Conference booklet - Combinatorics in Italy

Transcript

Giornate di
Geometria 2015
September 17–19, 2015
Caserta, Italy
Template and Style ©Luca Giuzzi 2014, 2015
[email protected]
Typeset in Garamond Premier Pro with LuaLATEX
Giornate di Geometria
Caserta, Italy
Dipartimento di Matematica e Fisica, S.U.N.
Contents
List of Participants
Invited talks . . . .
Contributed talks .
List of Talks . . . .
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43
Caserta, Italy
D
A
Marien Abreu
Università della Basilicata (Italy)
e-mail: [email protected]
Anneleeln De Schepper
Ghent University (Belgium)
Awss Al-Ogaidi
University of Sussex (UK)
Tai Do Duc
Nanyang Technological University (Singapore)
B
Laura Bader
Università di Napoli “Federico II” (Italy)
Giorgio Donati
Daniele Bartoli
Nicola Durante
Francesco Belardo
Marco Buratti
Università di Perugia (Italy)
G
Arrigo Bonisoli
Università di Modena e Reggio Emilia (Italy)
Massimo Giulietti
Emanuele Brugnoli
Luca Giuzzi
Università di Brescia (Italy)
C
Ilaria Cardinali
Università di Siena (Italy)
H
Benjamin Cooper
University of Virginia (USA)
Hans Havlicek
Vienna University of Technology (Austria)
Simone Costa
Università “Roma tre” (Italy)
Bence Csajbók
Università di Padova (Italy)
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Caserta, Italy
K
N
Fatma Karaoglu
University of Sussex (UK)
Vito Napolitano
Seconda Università di Napoli (Italy)
Ayse Karaoglu
Ege University of Türkiye (Turkey)
O
Gábor Korchmáros
Domenico Olanda
Mariusz Kwiatkowski
University of Warmia and Mazury (Poland)
Ferruh Özbudak
Middle East Technical University (Turkey)
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P
Domenico Labbate
Fernanda Pambianco
Michel Lavrauw
Mark Pankov
University of Warmia and Mazury (Poland)
Enzo Li Marzi
Università di Messina (Italy)
Francesco Pavese
Guglielmo Lunardon
Valentina Pepe
Università di Roma “La Sapienza” (Italy)
M
Giuseppe Marino
Sivlia Pianta
Università Cattolica del Sacro Cuore (Italy)
Francesco Mazzocca
Olga Polverino
Giuseppe Mazzuoccolo
Università di Modena e Reggio Emilia (Italy)
Alexander Pott
University of Madgeburg (Germany)
Francesca Merola
Università Roma tre (Italy)
4
Caserta, Italy
R
Morgan Rodgers
S
Alessandro Siciliano
Angelo Sonnino
Pietro Speziali
Péter Sziklai
Eötvös Loránd University (Hungary)
T
Hiroaki Taniguchi
Nat. Inst. of Technology, Kagawa College ( Japan)
Tommaso Traetta
Ryerson University (Canada)
Rocco Trombetti
V
Hendrik Van Maldeghem
Z
Corrado Zanella
Giovanni Zini
Università di Firenze (Italy)
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Invited talks
Caserta, Italy
Main speakers
Marco Buratti (Some results and open problems on cycle decompositions) . .
Ferruh Özbudak (Classification of Function Fields with Class Number Three)
Alexander Pott (Graphs and Difference Sets) . . . . . . . . . . . .
Peter Sziklai (Directions and related topics) . . . . . . . . . . . . .
Hendrik Van Maldeghem (Domestic automorphisms of buildings) . .
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Marco
Buratti
Caserta, Italy
Some results and open problems on cycle decompositions
Marco Buratti
University of Perugia
Abstract
A cycle decomposition of a simple graph Γ is a set of subcycles of Γ whose edges partition the edge-set of Γ. In this
talk I will survey some results on this topic. Also, I will try to show how algebra might be useful to attack some open
problems on the same topic.
Keywords: Graph; Cycle Decomposition; Group Action
11
Ferruh
Özbudak
Caserta, Italy
Classification of Function Fields with Class Number Three
Ferruh Özbudak
Middle East Technical University
( Joint work with Mehpare Bilhan and Dilek Buyruk)
Abstract
The classification of the algebraic function fields with class number one and two was completed before. In this study,
we give the full list of all algebraic function fields over a finite field with class number three up to isomorphism. This list
consists of explicit equations of algebraic function fields which are mutually non-isomorphic over the full constant field.
Keywords: Class number, function field, L-polynomial.
12
Alexander
Pott
Caserta, Italy
Graphs and Difference Sets
Alexander Pott
Otto-von-Guericke University Magdeburg
Abstract
The most famous difference sets are the Singer difference sets which describe Desarguesian projective planes. There
are slight variations (sometimes called generalized difference sets) which again describe projective planes. In my talk, I
will discuss some applications how these difference sets can be used to construct also other interesting combinatorial
objects like
• Costas arrays (joint work with Amela Muratović-Ribić, David Thomson and Qiang Wang).
• Strongly regular graphs (joint work with Tan Yin and Tao Feng).
• Construction of some extremal Abelian Cayley graphs (joint work with Yue Zhou).
Keywords: difference set, Cayley graph, Costas array, Moore bound, strongly regular graph
13
Peter
Sziklai
Caserta, Italy
Directions and related topics
Peter Sziklai
Eotvos L. University, Budapest
Abstract
Direction problems have played an important role in the research on finite (Galois) geometry for at least fourty years.
They are interesting enough themselves, and they have many applications and connections to other fields.
The definition is quite innocent: let U be a point set in the n-dimensional affine space AG(n, q) over the finite field
GF(q) of q elements. We say a direction d (i.e. a point in the hyperplane at infinity H∞ ) is determined by U if there is a
line with the point d at infinity, containing at least two points of U . The set of determined directions is usually denoted
by D.
A typical direction problem puts some restriction on U and asks about the properties of D, or vice versa, under some
condition on D, asks about the structure of U . E.g. Rédei and Megyesi proved that in AG(2, p), p prime, if |U | = p,
then U determines either 1 or at least p+3
2 directions. Another example is a result of Bruen and Levinger, classifying all
affine point sets U ⊂ AG(2, q) of size q, which determine directions corresponding to slopes being square elements of
GF(q).
