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July 2016 - present

February 2007 - January 2008

## Publications

Publications (72)

It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays, when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number o...

It is known that correlation-immune (CI) Boolean functions used in the framework of side-channel attacks need to have low Hamming weights. The supports of CI functions are (equivalently) simple orthogonal arrays when their elements are written as rows of an array. The minimum Hamming weight of a CI function is then the same as the minimum number of...

We show that the Ree unital R(q) has an embedding in a projective plane over a field F if and only if q=3 and F8 is a subfield of F. In this case, the embedding is unique up to projective linear transformations. Besides elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree groups.

Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of S...

Дано решение открытой проблемы олимпиады по криптографии NSUCRYPTO-2018: показано, что не существует ортогональных массивов OA (16L, 11, 2, 4) с L = 6 и 7. Этот результат позволяет определить минимальные веса некоторых корреляционно-иммунных булевых функций высокого порядка.

We show that no orthogonal arrays $OA(16 \lambda, 11, 2,4)$ exist with $\lambda=6$ and $\lambda=7$. This solves an open problem of the NSUCRYPTO Olympiad 2018.

In this paper, we study the behavior of the true dimension of the subfield subcodes of Hermitian codes. Our motivation is to use these classes of linear codes to improve the parameters of the McEliece cryptosystem, suchas key size and security level. The McEliece scheme is one of the promising alternative cryptographic schemes to the current public...

We show that the Ree unital $\mathcal{R}(q)$ has an embedding in a projective plane over a field $F$ if and only if $q=3$ and $\mathbb{F}_8$ is a subfield of $F$. In this case, the embedding is unique up to projective linear transformations. Beside elementary calculations, our proof uses the classification of the maximal subgroups of the simple Ree...

In this paper, we study the behavior of the true dimension of the subfield subcodes of Hermitian codes. Our motivation is to use these classes of linear codes to improve the parameters of the McEliece cryptosystem, such that key size and security level. The McEliece scheme is one of the promising alternative cryptographic schemes to the current pub...

The main result of this paper is a general construction which produces new Steiner systems (2-designs) from old ones with the same parameters. We call this construction paramodification of Steiner systems, since it modifies the parallelism of a subsystem. We study in more details the paramodifications of affine planes, Steiner triple systems and un...

The concept of full points of abstract unitals has been introduced by Korchmáros, Siciliano and Szőnyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order.

Hermitian functional and differential codes are AG-codes defined on a Hermitian curve. To ensure good performance, the divisors defining such AG-codes have to be carefully chosen, exploiting the rich combinatorial and algebraic properties of the Hermitian curves. In this paper, the case of differential codes C
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A group $G$ has a Frobenius graphical representation (GFR) if there is a simple graph $\varGamma$ whose full automorphism group is isomorphic to $G$ and it acts on vertices as a Frobenius group. In particular, any group $G$ with GFR is a Frobenius group and $\varGamma$ is a Cayley graph. The existence of an infinite family of groups with GFR whose...

Subfield subcodes of algebraic-geometric codes are good candidates for the use in post-quantum cryptosystems, provided their true parameters such as dimension and minimum distance can be determined. In this paper we present new values of the true dimension of subfield subcodes of $1$--point Hermitian codes, including the case when the subfield is n...

Embedded multinets are line arrangements of the projective plane with a rich combinatorial structure. In this paper, we first classify all abstract light dual multinets of order 6 which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero....

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of t...

The concept of full points of abstract unitals has been introduced by Korchm\'aros, Siciliano and Sz\H{o}nyi as a tool for the study of projective embeddings of abstract unitals. In this paper we give a more detailed description of the combinatorial and geometric structure of the sets of full points in abstract unitals of finite order.

The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak...

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of ${\rm PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredi...

We present a new method for the study of hemisystems of the Hermitian surface $\mathcal{U}_3$ of $PG(3,q^2)$. The basic idea is to represent generator-sets of $\mathcal{U}_3$ by means of a maximal curve naturally embedded in $\mathcal{U}_3$ so that a sufficient condition for the existence of hemisystems may follow from results about maximal curves...

