2016 Pecs Workshop on

Geometric and Algebraic Combinatorics

May 5–9, 2016

Pecs, Hungary

Abstracts

in alphabetical order

Keynote talks:

Aart Blokhuis 3

Gabor Korchmaros 8

Klavdija Kutnar 12

Leo Storme 19

Contributed talks:

Janos Barat 1

Daniele Bartoli 2

Bence Csajbok 4

Jan De Beule 5

Gabor Gevay 6

Robert Jajcay and Tatiana Jajcayova 7

Istvan Kovacs 11

Bostjan Kuzman 13

Gabor Nagy 14

Zoltan Lorant Nagy 15

Francesco Pavese 16

Angelo Sonnino 17

Primoz Sparl 18

Ferdinando Zullo 20

(In)stability of Drisko’s Theorem and Woolbright’s ideaJanos Barat

MTA–ELTE Geometric and Algebraic Combinatorics Research Group

Ryser [4] conjectured that every Latin square of odd order has a transversal.Equivalently, if the edges of the complete bipartite graph Kn,n is partitioned into ndisjoint perfect matchingM1, . . . ,Mn, then there exists a rainbow matching R. Thatis, R = {e1, . . . , en} such that ei ∈ Mi for 1 ≤ i ≤ n. One might drop the conditionof disjointness on M1, . . . ,Mn and consider the union (a bipartite multigraph on 2nvertices) of any n perfect matchings. Drisko [3] essentially proved that the unionof any 2n − 1 perfect matchings has a rainbow matching. He also showed 2n − 2perfect matchings without a rainbow matching.

Together with Gyarfas and Sarkozy, we used Woolbright’s idea [5] to prove anextension of Drisko’s theorem [2]. The aim was to determine the number of perfectmatchings such that their union has a rainbow matching of size n − k, where 0 <k < n. In particular, for n − 1, we can prove that 3

2n suffices. On the other hand,

Aharoni [1] conjectures that n might be the correct answer.

If time permits, we would like to mention another variant of the Latin theme.

References

[1] R. Aharoni, P. Charbit and D. Howard, On a generalization of theRyser-Brualdi-Stein conjecture, J Graph Theory 78 Issue 2 (2015), 143–156.

[2] J. Barat, A. Gyarfas, G.N. Sarkozy, Rainbow matchings in bipartitemultigraphs, Period. Math. Hungar. to appear.

[3] A.A. Drisko, Transversals in row-Latin rectangles, J. Combin. Theory Ser.A 84 (1998), 181–195.

[4] H.J. Ryser, Neuere Probleme der Kombinatorik. In: Vortrage uber Kombina-torik, Oberwolfach, Matematisches Forschungsinstitute Oberwolfach, Germany,24–29 July (1967), 69–91.

[5] D.E. Woolbright, An n × n Latin square has a transversal with at leastn−√

n distinct symbols, J. Combin. Theory Ser. A 24 (1978), 235–237.

1

On monomial complete permutation polynomialsDaniele Bartoli

Department of Mathematics and Computer Science,University of Perugia, Perugia, Italy

(joint work with M. Giulietti, L. Quoos Conte, and G. Zini)

Let Fℓ denote the finite field of order ℓ and characteristic p. A permutationpolynomial (or PP) f(x) ∈ Fℓ[x] is a bijection of Fℓ onto itself. A polynomialf(x) ∈ Fℓ[x] is a complete permutation polynomial (or CPP) if both f(x) and f(x)+xare permutation polynomials of Fℓ.

We classify complete permutation polynomials of type aXqn−1

q−1+1 over the finite

field with qn elements, for n + 1 a prime. For the case n + 1 a power of the char-acteristic we study some known families. When n + 1 ∈ {8, 9} we determine someparticular examples.

2

On almost small and almost large super-Vandermonde sets inGF(q)

Aart Blokhuis

Department of Mathematics and Computing Science, Eindhoven University ofTechnology, the Netherlands

A set T ⊂ GF (q), q = ph is a super-Vandermonde set if∑

y∈T yk = 0 for 0 < k <|T |. We determine the structure of super-Vandermonde sets of size p + 1 (almostsmall) and size q/p− 1 (almost large).