By the pigeon hole principle it is clear that if |U | > q n−1 then U determines every direction. Most of the classical
results consider the “extremal case”, i.e. affine pointsets of size q n−1 .
In my talk I will give a summary of the history and of the most important results, including some old and new results
about Rédei-polynomials, as their theory provided the key methods in this topic. I will also show some of the related
non-geometrical problems from group theory, graph theory, etc.
After this survey-like part we will deal with several extensions and generalizations. We get interesting cases when
changing the size of U , or if we alter the definition of determining a direction. Then we embed the topic into a broader
context and we define a large class of problems, which resemble to the direction questions.
Some of the results are joint work with Jan De Beule, Szabolcs Fancsali and Marcella Takáts.
Keywords: affine space, finite field, direction
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Hendrik
Van Maldeghem
Caserta, Italy
Domestic automorphisms of buildings
Hendrik Van Maldeghem
Ghent University
( Joint work with James Parkinson — University of Sydney)
Abstract
A domestic automorphism of a building is an automorphism that does not map any chamber to an opposite chamber.
Such automorphisms seem the be rare, especially for buildings with large panels. We explain the motivation, give an
overview of known results and present some new ones. The emphasis is on finite buildings, more exactly finite generalized
polygons, finite projective spaces and finite polar spaces.
Keywords: Buildings, Opposition, Generalized polygons, Projective spaces, Polar spaces
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Contributed talks
Caserta, Italy
List of contributors
Daniele Bartoli (Algebraic curves and Random Network Codes) . . . . . . . . . . . . . . . . . . . . . . .
21
Francesco Belardo (Balance, frustration and least Laplacian eigenvalue of signed graphs) . . . . . . . . . . . .
22
Emanuele Brugnoli (Enumerating the Walecki-type Hamiltonian cycle systems) . . . . . . . . . . . . . . . .
23
Ilaria Cardinali (Orthogonal and Symplectic Grassmann codes) . . . . . . . . . . . . . . . . . . . . . . .
24
Simone Costa (New i-perfect cycle decompositions via i-perfect SDFs) . . . . . . . . . . . . . . . . . . . . .
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Bence Csajbók (On scattered linear sets of pseudoregulus type in PG(1, q t )) . . . . . . . . . . . . . . . . . . .
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Anneleen De Schepper (A new dimension in the Magic Square) . . . . . . . . . . . . . . . . . . . . . . .
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Tai Do Duc (Unique Differences in Symmetric Subsets of Fp ) . . . . . . . . . . . . . . . . . . . . . . . . .
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Hans Havlicek (Harmonicity preservers) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Mariusz Kwiatkowski (On the distance between linear codes) . . . . . . . . . . . . . . . . . . . . . . . .
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Giuseppe Mazzuoccolo (On graphs with circular flow number 5) . . . . . . . . . . . . . . . . . . . . .
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Francesca Merola (Cyclic Hamiltonian cycle systems for the complete multipartite graph) . . . . . . . . . . . . .
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Mark Pankov (On embeddings of Grassmann graphs in polar Grassmann graphs) . . . . . . . . . . . . . . . . .
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Francesco Pavese (Intriguing sets of quadrics in PG(5, q)) . . . . . . . . . . . . . . . . . . . . . . . . . .
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Valentina Pepe (Symplectic semifield spreads of PG(5, q 2 )) . . . . . . . . . . . . . . . . . . . . . . . . . .
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Morgan Rodgers (Cameron–Liebler type sets and Completely regular codes in Grassmann graphs) . . . . . . . . .
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Alessandro Siciliano (On Delsarte’s linear MRD-codes and their automorphism group) . . . . . . . . . . . . .
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Pietro Speziali (Hermitian Codes with automorphism group isomorphic to P GL(2, q) with q odd) . . . . . . . .
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Hiroaki Taniguchi (On some bilinear dual hyperovals) . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Tommaso Traetta (Resolvability vs. absence of parallel classes) . . . . . . . . . . . . . . . . . . . . . . . .
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Giovanni Zini (Maximal curves which are not Galois-subcovers of the Hermitian curve) . . . . . . . . . . . . . .
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Daniele
Bartoli
Caserta, Italy
Algebraic curves and Random Network Codes
Daniele Bartoli
Ghent University
( Joint work with Matteo Bonini and Massimo Giulietti — University of Perugia)
Abstract
In their seminal paper [1], Koetter and Kschischang introduced a metric on the set of vector spaces and showed that
if the dimension of the intersections of the vector spaces is large enough then a minimal distance decoder for this metric
achieves correct decoding. In particular, given an r-dimensional vector space V over Fq , the set S(V ) of all subspaces of
V forms a metric space with respect to the subspace distance defined by
d(U, U 0 ) = dim(U + U 0 ) − dim(U ∩ U 0 ).
In this context the main problem asks for the determination of the larger size of codes in the space (S(V ), d) with given
minimum distance.
Recently, Hansen [2] presented a construction of random network codes based on Riemann-Roch spaces associated
to algebraic curves, describing the parameters of these codes.
We generalize this construction and we obtain new infinite families of random network codes from algebraic curves.
[1] R. Koetter, F.R. Kschischang. Coding for errors and erasures in random network coding. IEEE Transactions on
Information Theory 54(8), (2008) 3579–3591.
[2] J.P. Hansen. Riemann-Roch spaces and linear network codes,
http://arxiv.org/abs/1503.02386.
Keywords: Random Network Codes, Riemann-Roch spaces
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Francesco
Belardo
Caserta, Italy
Balance, frustration and least Laplacian eigenvalue of signed graphs
Francesco Belardo
University of Naples Federico II
Abstract
A signed graph Γ is a pair (G, σ), where G = (V (G), E(G)) is a simple graph and σ : E(G) → {+1, −1} is a
mapping on the edges of G called signature. A cycle in Γ is said to be positive if and only if the product of its edge signs is
positive. A signed graph Γ is balanced if all of its cycles are positive. If Γ is not balanced, then a suitable deletion of some
vertices or edges leads to a balanced graph. Let ν(Γ) (resp. (Γ)) be the minimum number of vertices (resp. edges) to be
deleted such that the obtained signed graph is balanced. The values ν(Γ) and (Γ) are called the frustration number and
frustration index, respectively. It is well-known that if Γ is a connected balanced graph then the least Laplacian eigenvalue
λn (Γ) equals zero. Some recent papers have recognized that λn is a good measure for the frustration of a connected
signed graph, and in particular it is λn (Γ) ≤ ν(Γ) ≤ (Γ). Hence, we refer to λn (Γ) as the algebraic frustration of Γ.
