
I. Cardinali, L. Giuzzi,
Enumerative Coding for Line Polar Grassmannians with applications to codes, Finite Fields Appl. 46 (2017) 107138, 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 nondegenerate 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 nondegenerate 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, to appear on J. Algebraic Combin., doi:10.1007/s1080101607306
(arXiv:1601.08115)
Abstract:
Given an \(n\)dimensional vector space \(V\) over a field \({\mathbb K}\), let \(2\leq k < n\). A natural onetoone 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 pointline 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 \({\mathcal G}_{k1}(V)\) formed by the \((k1)\)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 \((k1)\)subspaces \(A = \langle a_1,\ldots, a_{k1}\rangle\) of \(V\) such that \(\varphi(a_1,\ldots, a_{k1},x) = 0\) for all \(x\in V\). In principle, when \(nk\) is even it might happen that \(R^\uparrow(H) = \emptyset\). When \(nk\) is odd then \(R^\uparrow(H) \neq \emptyset\), since every \((k2)\)subspace of \(V\) is contained in at least one member of \(R^\uparrow(H)\), but it can happen that every \((k2)\)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{spreadlike}. 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 \(nk\) 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 \(nk\) is odd and at least \(5\) then \(R^\uparrow(H)\) is never spreadlike.

A. Aguglia, L. Giuzzi,
Intersection sets, threecharacter multisets and associated codes, Des. Codes Cryptogr. 83:269282 (2017),doi:10.1007/s1062301603028
(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 threecharacter 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 nonsingular 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): 19241934 (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: 124134 (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.
LagrangianGrassmannian 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
LagrangianGrassmannian codes of rank \(2\) and \(3\).

L. Giuzzi, V. Pepe,
On some subvarieties of the Grassmann variety, Linear Multilinear Algebra 63 (11): 21212134 (2015),
ISSN 03081087, doi:10.1080/03081087.2014.983449
(arXiv:1405.6926)
Abstract:
Let \(\mathcal S\) be a Desarguesian \((t1)\)spread of
\(\mathrm{PG}(rt1,q)\), \(\Pi\) a
\(m\)dimensional subspace of \(\mathrm{PG}(rt1,q)\) and
\(\Lambda\) the linear
set consisting of the elements of \(\mathcal S\) with nonempty intersection
with \(\Pi\). It is known that the Plücker embedding of the
elements of \(\mathcal S\) is a variety of \(\mathrm{PG}(r^t1,q)\),
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: 113 (2014), ISSN: 10715797, 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: 148169 (2013), ISSN: 10715797, 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: 375384 (2013), doi:10.1007/s1062301296136
(arXiv:1107.1066)
Abstract:
Using geometric properties of the
variety \({\mathcal V}_{r,t}\), the image
under the Grassmannian map
of a Desarguesian \((t1)\)spread of \(\mathrm{PG}(rt1,q)\),
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.

A. Benini, L. Giuzzi, A. Pasotti,
New results on pathdecompositions and their downlinks, Util. Math., 90: 369382 (2013), ISSN: 03153681 (arXiv:1106.1095)
Abstract:In (arXiv:1004.4127) the concept of downlink 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
pathdecompositions and embeddings between pathdecompositions playing a
fundamental role for the construction of downlinks are also presented. 
A. Benini, L. Giuzzi, A. Pasotti,
Downlinking \((K_v,\Gamma)\)designs to \(P_3\)designs, Util. Math., 90: 321 (2013), ISSN: 03153681 (arXiv:1004.4127)
Abstract:
Let \(\Gamma'\) be a subgraph of a graph \(\Gamma\).
We define a downlink 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 downlinks in general and prove that any
\((K_v,\Gamma)\)design might be downlinked 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 downlink
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 2point stabiliser, Discrete Math.
312 (3): 532535 (2012), ISSN: 0012365X, 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
\(q^21\).

L. Giuzzi, A. Pasotti,
Sampling complete graphs, Discrete Math. 312 (3): 488497 (2012), ISSN: 0012365X, 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): 31623167 (2010), ISSN: 0012365X, doi:10.1016/j.disc.2009.06.023
(arXiv:0810.2233)
Abstract:
We present a new construction of nonclassical unitals from a
classical unital \(\mathcal U\) in \(\mathrm{PG}(2,q^2)\). The resulting
nonclassical unitals are BM unitals. The idea is to find a
nonstandard 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 nonclassical BM unital in \(\Pi\).
Our construction also works for the BT 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: 17231728 (2009), ISSN: 0166218X, 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. 12: 1733 (2008), ISSN: 14226383, doi:10.1007/s000250070268y
(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 SuzukiTits
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 nonexistence 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: 531544 (2008), ISSN: 09259899, doi:10.1007/s1080100801227
(arXiv:math/0702770)
Abstract:
A lower bound on the minimum degree of the plane algebraic curves
containing every point in a large pointset \(\mathcal K\) 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 3Dimensional MDS code, Contrib. Discrete Math. 3 no. 1: 3946 (2007),
ISSN: 17150868 (arXiv:0708.1558)
Abstract:
In this paper, we describe a procedure for constructing \(q\)ary
\([N,3,N2]\)MDS codes, of length \(N\leq q+1\) (for \(q\) odd) or
\(N\leq q+2\) (for \(q\) even), using a set of nondegenerate Hermitian
forms in \(\mathrm{PG}(2,q^2)\).

A. Aguglia, L. Giuzzi,
Orthogonal arrays from Hermitian varieties, Innov. Incidence Geom. 5: 129144 (2007),
ISSN: 17816475, (arXiv:0705.3590)
Abstract:
A simple orthogonal array \(\mathrm{OA}(q^{2n1},q^{2n2}, q,2)\) is
constructed by using the action of a large subgroup of
\(\mathrm{PGL}(n+1,q^2)\) on a set of nondegenerate Hermitian varieties in
\(\mathrm{PG}(n,q^2)\).

L. Giuzzi,
On the intersection of Hermitian surfaces, J. Geom. 85: 4960 (2006), ISSN: 00472468, doi:10.1007/s0002200600424
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: 8188 (2006),
ISSN: 14226383, doi:10.1007/s0002500602108
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 nonclassical 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: S49S58 (2003),
ISSN: 1615715X.
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
\(\frac{1}{2}q(q1)(q+1)\).

L. Giuzzi, H. Karzel,
CoMinkowski spaces, their reflection structure and Kloops, Discrete Math. 255: 161179 (2002), ISSN: 0012365X, doi:10.1016/S0012365X(01)00396X
Abstract:
In this work an infinite family of Kloops is constructed from
the reflection structure of coMinkowski planes and their properties
are analysed.

L. Giuzzi,
Collineation groups of the intersection of two classical unitals, J. Combin. Des. 9: 445459 (2001), ISSN: 10638539, doi:10.1002/jcd.1023
Abstract:
Kestenband proved that there are only seven pairwise
nonisomorphic 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: 8689 (2002), ISSN: 00472468, 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
\(q\sqrt{q}+1\).