3

Classes and equivalence of linear sets in PG(1, qn)Bence Csajbok

Department of Mathematics and Physics, Second University of Naples, Italy

(joint work with Giuseppe Marino, Olga Polverino)

Linear sets are natural generalizations of subgeometries. Let Λ = PG(W,Fqn)= PG(r − 1, qn), where W is a vector space of dimension r over Fqn . A point setL of Λ is said to be an Fq-linear set of Λ of rank k if it is defined by the non-zerovectors of a k-dimensional Fq-vector subspace U of W , i.e.

L = LU = {〈u〉Fqn: u ∈ U \ {0}}.

In the recent years linear sets have been used to construct or characterize variousobjects in finite geometry, such as blocking sets and multiple blocking sets in finiteprojective spaces, two-intersection sets in finite projective spaces, translation spreadsof the Cayley Generalized Hexagon, translation ovoids of polar spaces, semifieldflocks and finite semifields.

One of the most natural questions about linear sets is their equivalence. Two linearsets LU and LV of PG(r−1, qn) are said to be PΓL-equivalent (or simply equivalent)if there is an element ϕ in PΓL(r, qn) such that Lϕ

U = LV . In the applications it iscrucial to have methods to decide whether two linear sets are equivalent or not.

In this talk the equivalence problem of Fq-linear sets of rank n of PG(1, qn) willbe investigated, also in terms of the associated variety, projecting configurations,Fq-linear blocking sets of Redei type, MRD-codes.

4

On i-tight sets of the Hermitian polar space with smallparameter iJan De Beule

Vrije Universiteit Brussel

(joint work with Klaus Metsch)

The finite classical polar spaces are the geometries that consists of totally isotropic,totally singular respectively, subspaces with relation to a chosen sesquilinear, quad-ratic respectively, form on a finite dimensional vectorspace over a finite field. Thesegeometries are fully embedded in a finite projective space. The rank of a finiteclassical polar space equals the Witt index of the underlying form, the generatorsare the maximal subspaces contained in it.

An i-tight set in a finite classical polar space, is a set of points that behavescombinatorially as if it was a disjoint union of i generators. When considering thepoint graph of a polar space, which is a strongly regular graph, tight sets correspondwith characteristic vectors orthogonal to one of the eigenspaces of the adjacencymatrix of the graph. This relation has been shown very useful to study the geometricinteraction between tight sets and other objects in polar spaces.

The geometric interaction between tight sets and other objects in polar spaces hasbeen one reason for the attention that tight sets received recently. When studyingthis interaction, one is often interested in finding non-trivial i-tight sets, i.e. tightsets with parameter i that are different from the disjoint union of generators.

In the talk, we discuss recent joint work with Klaus Metsch on tight sets withsmall parameter of the Hermitian polar space in rank 2 and 3 in even projectivedimension. We discuss some non-trivial examples, which determines the notionsmall in this case. Then we discuss a result which shows that a tight set of theparticular polar space, with a small parameter, is necessarily the disjoint union ofgenerators.

5

Pascal triangle made of configurationsGabor Gevay

University of Szeged, Hungary

We introduce an infinite family of geometric point-line configurations. They gen-eralize the Desargues–Cayley–Danzer configurations studied in a recent paper [M.Boben, G. Gevay and T. Pisanski, Danzer’s configurations revisited, Adv. Geom. 15(2015), 393–408]. The number of points and lines in these configurations are givenby binomial coefficients; hence, they can be arranged in a triangular array analogousto Pascal’s triangle. The array is generated by a recurrence relation which makesthe analogy to Pascal’s triangle even closer. We also mention some consequences;one of the most interesting of them is that each of these configurations representssome incidence theorem. Besides, each of them has a point-circle representation.