In this talk we discuss the limit case λn (Γ) = ν(Γ).
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Emanuele
Brugnoli
Caserta, Italy
Enumerating the Walecki-type Hamiltonian cycle systems
Emanuele Brugnoli
University of Perugia
Abstract
Let Kv be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly HCS(v)) is a set of
Hamiltonian cycles of Kv whose edges partition the edge-set of Kv . The earliest example of a HCS(2n + 1) is attributed
to Walecki; its vertex-set is V2n+1 := Z2n ∪ {∞} and it consists of all cycles belonging to the orbit of the starter cycle
(∞, 0, 1, −1, 2, −2, . . . , i, −i, . . . , n − 1, −(n − 1), n)
under the natural action of Z2n on V2n+1 .
By means of a slight modification of the HCS(4n + 1) of Walecki, we obtain 2n pairwise distinct HCS(4n + 1)
and we enumerate them up to isomorphism.
Keywords: Hamiltonian cycle system; permutation group
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Ilaria
Cardinali
Caserta, Italy
Orthogonal and Symplectic Grassmann codes
Ilaria Cardinali
University of Siena
( Joint work with Luca Giuzzi — University of Brescia)
Abstract
Polar Grassmann codes have been introduced in [1] as projective codes induced by projective systems associated with
polar Grassmannians. In this talk we shall recall the basic notions motivating the introduction of polar Grassmann codes
and report on some recent results on polar Grassmann codes of either orthogonal or symplectic type.
Keywords: Polar Grassmannians, Projective Codes.
References
[1] I. Cardinali and L. Giuzzi, Codes and caps from orthogonal Grassmannians, Finite Fields Appl. 24 (2013), 148-169.
[2] I. Cardinali, L. Giuzzi and A. Pasini, Line Polar Grassmann Codes of Orthogonal Type, arxiv:1407.6149.
[3] I. Cardinali and L. Giuzzi, Enumerative Coding for Line Polar Grassmannians, arxiv:1412.5466.
[4] I. Cardinali and L. Giuzzi, Minimum distance of Symplectic Grassmann codes, arxiv:1503.05456.
24
Simone
Costa
Caserta, Italy
New i-perfect cycle decompositions via i-perfect SDFs
Simone Costa
Università degli Studi Roma Tre
( Joint work with Marco Buratti and Xiaomiao Wang)
Abstract
A k-cycle decomposition of a graph G is a set C of k-cycles with vertices in V (G) whose edges partition E(G). Such
a decomposition is said to be i-perfect, with i an integer between 1 and bk/2c, if for any pair of distinct vertices x, y of
V (G) there is exactly one cycle of C where x and y occur at distance i (see [6]).
In this talk I will present a technique to construct regular i-perfect k-cycle decompositions of the m-partite graph
Km×k that generalizes the constructions of [3] and [1] and makes use of the new concept, inspired by [2], of i-perfect
(Zk , Ck , µ) strong difference family. This method is particularly effective in the case i ∈ {3, 5, 7, 9}; for these values
of i we proved that there exists an i-perfect k-cycle decomposition of the graph Km×k for k odd whenever we have
gcd(6, m) = 1. By using the result on 3-perfect Hamiltonian decompositions of a complete graph obtained in [4] and
[5] we also obtain the existence of 3-perfect k-cycle decompositions of the complete graph Kmk for any pair of integers
m, k such that gcd(6, m) = 1 and k ≥ 7 is odd.
References
[1] M. Buratti, S. Costa, X. Wang, New i-perfect cycle decompositions via graph colorings, preprint.
[2] M. Buratti, A. Pasotti, On perfect Γ-decompositions of the complete graph, J Combin Des. 17, Issue 2, (2009)
197-209.
[3] M. Buratti, F. Rania and F. Zuanni, Some constructions for cyclic perfect cycle system, Discrete Math. 299 (2005),
33-48.
[4] M. Buratti, G. Rinaldi and T. Traetta, Some results on 1-rotational Hamiltonian cycle systems, J Combin Des. 22,
Issue 6, (2014), 231-251.
[5] M. Kobayashi, B. McKay, N. Mutoh, G. Nakamura and C. Nara, 3-perfect Hamiltonian decomposition of the
complete graph, Australian J Combin. 56 (2013), 219-224.
[6] C.C. Lindner and C.A. Rodger, 2-perfect m-cycle systems, Discrete Math. 104 (1992), 83-90.
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Bence
Csajbók
Caserta, Italy
On scattered linear sets of pseudoregulus type in PG(1, q t)
Bence Csajbók
University of Padova
( Joint work with Corrado Zanella — University of Padua)
Abstract
Linear sets of a finite projective space generalize the concept of a subgeometry. In the recent years they have been
used to construct or characterize various objects in finite geometry. A point set L of PGqt (Fnqt ) ∼
= PG(n − 1, q t ) is
n
Fq -linear of rank k, if L = {huiqt : u ∈ U }, for some k-dimensional Fq -subspace U of Fqt . Lunardon and Polverino
characterized linear sets as projections of canonical subgeometries. In this talk we give a geometric characterization of
the projecting configurations (Γ, Σ), where Γ is a (t − 3)-space of Σ∗ ∼
= PG(t − 1, q t ) and Σ ∼
= PG(t − 1, q) is a
∗
subgeometry of Σ disjoint from Γ, such that the projection of Σ from Γ to the line Λ ∼
= PG(1, q t ) ⊂ Σ∗ \ Γ, that
is L := p Γ, Λ (Σ), is a maximum scattered linear set of pseudoregulus type. This family of linear sets has been defined
recently by Lunardon, Marino, Polverino and Trombetti; and also studied by Donati and Durante. We count the number
of q-order sublines contained in L and we show that the preimage p−1
Γ, Λ (r) ∩ Σ of any q-order subline r ⊆ L is a normal
rational curve. This extends a result of Lavrauw and Van de Voorde who proved the same in PG(1, q 3 ). In the proof we
use some properties of a degree t hypersurface of PG(2t − 1, q) studied by Lavrauw, Sheekey and Zanella. Our proof also
involves the study of point sets `d := {hz d iq : z ∈ Fqt , hziq ∈ `}, where ` is a line of PGq (Fqt ) and d is an integer of
a special form. It is well-known that `−1 is a rational normal curve. We generalize this result.