In this paper we investigate light dual multinets labeled by a finite group in the projective plane $PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$. We present two classes of new examples. Moreover, under some conditions on the characteristic $\mathbb{K}$, we classify group-labeled light dual multinets with lines of length least $9$.

Right Bol loops are loops satisfying the identity $((zx)y)x = z((xy)x)$, and right Bruck loops are right Bol loops satisfying the identity $(xy)^{-1} = x^{-1}y^{-1}$. Let $p$ and $q$ be odd primes such that $p>q$. Advancing the research program of Niederreiter and Robinson from $1981$, we classify right Bol loops of order $pq$. When $q$ does not di...

A 3-net of order n is a finite incidence structure consisting of points and three pairwise disjoint classes of lines, each of size n, such that every point incident with two lines from distinct classes is incident with exactly one line from each of the three classes. The current interest around 3-nets (embedded) in a projective plane \(\mathrm{PG}(...

A finite -net of order is an incidence structure consisting of pairwise disjoint classes of lines, each of size , such that every point incident with two lines from distinct classes is incident with exactly one line from each of the classes. Deleting a line class from a -net, with , gives a derived ( )-net of the same order. Finite -nets embedded i...

In this short paper, we survey the results on commutative automorphic loops
and give a new construction method. Using this method, we present new classes
of commutative automorphic loops of exponent 2 with trivial center.

The intrinsic structure of binary fields poses a challenging complexity
problem from both hardware and software point of view. Motivated by
applications to modern cryptography, we describe some simple techniques aimed
at performing computations over binary fields using systems with limited
resources. This is particularly important when such computa...

Matthews and Michel (2005) [29] investigated the minimum distances of certain algebraic-geometry codes arising from a higher degree place PP. In terms of the Weierstrass gap sequence at PP, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the He...

Korchmáros and Nagy [Hermitian codes from higher degree places. J Pure Appl Algebra, doi: 10. 1016/j.jpaa.2013.04.002, 2013] computed the Weierstrass gap sequence G(P) of the Hermitian function field
at any place P of degree 3, and obtained an explicit formula of the Matthews-Michel lower bound on the minimum distance in the associated differenti...

We investigate $k$-nets with $k\geq 4$ embedded in the projective plane
$PG(2,\mathbb{K})$ defined over a field $\mathbb{K}$; they are line
configurations in $PG(2,\mathbb{K})$ consisting of $k$ pairwise disjoint
line-sets, called components, such that any two lines from distinct families
are concurrent with exactly one line from each component. Th...

For quasifields, the concept of parastrophy is slightly weaker than isotopy.
Parastrophic quasifields yield isomorphic translation planes but not
conversely. We investigate the right multiplication groups of finite
quasifields. We classify all quasifields having an exceptional finite
transitive linear group as right multiplication group. The classi...

We prove that every finite, commutative automorphic loop is solvable. We also
prove that every finite, automorphic 2-loop is solvable. The main idea of the
proof is to associate a simple Lie algebra of characteristic 2 to a
hypothetical finite simple commutative automorphic loop. The "crust of a thin
sandwich" theorem of Zel'manov and Kostrikin lea...

A loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 2 12 , and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples o...

In this paper, we investigate dual 3-nets realizing the groups $C_3 \times
C_3$, $C_2 \times C_4$, $\Alt_4$ and that can be embedded in a projective plane
$PG(2,\mathbb K)$, where $\mathbb K$ is an algebraically closed field. We give
a symbolically verifiable computational proof that every dual 3-net realizing
the groups $C_3 \times C_3$ and $C_2 \...

In a projective plane PG(2,K) defined over an algebraically closed field K of
characteristic 0, we give a complete classification of 3-nets realizing a
finite group. An infinite family, due to Yuzvinsky, arises from plane cubics
and comprises 3-nets realizing cyclic and direct products of two cyclic groups.
Another known infinite family, due to Per...

This paper is a relatively short survey the aim of which is to present the theory of semifields and the related areas of finite geometry to loop theorists.

For a flnite group G let ¡(G) denote the graph deflned on the non- identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. In this paper it is shown that the graph ¡(G) contains a Hamiltonian cycle for many flnite groups G. In the literature many deep results about flnite simple group...