6

Regular representations of finite groups as automorphismsgroups of hypergraphs

Robert Jajcay* and Tatiana Jajcayova

Comenius University, Bratislava and University of Primorska, Koper

The regular action of a group on itself via left or right multiplication is probablythe most natural group action. Interest in the regular actions goes all the way to theclassical result of Cayley who used the regular action to show that every group canbe faithfully represented as a permutation group. The main topic of our talk grewout of the classical problem of Graphical Regular Representations – the problemof finding a graph on which a given group acts regularly as its full automorphismgroup. Another variation of the problem is the Digraphical Regular RepresentationProblem of finding a digraph with a given regular automorphism group. In the mostgeneral setting, the problem can be stated as follows: Given a finite group G, finda set of subsets B of G with the property Aut(G,B)) = GL, where GL is the groupof permutations obtained from left multiplications by the elements of G. Thus, weare looking for a combinatorial structure whose full automorphism group is not justisomorphic to G, but is equal to a specific permutation representation. Specifically,for any given finite group G, we attempt to classify all k, 1 ≤ k ≤ |G|, for whichthere exists a k-uniform hypergraph on G whose full automorphism group is equalto GL.

7

Automorphism groups of algebraic curvesGabor Korchmaros

Dipartimento di Matematica ed Informatica, Universita della Basilicata, CampusUniversitario di Macchia Romana

85100 Potenza, Italy

(joint work with Massimo Giulietti)

Let X be a (projective, geometrically irreducible, non-singular) algebraic curvedefined over an algebraically closed field K of characteristic p ≥ 0. We mainly focuson positive characteristic, and in particular on the case where K is the algebraic clo-sure of a finite field. Let K(X ) be the field of rational functions (the function fieldof transcendency degree one over K) of X . The K–automorphism group Aut(X ) ofX is defined to be the automorphism group Aut(K(X )) consisting of those auto-morphisms of K(X ) which fix each element of K. Aut(X ) has a faithful action onthe set of points of X .

By a classical result, Aut(X ) is finite if the genus g of X is at least two.

It has been known for a long time that every finite group occurs in this way, sincefor any ground field K and any finite group G, there exists X such that Aut(X ) ∼= G,

This result raised a general problem for groups and curves: Determine the finitegroups that can be realized as the K-automorphism group of some curve with agiven invariant. The most important such invariant is the genus g of the curve, andthere is a long history of results on the interaction between the automorphism groupof a curve and its genus.

In positive characteristic, another important invariant is the p-rank of the curve(also called the Hasse-Witt invariant), which is the integer γ so that the Jacobianof X has pγ points of order p. It is known that 0 ≤ γ ≤ g.

In this survey we focus on the following issues:

(i) Upper bounds on the size of G depending on g.

(ii) Examples of curves defined over a finite field with very large automorphismgroups.

(iii) The possibilities for G when the p-rank is 0.

(iv) Upper bounds on the size of the p-subgroups of G depending on the p-rank.

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(v) Large automorphism group implies zero p-rank.

The study of the automorphism group of an algebraic curve is mostly carried outby using Galois Theory, via the fundamental group of the curve. Here, we adopt adifferent approach in order to exploit the potential of Finite Group Theory.

References

[1] N. Anbar, D. Bartoli, S. Fanali, and M. Giulietti, On the size of the automorphismgroup of a plane algebraic curve, J. Pure Appl. Algebra 217(7) (2013), 1224–1236.

[2] C. Guneri, M. Ozdemir, and H. Stichtenoth, The automorphism group of the gener-alized Giulietti-Korchmaros function field, Adv. Geom. 13 (2013), 369–380.

[3] M. Giulietti and G. Korchmaros, A new of family of maximal curves over a finitefield, Math. Ann. 343 (2009), 229-245.

[4] M. Giulietti and G. Korchmaros, Algebraic curves with a large non-tame automor-phism group fixing no point, Trans. Amer. Math. Soc. 362(11) (2010), 5983–6001.