One of the most natural questions about linear sets is their equivalence. Let L be a linear set of pseudoregulus type
in Λ = PG(1, q t ), t = 5 or t > 6. We show that there exist projective configurations (Γ, Σ) and (Γ0 , Σ0 ) such that
L = p Γ, Λ (Σ) = p Γ0 , Λ (Σ0 ) and there is no collineation mapping Γ to Γ0 and Σ to Σ0 .
Keywords: Linear set, normal rational curve
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Anneleen
De Schepper
Caserta, Italy
A new dimension in the Magic Square
Anneleen De Schepper
Ghent University
( Joint work with Hendrik Van Maldeghem — University of Ghent)
Abstract
Veronesean varieties are connected with the geometries of the second row of the Freudenthal-Tits Magic Square.
Traditionally, there is a split and a nonsplit version of this square, and for the first column, these two versions coincide,
yielding the quadratic Veronesean. We discuss a possibility to introduce “degenerate” versions of the Magic Square that
connect the split with the nonsplit version. This would turn the square into a triangular prism. In particular, the second
column admits 3 different versions. We will emphasize the geometries behind this extension of the Magic Square.
Keywords: Veronesean varieties, projective remoteness planes
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Tai
Do Duc
Caserta, Italy
Unique Differences in Symmetric Subsets of Fp
Tai Do Duc
Division of Mathematical Sciences
Nanyang Technological University
Republic of Singapore
[email protected]
Abstract
Let p be a prime and let A be a subset of Fp with A = −A and |A \ {0}| ≤ 2 log3 (p). Then there is an element of
Fp which has a unique representation as a difference of two elements of A. This result can be used to show that under
certain conditions, Weil numbers in Z[ζm ] are necessarily contained in proper subfileds Q[ζm ].
Keywords: Weil numbers; Subfields.
References
[1] J. Browkin, B. Divis, and A. Schinzel: Addition of sequences in general fields, Monatsh. Math. 82 (1976), 261–268.
[2] E. Croot, T. Schoen: On sumsets and spectral gaps. Acta Arith. 136 (2009), 47–55.
[3] J. W. S. Cassels: On a conjecture of R. M. Robinson about sums of roots of unity. J. Reine Angew. Math. 238 (1969),
112–131.
[4] K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics 84, Springer
1990.
[5] K. H. Leung, B. Schmidt: Unique Sums and Differences in Finite Abelian Groups. Submitted.
[6] V. F. Lev: The rectifiability threshold in abelian groups. Combinatorica 28 (2008), 491–497.
[7] J. H. Loxton: On two problems of E. M. Robinson about sums of roots of unity. Acta Arith. 26 (1974), 159–174.
[8] Z. Nedev: An algorithm for finding a nearly minimal balanced set in Fp . Math. Comp. 268 (2009), 2259–2267.
[9] Z. Nedev: Lower bound for balanced sets. Theoret. Comput. Sci. 460 (2012), 89–93.
[10] Z. Nedev, A. Quas: Balanced sets and the vector game. Int. J. Number Theory 4 (2008), 339–347.
[11] C. Norman: Finitely Generated Abelian Groups and Similarity of Matrices over a Field. Springer 2012.
[12] B. Schmidt: Cyclotomic integers and finite geometry. J. Am. Math. Soc. 12 (1999), 929–952.
[13] E. G. Straus: Differences of residues (mod p). J. Number Th. 8 (1976), 40–42.
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Hans
Havlicek
Caserta, Italy
Harmonicity preservers
Hans Havlicek
Institute of Discrete Mathematics and Geometry
Vienna University of Technology
Wiedner Hauptstraße 8–10, 1040 Wien, Austria,
[email protected]
Abstract
In the first edition of the seminal book Geometrie der Lage by Karl Georg Christian von Staudt projectivities are
defined by the invariance of harmonic quadruples as follows [4, p. 49]: ”Zwei einförmige Grundgebilde heissen zu einander
projektivisch (∧), wenn sie so auf einander bezogen sind, dass jedem harmonischen Gebilde in dem einen ein harmonisches
Gebilde im andern entspricht.”
In this talk we are concerned with projective lines over rings R and R0 (associative with a unit element); see, among
others, [1] or [3]. Our aim is the algebraic description of their harmonicity preservers, i. e., mappings which take all
harmonic quadruples of the projective line P(R) to harmonic quadruples of the projective line P(R0 ).
There are numerous contributions to this topic, and results were obtained for certain classes of rings, first of all
for commutative and non-commutative fields of characteristic different from two. On the other hand, as follows from
various examples, a uniform algebraic description of harmonicity preservers is out of reach in the general case, even if the
underlying rings are commutative. A brief historical overview together with a comprehensive list of references can be
found in [2].
Recall that a mapping α : R → R0 is called a Jordan homomorphism if it satisfies
(x + y)α = xα + y α ,
1α = 1 0 ,
(xyx)α = xα y α xα
for all
x, y ∈ R.
(Homomorphisms and antihomomorphisms of rings are well known examples of Jordan homomorphisms.) Suppose
now that the ring R contains “sufficiently many units”, in particular the ring element 1 + 1 has to be a unit. We consider
the distant graph on the projective line P(R). Its vertices are the points of P(R). Two points of P(R) form an edge if,
and only if, they have representatives that form a basis of R2 . Let C ⊂ P(R) denote any connected component of the
distant graph. Under these assumption our main result is as follows:
Theorem [2, Thm. 1]: If µ : P(R) → P(R0 ) is a harmonicity preserver then the restriction of µ to C can be described in
terms of a Jordan homomorphism R → R0 .
References
[1] A. Blunck and A. Herzer. Kettengeometrien – Eine Einführung. Shaker Verlag, Aachen, 2005.