In this short note we present a simple combinatorial trick which can be effectively applied to show the non--existence of sharply transitive sets of permutations in certain finite permutation groups. Comment: 8 pages

We investigate the multiplicative loops of finite semifields. We show that the group generated by the left and right multiplication maps contains the special linear group. This result solves a BCC18 problem of A. Drápal. Moreover, we study the question of whether the big Mathieu groups can occur as multiplication groups of loops.

In this paper, we study the category of algebraic Bol loops over an algebraically closed field of definition. On the one hand, we apply techniques from the theory of algebraic groups in order to prove structural theorems for this category. On the other hand, we present some examples showing that these loops lack some nice properties of algebraic gr...

We investigate the relation between the structure of a Moufang loop and its
inner mapping group. Moufang loops of odd order with commuting inner mappings
have nilpotency class at most two. $6$-divisible Moufang loops with commuting
inner mappings have nilpotency class at most two. There is a Moufang loop of
order $2^{14}$ with commuting inner mappi...

Code loops were introduced by R.L. Griess Jr. [Code loops, J. Algebra 100 (1) (1986) 224–234] and T. Hsu [Explicit constructions of code loops as centrally twisted products, Math. Proc. Cambridge Philos. Soc. 128 (2) (2000) 223–232] gave methods to construct the corresponding code loop from any given doubly even binary code; both these methods used...

In this paper we make some remarks on simple Bol loops which were motivated by questions at the LOOPS'07 conference. We also list some open problems on simple loops.

In this short note we show that the group of projectiv- ities of a projective plane of order 23 cannot be isomorphic to the Mathieu group M24. By a result of T. Grundhofer (5), this implies that the group of projectivities of a non-desarguesian projective plane of finite order n is isomorphic either to the alternating group An+1 or to the symmetric...

In this paper we give an infinite class of finite simple right Bol loops of exponent 2. The right multiplication group of these loops is an extension of an elementary Abelian 2-group by $S_5$. The construction uses the description of the structure of such loops given by M. Aschbacher. These results answer some questions of M. Aschbacher.

We classify Moufang loops of order 64 and 81 up to isomorphism, using a
linear algebraic approach to central loop extensions. In addition to the 267
groups of order 64, there are 4262 nonassociative Moufang loops of order 64. In
addition to the $15$ groups of order $81$, there are $5$ nonassociative Moufang
loops of order $81$, $2$ of which are com...

The existence of finite simple non-Moufang Bol loops was considered as one of the main open problems in the theory of loops and quasigroups. In this paper, we present a class of proper simple Bol loops. This class also contains finite and new infinite simple proper Bol loops.

In this paper, we investigate Moufang p-loops of nilpotency class at least three for p>3. The smallest examples have order p5 and satisfy the following properties: (1) They are of maximal nilpotency class, (2) their associators lie in the center, and (3) they can be constructed using a general form of the semidirect product of a cyclic group and a...

This is a companion to our lectures GAP and loops, to be delivered at the
Workshops Loops 2007, Prague, Czech Republic. In the lectures we introduce the
GAP package LOOPS, describe its capabilities, and explain in detail how to use
it. In this paper we first outline the philosophy behind the package and its
main features, and then we focus on three...

Analogously to extraspecial $p$-groups, we define the class of small Frattini $p$-loops and investigate some properties of small Frattini Bol 2-loops. We show that they are related to certain invariant transversals in extraspecial 2-groups. This fact enables us to describe the automorphisms and isomorphisms of small Frattini Bol 2-loops by linear a...

Some associativity properties of a loop can be interpreted as certain closure configuration of the corresponding 3-net. It was known that the smallest non-associative loops with the so called left Bol property have order 8. In this paper, we determine the direction preserving collineation groups of the 3-nets belonging to these smallest Bol loops....

In this paper, we determine the collineation groups generated by the Bol reflections, the core, the automorphism groups and the full direction preserving collineation groups of the loops $B_{4n}$ and $C_{4n}$ given by R.P. Burn. We also prove some lemmas and use new methods in order to simplify the calculations in these groups.