[5] M. Giulietti and G. Korchmaros, Automorphism groups of algebraic curves withp-rank zero, J. Lond. Math. Soc. (2) 81(2) (2010), 277–296.

[6] M. Giulietti and G. KorchmarosLarge 2-groups of automorphisms of algebraic curvesover a field of characteristic 2, J. Algebra 427 (2015), 264-294.

[7] M. Giulietti, G. Korchmaros and F. Torres, Quotient curves of the Deligne–Lusztigcurve of Suzuki type, Acta Arith. 122 (2006), 245–274.

[8] R. Guralnick, B. Malmskog, and R. Pries, The automorphism groups of a family ofmaximal curves, J. Algebra 361 (2012), 92–106.

[9] H.W. Henn, Funktionenkorper mit groβer Automorphismengruppe, J. Reine Angew.Math. 302 (1978), 96–115.

[10] J.W.P. Hirschfeld, G. Korchmaros, and F. Torres, Algebraic Curves Over a FiniteField, Princeton Univ. Press, Princeton and Oxford, 2008.

[11] K. Iwasawa and T. Tamagawa, On the group of automorphisms of a function field J.Math. Soc. Japan 3 (1951), 137–147.

[12] C. Lehr and M. Matignon, Automorphism groups for p-cyclic covers of the affine line,Compositio Math. 141 (2005), 1213–1237.

[13] M. Madan and M. Rosen, The automorphism group of a function field, Proc. Amer.Math. Soc. 115 (1992), 923–929.

[14] D.J. Madden and R.C. Valentini, The group of automorphisms of algebraic functionfields, J. Reine Angew. Math. 343 (1983), 162–168.

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[15] M. Matignon and M. Rocher, On smooth curves endowed with a large automorphismp-group in characteristic p > 0, Algebra Number Theory 2 (2008), 887–926.

[16] S. Nakajima, p-ranks and automorphism groups of algebraic curves, Trans. Amer.Math. Soc. 303 (1987) 595–607.

[17] S. Nakajima, On automorphism groups of algebraic curves, Current Trends in NumberTheory, Hindustan Book Agency, New Delhi, 2002, 129–134.

[18] R. Pries and K. Stevenson, A survey of Galois theory of curves in Characteristic p. InWIN - Women in Numbers: Research Directions in Number Theory, A. C. Cojocaru,K. Lauter, R. Pries, and R. Scheidler Eds., Fields Inst. Commun., 60, Amer. Math.Soc., Providence, RI, 2011, pp. 169–191.

[19] M. Rocher, Large p-groups actions with a p-elementary abelian second ramificationgroup, J. Algebra 321 (2009), 704–740.

[20] M. Rocher, Large p-group actions with a p-elementary abelian derived group, J.Algebra 321(2) (2009), 704–740.

[21] P. Roquette, Uber die Automorphismengruppe eines algebraischen Funktio-nenkorpers, Arch. Math. 3 (1952), 343–350.

[22] H.I. Schmid, Uber Automorphismen eines algebraische Funktionenkorpern vonPrimzahlcharakteristic, J. Reine Angew. Math. 179 (1938), 5–15.

[23] H. Stichtenoth, Uber die Automorphismengruppe eines algebraischen Funktio-nenkorpers von Primzahlcharakteristik. I. Eine Abschatzung der Ordnung der Auto-morphismengruppe, Arch. Math. 24 (1973), 527–544.

[24] H. Stichtenoth, Uber die Automorphismengruppe eines algebraischen Funktio-nenkorpers von Primzahlcharakteristik. II. Ein spezieller Typ von Funktio-nenkorpern, Arch. Math. 24 (1973), 615–631.

[25] H. Stichtenoth, Die Hasse–Witt–Invariante eines Kongruenzfunktionenkorpers, Arch.Math. (Basel) 33 (1980), 357–360.

[26] H. Stichtenoth, Zur Realisierbarkeit endlicher Gruppen als Automorphismengruppenalgebraischer Funktionenkorper, Math. Z. 187 (1984), 221–225.