[2] H. Havlicek. Von Staudt’s theorem revisited. Aequationes Math. 89 (2015), 459–472. DOI 10.1007/s00010013-0218-6.
[3] A. Herzer. Chain geometries. In F. Buekenhout, editor, Handbook of Incidence Geometry, pages 781–842. Elsevier,
Amsterdam, 1995.
[4] K. G. Chr. von Staudt. Geometrie der Lage. Bauer und Raspe ( Julius Merz), Nürnberg, 1847.
29
Mariusz
Kwiatkowski
Caserta, Italy
On the distance between linear codes
Mariusz Kwiatkowski
University of Warmia and Mazury, Olsztyn, Poland
( Joint work with Mark Pankov — University of Warmia and Mazury)
Abstract
Let V be an n-dimensional vector space over the finite field consisting of q elements and let Γk (V ) be the Grassmann
graph formed by k-dimensional subspaces of V , 1 < k < n − 1. Denote by Γ(n, k)q the restriction of Γk (V ) to the
set of all non-degenerate linear [n, k]q codes. We show that for any two codes the distance in Γ(n, k)q coincides with
the distance in Γk (V ) only in the case when n < (q + 1)2 + k − 2, i.e. if n is sufficiently large then for some pairs of
codes the distances in the graphs Γk (V ) and Γ(n, k)q are distinct. We describe one class of such pairs.
Keywords: Grassmannian, Nondegenerate linear code, Distance graph
30
Giuseppe
Mazzuoccolo
Caserta, Italy
On graphs with circular flow number 5
Giuseppe Mazzuoccolo
University of Modena and Reggio Emilia.
( Joint work with Louis Esperet and Michael Tarsi.)
Abstract
For some time the Petersen graph has been the only known Snark with circular flow number 5 (or more, as long as
the assertion of Tutte’s 5-flow Conjecture is in doubt). Although infinitely many such snarks were presented in 2006 by
E. Máčajová and A. Raspaud, the variety of known methods to construct them and the structure of the obtained graphs
were still rather limited. After a brief introduction to the problem, we present an analysis of sets of flow values, which can
be transferred through flow networks with the flow on each edge restricted to the open interval (1, 4) modulo 5. All
these sets are symmetric unions of open integer intervals in the ring R/5Z. We use the results to design an arsenal of
methods for constructing snarks S with circular flow number at least 5. As one indication to the diversity and density
of the obtained family of graphs, we show that it is sufficiently rich so that the corresponding recognition problem is
NP-complete. Finally, we discuss the possible relations with other well-known conjectures about snarks.
Keywords: Snarks, Circular flows, Nowhere-zero flows, NP-Completeness.
31
Francesca
Merola
Caserta, Italy
Cyclic Hamiltonian cycle systems for the complete multipartite graph
Francesca Merola
Università Roma Tre, Italy
( Joint work with Anita Pasotti, Marco A. Pellegrini and Michael W. Schroeder)
Abstract
A Hamiltonian cycle system (HCS) for a graph or multigraph Γ is a set B of Hamiltonian cycles of Γ whose edges
partition the edge set of Γ. A cycle system is regular if there is an automorphism group G of the graph Γ acting sharply
transitively on the vertices of Γ and permuting the cycles of B, and it is called cyclic if G is the cyclic group.
The existence problem for cyclic HCS for the complete graph Kn , n odd, and for the graph K2n − I, I a 1-factor,
(the so-called cocktail party graph), has been solved in [2] and [4] respectively. In the talk I will consider existence results
for cyclic Hamiltonian cycle systems for the graph Km×n , the complete multipartite graph with m parts, each of size n.
I will present a complete solution to the existence problems when the number of parts m is even [5], and more generally
for mn even, and discuss some work in progress for the case in which both m and n are odd [6].
I will also touch on the symmetric HCS introduced by Brualdi and Schroeder [1] for the cocktail party graph and
recently generalized by Schroeder to the complete multipartite graphs [7]; indeed we may note that cyclic cycle systems
often turn out to possess this additional symmetry requirement [3], so that it makes sense to discuss these results in this
context.
References
[1] R.A. Brualdi, M.W. Schroeder, Symmetric Hamilton cycle decompositions of complete graphs minus a 1-factor, J.
Combin. Des. 19 (2011), 1–15.
[2] M. Buratti, A. Del Fra, Cyclic Hamiltonian cycle systems of the complete graph, Discrete Math. 279 (2004), 107–119.
[3] M. Buratti, F. Merola, Hamiltonian cycle systems which are both cyclic and symmetric, J. Combin. Des. 22 (2014),
367–390.
[4] H. Jordon, J. Morris, Cyclic hamiltonian cycle systems of the complete graph minus a 1-factor, Discrete Math. 308
(2008), 2440–2449.
[5] F. Merola, A. Pasotti, M.A. Pellegrini, Cyclic hamiltonian cycle systems of the complete multipartite graph: even
number of parts, preprint.
[6] F. Merola, A. Pasotti, M.A. Pellegrini, M.W. Schroeder, Cyclic hamiltonian cycle systems of the complete multipartite
graph: odd number of parts, in preparation.
[7] M.W. Schroeder, φ-symmetric Hamilton cycle decompositions of graphs, to appear in Discrete Math.
32
Mark
Pankov
Caserta, Italy
On embeddings of Grassmann graphs in polar Grassmann graphs
Mark Pankov
University of Warmia and Mazury
Abstract
Our first result says that every embedding of a Grassmann graph in a polar Grassmann graph can be reduced to an
embedding in a Grassmann graph or to an embedding in the collinearity graph of a polar space. As a simple consequence
of this theorem, we get a statement closely connected to Blok–Coperstein result.
Also, we consider 3-embeddings (embeddings preserving all distances not greater than 3) of dual polar graphs whose
diameter is not less than 3 in polar Grassmann graphs formed by non-maximal singular subspaces. The second result
states that every such an embedding can be reduced to an embedding in a Grassmann graph.
