\). As a byproduct, we determine the intersection sizes of the Hermitian curve defined over \(\mathrm{GF}(q^2)\) with suitable rational curves of degree \(\sqrt{q}\) and we obtain \(\sqrt{q}\)-divisible codes with \(5\) non-zero weights. We also determine the weight enumerator of the codes arising from the general constructions modulus some \(q\)-powers.
-
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 \(\cal P\) 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.
-
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 \(\mathcal{M}_{\alpha,\beta}\) in \(\mathrm{PG}(r,q^2)\) depending on a pair of parameters \(\alpha,\beta\) from the underlying field \(\mathrm{GF}(q^2)\) have been constructed. In the present paper we determine the projective equivalence classes of such varieties for \(r=3\) and \(q\) odd.
-
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 \(X_n({\mathbb K})\) be a building of Coxeter type \(X_n = A_n\) or \(X_n = D_n\) defined over a given division ring \(\mathbb K\) (a field when \(X_n = D_n\)). For a non-connected set \(J\) of nodes of the diagram \(X_n\), let \(\Gamma({\mathbb K}) = \mathrm{Gr}_J(X_n({\mathbb K}))\) be the \(J\)-Grassmannian of \(X_n({\mathbb K})\). We prove that \(\Gamma({\mathbb K})\) cannot be generated over any proper sub-division ring \({\mathbb K}_0\) of \({\mathbb K}\). As a consequence, the generating rank of \(\Gamma({\mathbb K})\) is infinite when \(\mathbb K\) is not finitely generated. In particular, if \(\mathbb K\) is the algebraic closure of a finite field of prime order then the generating rank of \(\mathrm{Gr}_{1,n}(A_n({\mathbb K}))\) is infinite, although its embedding rank is either \((n+1)^2-1\) or \((n+1)^2\).
-
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 \(\cal S\) is said to be symplectic if it admits an embedding \(\varepsilon\) in a projective geometry \(\mathrm{PG}(V)\) such that the \(\varepsilon\)-image \(\varepsilon({\cal S})\) of \(\cal S\) 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 \(\cal S\) admits different (non isomorphic) embeddings, as it is the case (precisely) when \(\cal S\) is defined over a field of characteristic \(2\).
-
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.
-
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 \(\Gamma\) be an embeddable non-degenerate polar space of finite rank \(n\geq 2\). Assuming that \(\Gamma\) admits the universal embedding (which is true for all embeddable polar spaces except grids of order at least \(5\) and certain generalized quadrangles defined over quaternion division rings), let \(\varepsilon:\Gamma\to\mathrm{PG}(V)\) be the universal embedding of \(\Gamma\). Let \(\cal S\) be a subspace of \(\Gamma\) and suppose that \(\cal S\), regarded as a polar space, has non-degenerate rank at least \(2\). We shall prove that \(\cal S\) is the \(\varepsilon\)-preimage of a projective subspace of \(\mathrm{PG}(V)\).
-
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 \(\Gamma(n,k)\) be the Grassmann graph formed by the \(k\)-dimensional subspaces of a vector space of dimension \(n\) over a field \(\mathbb F\)$ and, for \(t\in \mathbb{N}\setminus \{0\}\), let \(\Delta_t(n,k)\) be the subgraph of \(\Gamma(n,k)\) formed by the set of linear \([n,k]\)-codes having minimum dual distance at least \(t+1\).
We show that if \(|{\mathbb F}|\geq{n\choose t}\) then \(\Delta_t(n,k)\) is connected and it is isometrically embedded in \(\Gamma(n,k)\).
-
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 \(V(N,{\mathbb F}_q)\) 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.
-
A. Aguglia, L. Giuzzi, M. Homma,
On Hermitian varieties in \(\mathrm{PG}(6,q^2)\), 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 \(\mathcal{H}(6,q^2)\) of \(\mathrm{PG}(6,q^2)\),
\( q \neq 2 \)
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.
-
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 \( [n,9,n-9]_q \) 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.
-
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.
-
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.
-
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)
-
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 \({\mathbb F}_q\)-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 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.
-
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.
-
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.
-
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
\(\Gamma = ({\mathcal P},{\mathcal L})\) 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.
-
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.
-
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 \({\mathbb K}\), 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.
-
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.
-
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.
-
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.
-
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 \(3\).
-
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 \({\mathcal V}_{rt}\).
-
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
\( P\in{\mathcal Q}\cap{\mathcal H}\).
-
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.
-
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.
-
A. Benini, L. Giuzzi, A. Pasotti,
New results on path-decompositions and their down-links, Util. Math., 90: 369-382 (2013), ISSN: 0315-3681 (arXiv:1106.1095)
Abstract:In (arXiv:1004.4127) the concept of down-link from a
\((K_v, \Gamma)\)-design \(\mathcal B\) to a
\((K_n,\Gamma')\)-design \({\mathcal B}'\) has been introduced.
In the present paper the spectrum
problems for \(\Gamma'= P_4\) 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 also presented. -
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.
-
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
\(q^2-1\).
-
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.
-
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:
- points of \(\Pi\) are those of \(\mathrm{PG}(2,q^2)\);
- lines of \(\Pi\) are certain lines and conics of \(\mathrm{PG}(2,q^2)\);
- 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.
-
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.
-
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: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 \(\mathrm{PG}(2,q^2)\).
-
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 \(q^2\).
-
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 \(\mathcal K\) 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.
-
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
\([N,3,N-2]\)-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 \(\mathrm{PG}(2,q^2)\).
-
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
\(\mathrm{PG}(n,q^2)\).
-
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.
-
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.
-
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
\(\frac{1}{2}q(q-1)(q+1)\).
-
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.
-
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.
-
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
\(q-\sqrt{q}+1\).
![[picture of mine]](../img/jpg/foto-ric.jpg)