[27] E. Witt, Der Existenzsatz fur abelsche Funktionenkorper, J. Reine Angew. Math.173 (1935), 43–51.

[28] E. Witt, Konstruktion von galoischen Korpern der Characteristik p zu vorgegebenerGruppe der Ordnung pf , J. Reine Angew. Math. 174 (1936), 237–245.

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Integral automorphisms of affine spaces over finite fieldsIstvan Kovacs

University of Primorska, Koper, Slovenia

(based on joint works with Klavdija Kutnar, Janos Ruff and Tamas Szonyi)

Let GF(q) denote the finite field with q elements, and let S denote the set of allsquare elements of GF(q). The norm N of a point x = (x1, . . . , xn) of the affinespace AG(n, q) is defined by N(x) = x2

1 + · · ·+ x2n, and two points x and y are said

to be at integral distance if N(x−y) is in S. An integral automorphism of AG(n, q)is a permutation of the point set which preserves integral distances. The integralautomorphisms which are also semiaffine transformations were determined by Kurzand Meyer (JCTA, 2009). In this talk, I present some recent results on integralautomorphisms which are not semiaffine transformations.

11

Odd Automorphisms in Vertex-transitive GraphsKlavdija Kutnar

University of Primorskaklavdija.kutnar@gmail.com

(joint work with Ademir Hujdurovic and Dragan Marusic)

An automorphism of a graph is said to be even/odd if it acts on the set of ver-tices as an even/odd permutation. In this talk some recent results in regards tothe problem about the existence of odd automorphisms in vertex-transitive graphswill be considered. Partial results for certain classes of vertex-transitive graphs, inparticular for Cayley graphs, will be presented.

12

Symmetric tetravalent graphs as covers of doubled cyclesBostjan Kuzman

University of Ljubljana and IMFM

(joint work with A. Malnic and P. Potocnik)

A specific class of symmetric tetravalent graphs was partially classified by Gar-diner and Praeger (Europ. J. Comb, 1994) by applying the quotient reduction overelementary abelian normal subgroup of automorphisms. Inspecting the problemfrom the other end, we complete this classification by applying the automorphismlifting method to regular covers over doubled cycles. Moreover, we present a unifieddescription of the graphs in question in terms of cyclic codes and their generatingpolynomials.

13

Solvability of automorphic commutative loopsGabor Nagy

University of Szeged

We prove that every finite, commutative automorphic loop is solvable. We alsoprove that every finite, automorphic 2-loop is solvable. We use results of Guralnick-Saxl on primitive permutation groups of prime power degree to show that the mul-tiplication group of the loop is of affine type. Then, we associate a simple Liealgebra of characteristic 2 to a hypothetical finite simple commutative automorphicloop. The ,,crust of a thin sandwich” theorem of Zel’manov and Kostrikin leads toa contradiction.

14

On the number of k-dominating independent setsZoltan Lorant Nagy

MTA–ELTE Geometric and Algebraic Combinatorics Research Groupnagyzoli@cs.elte.hu

In this talk we study the existence and the number of k-dominating independentsets in certain graph families. While the case k = 1 namely the case of maximalindependent sets - which is originated from Erdos and Moser - is widely investigated,much less is known in general. In this talk we settle the question for trees and provethat the maximum number of k-dominating independent sets in n-vertex graphsis between ck · 2k

√2nand c′k · k+1

√2nif k ≥ 2, moreover the maximum number of

2-dominating independent sets in n-vertex graphs is between c ·1.22n and c′ ·1.246n.

Graph constructions containing a large number of k-dominating independent setsare coming from product graphs, complete bipartite graphs and finite geometries.The product graph construction is associated with the number of certain MDS codes.