Keywords: Grassmann graph, polar Grassmann graph, embedding
33
Francesco
Pavese
Caserta, Italy
Intriguing sets of quadrics in PG(5, q)
Francesco Pavese
University of Basilicata
( Joint work with A. Cossidente — University of Basilicata)
Abstract
Let P be a finite classical polar space of rank r ≥ 2 over a finite field of order q. We say that a set of points I of P is
intriguing if
(
h1 if P ∈ I
⊥
|P ∩ I| =
,
h2 if P 6∈ I
for some constants h1 and h2 (where P ranges over the points of P). The integers h1 and h2 are called the intersection
numbers of I. Here ⊥ is the polarity of P.
It turns out that an intriguing set of P is either an m–ovoid or an i–tight set of P [1]. An m–ovoid O of P is a
subset of points of P such that every maximal of P meets O in m points. If T is a subset of points of P, then the average
number of points of T collinear with a given point is bounded above by i(q r−1 − 1)/(q − 1) + q r−1 − 1, where i is
determined by the size of T . If equality occurs, T is said to be i–tight.
There exist two very nice and important objects in finite geometry admitting the same automorphism group: a
hemisystem of the Hermitian generalized quadrangle H(3, q 2 ), q odd [3], and a Cameron–Liebler line class in PG(3, q),
q odd, of parameter (q 2 + 1)/2 [2]. They both admit the classical group G := P Ω− (4, q) stabilizing an elliptic quadric
Q− (3, q) of PG(3, q) as an automorphism group. Under the Klein correspondence, they are both examples of intriguing
sets of quadrics of PG(5, q).
We pose the following question. There does exist a suitable chosen subgroup of G producing new hemisystems of the
Hermitian generalized quadrangle H(3, q 2 ) and new Cameron–Liebler line classes of PG(3, q)? In this talk I will show
that the answer is affirmative. In particular, by considering a suitable subgroup of the stabilizer of a point of Q− (3, q) in
G, a new family of hemisystems of H(3, q 2 ), q ≥ 7 odd, and a new family of Cameron–Liebler line classes of PG(3, q)
with parameter (q 2 + 1)/2, q ≥ 5 odd, arise.
Keywords: Hermitian surface; hemisystem; Cameron–Liebler line class; strongly regular graph; partial quadrangle, intriguing sets; quadrics.
References
[1] J. Bamberg, S. Kelly, M. Law, T. Penttila, Tight sets and m–ovoids of finite polar spaces, J. Combin. Theory Ser. A 114
(2007), no. 7, 1293-1314.
[2] A. A. Bruen, K. Drudge, The construction of Cameron–Liebler line classes in PG(3, q), Finite Fields Appl. 5 (1999), no.
1, 35-45.
[3] A. Cossidente, T. Penttila, Hemisystems on the Hermitian surface, J. London Math. Soc. (2) 72 (2005), no. 3, 731-741.
34
Valentina
Pepe
Caserta, Italy
Symplectic semifield spreads of PG(5, q 2)
Valentina Pepe
Sapienza University of Rome
( Joint work with S.Capparelli — Sapienza University of Rome)
Abstract
I will present some recent results about symplectic semifield spreads of PG(5, q 2 ), whose associated semifield has
center containing Fq . In particular, for even q, we have the following:
Theorem[S.Capparelli,V.P.] There do not exist non-Desarguesian symplectic semifield spreads of PG(5, q 2 ), q ≥ 214
even, whose associated semifield has center containing Fq .
35
Morgan
Rodgers
Caserta, Italy
Cameron–Liebler type sets and Completely regular codes in Grassmann graphs
Morgan Rodgers
University of Padova
( Joint work with Leo Storme and Andries Vansweevelt)
Abstract
A Cameron-Liebler line class in PG(3, q) can be defined as a set L of lines whose characteristic vector lies in row(A),
where A is the point-line incidence matrix of PG(3, q). These objects are connected to collineation groups of PG(n, q)
having the same number of orbits on points and lines, as well as to symmetric tactical decompositions of the point-line
design PG(n, q). These objects also provide examples of completely regular codes in the Grassmann graph Gq (4, 2);
these are sets of vertices that induce an equitable partition in the graph, and provide generalizations of the classical
concept of perfect codes.
We generalize the concept of a Cameron-Liebler line class to sets of k-spaces in PG(2k + 1, q). After looking at
various characterizations of these sets and explaining some of the difficulties that arise in contrast to the known results for
line classes, we will give some connections to completely regular codes in Gq (2k + 2, k + 1), and prove some preliminary
results concerning the existence of these objects.
Keywords: Cameron–Liebler line classes, Completely regular codes, Grassmann graph
36
Alessandro
Siciliano
Caserta, Italy
On Delsarte’s linear MRD-codes and their automorphism group
Alessandro Siciliano
Università degli Studi della Basilicata
Abstract
In the vector space Mn,m (Fq ), n ≤ m of all (n × m)-matrices over Fq , let Zt , 1 ≤ t ≤ n, denote the collection of
all 1-dimensional spaces hvi, such that rank(v) ≤ t. The set Zt is actually a projective variety in the projective space
PG(Mn,m (Fq )) defined by Mn,m (Fq ).
Though the study of linear subspaces external to Zt has its own interest from the point of algebraic geometry over a
finite field, in the last decades these subspaces became object of active research because of their applications in coding
theory. In particular, q-ary codes derived from subspaces external to Zt of size q m(n−t) are called maximum linear rank
distance codes.
Such subspaces were firstly constructed by Delsarte [1], and later by Gabidulin [2]. Recently, Sheekey [3] examined
in details the case n = m.
In the talk I will present a geometric construction for the subspaces constructed by Delsarte. The automorphism
group of these will also be discussed in the case m = 2n.
Keywords: External flats, linear codes
References
[1] Ph. Delsarte, Bilinear Forms over a finite field, with applications to coding theory, J. Comb. Theory, Ser. A 25 (1978),
226–241 .
[2] E. Gabidulin, Theory of codes with maximum rank distance, Probl. Inf. Trasm., 1 (1985), 1–12.
[3] J. Sheekey, A new family of linear maximum rank distance codes, preprint.