It is known that the concept of Moufang loops, Moufang 3-nets and groups with triality are strongly related. Due to S. Doro, a group with a splitting automorphism of order 3 can lead to a group with triality. This construction naturally appears in the classification of simple Moufang loops. In this paper, we consider groups with triality related to...

Let $F$ be a perfect field and $M(F)$ the non-associative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then ${\rm Aut}(M(F))$ is equal to $G_2(F) \rtimes {\rm Aut}(F)$. In particular, every automorphism of $M(F)$ is induced by a semilinear automorphism of $O(F)$. The proof combines res...

In this paper, we study the category of algebraic commutative Moufang loops over an algebraically closed field of definition. The most important properties of this class of loops are their central nilpotence, the characteristic 3 of the field of definition and that their multiplication group has a unique structure of an algebraic transformation gro...

Nonassociative finite simple Moufang loops are exactly the loops constructed by Paige from Zorn vector matrix algebras. We prove this result anew, using geometric loop theory. In order to make the paper accessible to a broader audience, we carefully discuss the connections between composition algebras, simple Moufang loops, simple Moufang 3-nets, $...

The class of local analyitic Bruck loops (or equivalently K-loops) is strongly related to locally symmetric spaces. In particular,
both have Lie triple systems as their tangent algebra. In this paper, we consider the existence and some properties of the
Campbell-Hausdorff series of local analytic Bruck loops (K-loops). This formula can be used to d...

The group theoretical problem of the existence of a system of representativesT of the subgroup H of G such that T consists of conjugacy classes of involutions leads to the theory of Bol loops of exponent
2.
In this paper, we develop a theory of extensions of such loops and give two applications of the theory. First, we classify
all (left) Bol loop...

Let p be a prime, L a finite loop of p-power order and F a field of characteristic p. We show that the fundamental ideal of the loop ring FL is nilpotent if and only if the multiplication group of L is a p-group. We apply this theorem to answer a question of E.G. Goodaire.

In the theory of loops, the class of local analytic Bruck loop plays a substantial role, mainly because of its strong relation with symmetric spaces. Like for formal groups, one can derive the concept of formal loops from the classical theory of local analytic loops in a natural way. Also the process of localization of algebraic loops leads to form...

In his paper [6], S. Doro constructed a partial relationship between Mo-ufang loops and groups with triality. We extend this relationship by showing that the following concepts are equivalent: Groups with triality and trivial centre, Moufang 3-nets, Latin square designs in which every point is the cen-tre of an automorphism, isotopy classes of Mouf...

In this paper we consider the Burnside problems for the class of Moufang and Bol loops. We show that a free finitely generated Moufang loop of exponent 3 is finite and that the orders of the finite 2-generated Bol loops of exponent 2 are not bounded.

In this paper upper bounds on the number of points of large complete caps in PG(n, q), (q odd,n3) are derived. The previously known bounds of Hirschfeld and Segre are improved using recent bounds on the cardinality of the second largest complete cap in the plane PG(2,q).

We consider the special properties of the tangent Lie triple systems of algebraic commutative Moufang loops and relate them with the tangent Lie algebras of the multiplication group of the loops. Using this relation, we obtain an algebraic homomorphism of the group of inner mappings of the loop in the (linear) group of automorphisms of the tangent...

Nagy G. olyan geometriai struktúrákat vizsgált, melyek Moufang-féle és Bol-féle egységelemes kvázicsoportokkal (loopokkal) koordinátázhatóak. a) Kis Frattini 2-loopok, azaz melyeknél L/A elemi Abel 2-csoport valamely 2-rendű A normális részloopra. A Bol-esetben explicit formulát, a Moufang-esetben új globális konstrukciót adott. b) Moufang-féle p-l...

Megmutattuk, hogy négyzet q-ra PG(2,q)-ban 4qlog q és q^(3/2)-q+2q^(1/2) között minden méretű minimális lefogó ponthalmaz létezik, sőt egy kicsit szűkebb intervallum minden értékére q-ban több, mint polinomnyi. Magasabb dimenziós projektív terekben a hipersíkokat r modulo p pontban metsző halmazok méretére bizonyos esetekben éles alsó becslést adtu...