15

On maximal partial spread of finite classical polar spacesFrancesco Pavese

Politecnico di Bari, Italyfrancesco.pavese@poliba.it

(joint work with A. Cossidente)

Let P be a finite classical polar space. A partial spread S of P is a set of pairwisedisjoint generators of P . A partial spread is said to be maximal if it maximalwith respect to set-theoretic inclusion. A partial spread S is called a spread if Spartitions the point set of P . If a polar space does not admit spreads, the questionon the size of a maximal partial spread in such a space naturally arises. In generalconstructing maximal partial spreads and obtaining reasonable upper and lowerbounds for the size of such partial spreads is an interesting problem. Recentlymaximal partial spreads of symplectic polar spaces received particular attention dueto their applications in quantum information theory. In fact they correspond to so-called weakly unextendible mutually unbiased bases [1], [2]. In this talk I will showthat, for n ≥ 1,H(4n−1, q2) has a maximal partial spread of size q2n+1,H(4n+1, q2)has a maximal partial spread of size q2n+1 + 1 and, for n ≥ 2, Q+(4n − 1, q),Q(4n − 2, q), W(4n − 1, q), q even, W (4n − 3, q), q even, have a maximal partialspread of size qn + 1. These results are obtained by investigating particular Segrevarieties S1,n “embedded” in polar spaces of Hermitian or hyperbolic type.

References

[1] W. M. Kantor, MUB inequivalence and affine planes, J. Math. Phys. 53 (3)(2012), 9 pp.

[2] P. Mandayam, S. Bandyopadhyay, M. Grassl, W.K. Wootters, Unextendible mu-tually unbiased bases from Pauli classes, Quantum Inf. Comput. 14 (2014), no.9-10, 823-844.

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Automorphisms and permutation decoding of linear codesAngelo Sonnino

Dipartimento di Matematica, Informatica ed EconomiaUniversita degli Studi della Basilicata

Potenza (Italy)

(joint work with Nicola Pace)

Linear codes with large automorphism groups often arise from nice combinatorialobjects embedded in a projective space over a finite field. Such codes are of interestfrom various points of view, and certainly advantageous since the amount of com-putations needed for encoding and decoding can be considerably reduced when theautomorphism group of the code is sufficiently large. The decoding is even moreefficient when the code has a suitable permutation decoding set of automorphisms,and in many cases a large automorphism group grants the existence of such permu-tation decoding sets. Here we present a construction procedure based on varioustypes of groups—for instance low degree projective linear groups, alternating groups,unitary groups and the Higman Sims group—to obtain new linear codes admittinglarge automorphism groups known in advance. Some of these codes turn out to beoptimal, while others either are very close to, or achieve, the known lower bound onthe theoretically largest minimum distance.

17

Symmetric generalizations of Generalized Petersen graphs tohigher valencies

Primoz Sparl

University of Ljubljana, Slovenia

(joint work with Gorazd Vasiljevic)

In 1971 Frucht, Graver and Watkins classified all symmetric generalized Petersengraphs (GPGs) and proved that there are only seven of them. In 2008 Wilson pro-posed to study symmetry properties of tetravalent generalizations of GPGs, namelythe so called Rose-window graphs (obtained from GPGs by inserting an additionalperfect matching between the two orbits of the natural semiregular automorphismwith two orbits). The classification of symmetric Rose-window graphs, completed in2010, revealed that there are four infinite families of such graphs. In 2015 all sym-metric 5-valent generalizations of GPGs (adding another perfect matching) wereclassified. There are just three such graphs.

A natural question thus arises. Is the fact that there is such a drastic differencein the number of symmetric generalizations of GPGs for valencies 3, 4 and 5 dueto the fact that 4 is not a prime or perhaps that it is a power of 2? Is there anyother valency k ≥ 6 such that there exist infinitely many symmetric generalizationsof GPGs of this valency?

In this talk we present a partial answer to the above questions. We show thatthere are infinitely many symmetric 6-valent generalizations of GPGs, discuss someof their properties and point towards a possible classification. We also discuss thesituation with higher valencies.