37
Pietro
Speziali
Caserta, Italy
Hermitian Codes with automorphism group isomorphic to P GL(2, q) with q
odd
Pietro Speziali
University of Basilicata
( Joint work with Gábor Korchmáros — University of Basilicata)
Abstract
An algebraic-geometry (shortly AG) code on an algebraic curve X defined over a finite field F` arises by evaluating
certain rational functions that are regular outside a given F` -rational divisor G at some set D of F` -rational places that are
disjoint from the support of G. These codes, also named functional codes, may have good performance provided that X , G
and D are chosen in an appropriate way. The best known such codes are the 1-point Hermitian functional codes CL (D, G)
where ` = q 2 , X is the Hermitian curve H in P G(2, q 2 ), G = mP for an Fq2 -rational point of H, and D consists of
all Fq2 -rational points of H other than P . Also, Hermitian two-point codes arise whenever G = m1 P1 + m2 P2 and
D = H(Fq2 ) \ {P1 , P2 }. Several authors pointed out that even better parameters can be obtained for some values of m
whenever P is replaced by a higher degree place P .
In this talk we investigate the case where q is odd, m ≤ q 2 − 3, P = P1 + . . . + Pq + P∞ where {P1 , . . . , P∞ } is
a 2-transitive point-orbit of a subgroup G ∼
= P GL(2, q) of the automorphism group P GU (3, q) of H. If H is given
by its affine equation y q + y − xq+1 = 0 then P consists of all Fq -rational points of H. Using classical tools from
both Algebraic and Finite Geometry, such as linear series on curves and the action of P GL(2, q) on points and lines
in the projective plane P G(2, q), we completely determine the parameters of those codes for m ≤ q 2 − 12 (3q − 1).
Remarkably, in several cases the Goppa designed minimum distance is beaten. Since both D and P are preserved by
a subgroup G ∼
= P GL(2, q) of P GU (3, q), the inherited permutation automorphism group of CL (D, mP) has a
subgroup isomorphic to P GL(2, q). Furthermore, for m < q − 2, we show that this subgroup is the whole permutation
automorphism group of CL (mP, D).
Keywords: hermitian curve, goppa code, automorphism
38
Hiroaki
Taniguchi
Caserta, Italy
On some bilinear dual hyperovals
Hiroaki Taniguchi
National Institute of Technology, Kagawa College, Japan
Abstract
The concept of dimensional dual hyperoval (DHO) was introduced by Huybrechts and Pasini in [2]. If a DHO S
of rank n generate the vector space U , we say that S is a DHO in U . (In this talk, we assume that n > 4, and DHOs
are defined over GF (2).) It is proved [5] that 2n − 1 ≤ rank U ≤ n(n + 1)/2 + 2, and conjectured that 2n − 1 ≤
rank U ≤ n(n + 1)/2. Known DHOs in V (n(n + 1)/2, 2) are the Huybrechts DHO, the Buratti-Del Fra DHO,
the Veronesean DHO and the deformation of Veronesean DHO. A DHO S := {X(e) | e ∈ V } in V ⊕ W (V, W :
GF (2)-vector spaces) is said to be a bilinear DHO if there is a GF (2)-bilinear mapping B : V ⊕ V → W such that
X(e) := {(x, B(x, e)) | x ∈ V } ⊂ V ⊕ W for any e ∈ V . The Huybrechts DHO and the Buratti-Del Fra DHO
are bilinear DHOs, and the Veronesean DHO and the deformation of Veronesean DHO are non-bilinear DHOs. Let
Si be DHOs of rank n and Ui = hSi i for i = 1, 2. We say S1 is a quotient of S2 if there is a surjective GF (2)-linear
mapping (a covering map) π : U2 → U1 such that π(S2 ) = S1 .
In [4], we give a new construction for the Buratti-Del Fra DHO. As a generalization of this construction, we construct
DHOs Sc (l, GF (2r )) of rank n = rl + 1 in V (((n − 1)2 /r + 3(n − 1) + 2)/2, 2)) ( or V ((d2 /r + 3d + 2)/2, 2))
if we put d := n − 1 = rl), where l, r positive integers, and c ∈ GF (2r ) with the absolute trace T r(c) = 1 in [3].
In this talk, we first present an example of a bilinear quotient of the Buratti-Del Fra DHO of rank n in V (2n − 1, 2)
for d := n − 1 odd, and an example of a bilinear quotient of Sc (l, GF (2r )) of rank n in V (2n − 1, 2) for d :=
n − 1 = rl, l odd. Next, using the methods of [1], [6] and [3], we show that the Huybrechts DHO of rank n has
no bilinear quotient in V (2n − 1, 2). Using this result, for any quadratic APN function f on GF (2n ), we show that
the set {f (x + t) + f (x) + f (t) + f (0) | x, t ∈ GF (2n )} = GF (2n ). (Recall that “the GF (2)-vector space
h{f (x + t) + f (x) + f (t) + f (0) | x, t ∈ GF (2n )}i = GF (2n )” is proved by Yoshiara [6] in 2009.)
r
Next (if there remains some time), we talk about
We show that Sc (l, GF (2r )) is simply connected
P Sc (l, GF (2 )).
1
r
r/s
iff GF (2 ) = GF (2)(c), which gives us 2r s|r,s:odd µ(s)2
non-isomorphic examples of simply connected
dual hyperovals (for fixed l and r), where µ is the möbius function. And show that Sc (l, GF (2r )) is a quotient of
0
0
0
0
0
Sc (l , GF (2r )) if lr = l r and r is a divisor of r. We also determine the automorphism group of Sc (l, GF (2r )),
0
0
0
0
0
and show that Aut(Sc (l, GF (2r ))) is a subgroup of Aut(Sc (l , GF (2r ))) if lr = l r and r |r.
Keywords: dual hyperoval, quadratic APN function
References
[1] C. Carlet, P. Charpin and V. Zinoviev, Codes, Bent functions and Permutations Suitable For DES-like Cryptsystems,
Designs, Codes and Cryptgraphy 15 (1998), 125–156.
[2] C. Huybrechts and A. Pasini, Flag-transitive extensions of dual affine spaces, Contribution to Algebra and Geometry, 40
(1999), 503–532.
[3] H. Taniguchi, New dimensional dual hyperovals, which are not quotients of the classical dual hyperovals, Discrete Mathematics, 337 (2014), 65–75.
[4] H. Taniguchi and S. Yoshiara, A new construction of the d-dimensional Buratti-Del Fra dual hyperoval, European Journal
of Combinatorics, 33 (2012), 1030–1042.