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Cameron-Liebler sets of generators in finite classical polarspaces

Leo Storme

Ghent University, Department of Mathematics, Krijgslaan 281, 9000 Ghent,Belgium

ls@cage.ugent.be, http://cage.ugent.be/∼ls

(joint work with M. De Boeck, M. Rodgers, and A. Svob)

Cameron-Liebler structures were originally defined by P.J. Cameron and R.A.Liebler. One of the best known definitions is: a Cameron-Liebler set of lines inPG(3, q) is a set of lines having a constant number x of lines in common withevery spread of PG(3, q). But the nice fact is that many equivalent definitions forCameron-Liebler sets of lines in PG(3, q) have been found; thus showing that theyare interesting objects to investigate.

The above mentioned definition was extended by M. Rodgers, L. Storme and A.Vansweevelt to Cameron-Liebler sets of k-spaces in PG(2k + 1, q), as being sets ofk-spaces having a constant number x of k-spaces in common with every k-spread ofPG(2k + 1, q) [2]. Again, several equivalent definitions were proven.

Now, M. De Boeck, M. Rodgers, L. Storme and A. Svob are investigating Cameron-Liebler sets of generators in finite classical polar spaces [1]. Again, the goal is tofind as much as possible equivalent definitions for these Cameron-Liebler sets ofgenerators, and to obtain characterization results on these Cameron-Liebler sets ofgenerators.

In this talk, we present the main ideas about Cameron-Liebler sets of generatorsin finite classical polar spaces, show links with Erdos-Ko-Rado sets of generators infinite classical polar spaces, and also mention open problems for future research.

References

[1] M. De Boeck, M. Rodgers, L. Storme and A. Svob, Cameron-Liebler sets ofgenerators in finite classical polar spaces. (In preparation).

[2] M. Rodgers, L. Storme and A. Vansweevelt, Cameron-Liebler sets of k-spacesin PG(2k + 1, q). Combinatorica, submitted.

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Codes arising from incidence matrices of points andhyperplanes in PG(n, q)

Ferdinando Zullo

Dipartimento di Matematica e Fisica, Seconda Universita degli Studi di Napoli,Italy

ferdinando.zullo@unina2.it

(joint work with Olga Polverino)

Let consider the projective space PG(n, q), with q = ph, h ≥ 1 and p prime. Wedefine the incidence matrix A = (ai,j) of PG(n, q) as the matrix whose rows areindexed by hyperplanes of the space and whose columns are indexed by points ofthe space, and with entry

ai,j =

{

1 if the point j belongs to hyperplane i,0 otherwise

.

The linear code over Fp generated in Fθnp by the rows of the matrix A is denoted

by C(n, q). The minimum weight and the characterisation of the codewords withthis weight are well known (see e.g. [1] and [6]). In [3] the second minimum weightof C(n, q) is determined for p > 5. In this paper we determine for each prime pthe second minimum weight of C(n, q) and we characterise the codewords with thisweight as the scalar multiples of the difference of the incidence vectors of two distincthyperplanes of PG(n, q).

References

[1] Assmus E. F., Key J. D., Designs and their codes, Cambridge University Press,1992.

[2] Bagchi B., Inamdar S.P., Projective geometric codes, J. Combin. Theory Ser.A 99, pp. 128-142, 2002.

[3] Lavrauw M., Storme L., Sziklai P., Van de Voorde G., An empty interval inthe spectrum of small weight codewords in the code from points and k−spacesof PG(n, q), J. Combin. Theory, Ser. A 116, pp. 996-1001, 2009.

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[4] Lavrauw M., Storme L., Van de Voorde G., On the code generated by theincidence matrix of points and hyperplanes in PG(n, q) and its dual, Des.Codes Cryptogr., 48, pp. 231-245, 2008.

[5] Lavrauw M., Storme L., Van de Voorde G., On the code generated by theincidence matrix of points and k−spaces in PG(n, q) and its dual, Finite FieldsAppl. 14 (2008), pp. 1020-1038.

[6] MacWilliams F.J., Sloane N.J.A., The Theory of Error-Correcting Codes, BellLaboratories, Murray Hill, NJ 07974 U.S.A., 1977.

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