[5] S. Yoshiara, Ambient spaces of dimensional dual arcs, Journal of Algebraic Combinatorics, 19 (2004), 5–23.
[6] S. Yoshiara, Dimensional dual hyperovals associated with quadratic APN functions, Innovations in Incidence Geometry. 8
(2009).
39
Tommaso
Traetta
Caserta, Italy
Resolvability vs. absence of parallel classes
Tommaso Traetta
Department of Mathematics, Ryerson University
Abstract
A set of cycles of a simple graph Γ whose vertices partition the vertex-set of Γ is called a parallel class. A set of cycles
whose edges partition the edge-set of Γ is a cycle decomposition of Γ. Such a decomposition is resolvable if the cycle-set can
be partitioned into parallel classes. The Oberwolfach Problem and the Hamilton-Waterloo Problem are two well-known
open problems on the existence of resolvable cycle decompositions. Both have been the subject of an extensive research
activity over the last few years [1,2,4,5,6,7,8].
A problem opposite to the resolvability concerns the construction of cycle decompositions, with a given structure,
free from parallel classes. This problem is still open, for example, for Steiner triple systems [3].
In this talk I will present some recent results on cycle decompositions which are either resolvable or free from parallel
classes.
References
[1] A. Bonisoli, M. Buratti, G. Mazzuoccolo: “Doubly transitive 2-factorizations”, J. Combin. Des. 15 (2007), 120–132.
[2] D. Bryant, P. Danziger: “On bipartite 2−factorizations of Kn − I and the Oberwolfach problem”, J. Graph Theory
68 (2011), 22–37.
[3] D. Bryant, D. Horsley: “Steiner triple systems without parallel classes”, arXiv:1407.5766.
[4] D. Bryant, V. Scharaschkin: “Complete solutions to the Oberwolfach problem for an infinite set of orders”, J. Combin.
Theory Ser. B 99 (2009), 904–918.
[5] M. Buratti, S. Capparelli, A. Del Fra: “Cyclic Hamiltonian cycle systems of the λ-fold complete and cocktail party
graphs”, European J. Combin. 31 (2010), 1484–1496.
[6] M. Buratti, G. Rinaldi: “1-rotational k-factorizations of the complete graph and new solutions to the Oberwolfach
problem”, J. Combin. Des. 16 (2008), 87–100.
[7] T. Traetta: “Some new results on 1-rotational 2-factorizations of the complete graph”, J. Combin. Des. 18 (2010),
237–247.
[8] T. Traetta: “A complete solution to the two-table Oberwolfach problems”, J. Combin. Theory Ser. A 120 (2013),
984–997.
40
Giovanni
Zini
Caserta, Italy
Maximal curves which are not Galois-subcovers of the Hermitian curve
Giovanni Zini
Universitá di Firenze
( Joint work with Massimo Giulietti, Maria Montanucci, and Luciane Quoos)
Abstract
For every q = n3 with n a prime power, the so-called GK-curve
2
2
Z n −n+1 = Y n − Y
X :
X n + X = Y n+1
is an Fq2 -maximal curve, that is a curve of genus g attaining the Hasse-Weil upper bound
q 2 + 1 + 2gq
on the number of Fq2 -rational places; for n > 2, X is not Fq2 -covered by the Hermitian curve
Hq :
X q + X = Y q+1
maximal over Fq2 . The problem of giving explicit equations for maximal curves is relevant for applications to Coding
Theory.
In the first part we compute explicit equations for some families of maximal curves that are Galois-covered by the
GK-curve. We also determine the genera of the curves; some of them are new values in the spectrum of genera of
Fq2 -maximal curves. We provide some further examples of Fq2 -maximal curves that cannot be covered by Hq , as well as
infinite families of Fq2 -maximal curves not Galois-covered by Hq .
In the second part we deal with the Galois-covering problem for two other important maximal curves: the ÁbdonBezerra-Quoos curve
2
2
A : X q − X = Y q −q+1 ,
which is Fq2 -maximal for every prime power q, and the generalized GK-curve
(
n` +1
2
Z n+1 = Y n − Y ,
G:
X n + X = Y n+1
which is Fn6 -maximal. We solve the open cases by showing that A is not Galois-covered by Hq for q > 3, and G is not
Galois-covered by H2` for n = 2, ` ≥ 5. To this end we use group-theoretical arguments concerning the unitary groups.
The first part is a joint work with Massimo Giulietti and Luciane Quoos, the second part is a joint work with Massimo
Giulietti and Maria Montanucci.
Keywords: Maximal curve, Galois-covering
41
List of Talks
Caserta, Italy
List of talks
A new dimension in the Magic Square, 27
Algebraic curves and Random Network Codes, 21
Balance, frustration and least Laplacian eigenvalue of signed graphs, 22
Cameron–Liebler type sets and Completely regular codes in Grassmann graphs, 36
Classification of Function Fields with Class Number Three, 12
Cyclic Hamiltonian cycle systems for the complete multipartite graph, 32
Directions and related topics, 14
Domestic automorphisms of buildings, 15
Enumerating the Walecki-type Hamiltonian cycle systems, 23
Graphs and Difference Sets, 13
Harmonicity preservers, 29
Hermitian Codes with automorphism group isomorphic to P GL(2, q) with q odd, 38
Intriguing sets of quadrics in PG(5, q), 34
Maximal curves which are not Galois-subcovers of the Hermitian curve, 41
New i-perfect cycle decompositions via i-perfect SDFs, 25
On Delsarte’s linear MRD-codes and their automorphism group, 37
On embeddings of Grassmann graphs in polar Grassmann graphs, 33
On graphs with circular flow number 5, 31
On scattered linear sets of pseudoregulus type in PG(1, q t ), 26
On some bilinear dual hyperovals, 39
On the distance between linear codes, 30
Orthogonal and Symplectic Grassmann codes, 24
Resolvability vs. absence of parallel classes, 40
Some results and open problems on cycle decompositions, 11
Symplectic semifield spreads of PG(5, q 2 ), 35
Unique Differences in Symmetric Subsets of Fp , 28
45

Conference booklet - Combinatorics in Italy

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