104 research outputs found
Classification of Cyclic Steiner Quadruple Systems
The problem of classifying cyclic Steiner quadruple systems (CSQSs) is considered. A computational approach shows that the number of isomorphism classes of such designs with orders 26 and 28 is 52,170 and 1,028,387, respectively. It is further shown that CSQSs of order 2p, where p is a prime, are isomorphic iff they are multiplier equivalent. Moreover, no CSQSs of order less than or equal to 38 are isomorphic but not multiplier equivalent
Inequivalence of difference sets: on a remark of Baumert
An often cited statement of Baumert in his book Cyclic difference sets asserts that four well known families of cyclic (4t - 1,2t - 1,t - 1) difference sets are inequivalent, apart from a small number of exceptions with t ≤ 8. We are not aware of a proof of this statement in the literature. Three of the families discussed by Baumert have analogous constructions in non-cyclic groups. We extend his inequivalence statement to a general inequivalence result, for which we provide a complete and self-contained proof. We preface our proof with a survey of the four families of difference sets, since there seems to be some confusion in the literature between the cyclic and non-cyclic cases
Automorphism groups of Wada dessins and Wilson operations
Dessins d'enfants (children's drawings) may be defined as hypermaps, i.e. as bipartite graphs embedded in compact Riemann surfaces. They are very important objects in order to describe the surface of the embedding as an algebraic curve. Knowing the combinatorial properties of the dessin may, in fact, help us determining defining equations or the field of definition of the surface. This task is easier if the automorphism group of the dessin is "large". In this thesis we consider a special type of dessins, so-called Wada dessins, for which the underlying graph illustrates the incidence structure of points and of hyperplanes of projective spaces. We determine under which conditions they have a large orientation-preserving automorphism group. We show that applying algebraic operations called "mock" Wilson operations to the underlying graph we may obtain new dessins. We study the automorphism group of the new dessins and we show that the dessins we started with are coverings of the new ones.Dessins d'enfants (Kinderzeichnungen) wurden zuerst von Grothendieck (1984) als Objekte eingeführt, die sehr einfach, aber sehr wichtig sind, um kompakte Riemannsche Flächen als glatte algebraische Kurven über einem Zahlenkörper zu beschreiben. Dessins d'enfants können durch ihre Walsh-Darstellung definiert werden und entsprechen bipartiten Graphen, die in Riemannschen Flächen eingebettet sind. Ein grundlegendes Problem ist es, wie man aus den kombinatorischen Eigenschaften des Dessins auf die algebraischen Eigenschaften der Fläche, wie z.B. auf definierende Gleichungen und auf den Definitionskörper, schließen kann. Die Aufgabe ist normalerweise sehr schwierig, aber sie ist einfacher, wenn die Automorphismengruppe des Dessins besonders "groß" ist. In dieser Arbeit beschäftigen wir uns mit einem speziellen Typ von Dessins, mit sogenannten Wada-Dessins. Der zugrundeliegende Graph stellt die Inzidenzstruktur von Punkten und von Hyperebenen projektiver Räume dar. Wir bestimmen, unter welchen Bedingungen die orientierungserhaltende Automorphismengruppe "groß" ist. Wir zeigen, daß sich neue Dessins aus den ursprünglichen konstruieren lassen, wenn wir auf dem zugrundeliegenden Graphen sogenannte "mock" Wilson-Operationen anwenden. Wir analysieren die Automorphismengruppe der neuen Dessins und zeigen, daß die ursprünglichen Dessins Überlagerungen der neuen Dessins sind
Orbifolds of Lattice Vertex Operator Algebras at and
Motivated by the notion of extremal vertex operator algebras, we investigate
cyclic orbifolds of vertex operator algebras coming from extremal even
self-dual lattices in and . In this way we construct about one
hundred new examples of holomorphic VOAs with a small number of low weight
states.Comment: 18 pages, LaTe
Sets of three pairwise orthogonal Steiner triple systems
AbstractTwo Steiner triple systems (STS) are orthogonal if their sets of triples are disjoint, and two disjoint pairs of points defining intersecting triples in one system fail to do so in the other. In 1994, it was shown (Canad. J. Math. 46(2) (1994) 239–252) that there exist a pair of orthogonal Steiner triple systems of order v for all v≡1,3 (mod6), with v⩾7, v≠9. In this paper we show that there exist three pairwise orthogonal Steiner triple systems of order v for all v≡1(mod6), with v⩾19 and for all v≡3(mod6), with v⩾27 with only 24 possible exceptions
Folding Alternant and Goppa Codes with Non-Trivial Automorphism Groups
The main practical limitation of the McEliece public-key encryption scheme is
probably the size of its key. A famous trend to overcome this issue is to focus
on subclasses of alternant/Goppa codes with a non trivial automorphism group.
Such codes display then symmetries allowing compact parity-check or generator
matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC)
or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such
symmetric alternant/Goppa codes in cryptography introduces a fundamental
weakness. It is indeed possible to reduce the key-recovery on the original
symmetric public-code to the key-recovery on a (much) smaller code that has not
anymore symmetries. This result is obtained thanks to a new operation on codes
called folding that exploits the knowledge of the automorphism group. This
operation consists in adding the coordinates of codewords which belong to the
same orbit under the action of the automorphism group. The advantage is
twofold: the reduction factor can be as large as the size of the orbits, and it
preserves a fundamental property: folding the dual of an alternant (resp.
Goppa) code provides the dual of an alternant (resp. Goppa) code. A key point
is to show that all the existing constructions of alternant/Goppa codes with
symmetries follow a common principal of taking codes whose support is globally
invariant under the action of affine transformations (by building upon prior
works of T. Berger and A. D{\"{u}}r). This enables not only to present a
unified view but also to generalize the construction of QC, QD and even
quasi-monoidic (QM) Goppa codes. All in all, our results can be harnessed to
boost up any key-recovery attack on McEliece systems based on symmetric
alternant or Goppa codes, and in particular algebraic attacks.Comment: 19 page
Primitive symmetric designs with at most 255 points
In this paper we either prove the non-existence or give explicit construction of all (v, k, λ) symmetric designs with primitive automorphism groups of degree v ≤ 255. We prove that, up to isomorphism, there exist exactly 142 such designs. The research involves programming and wide-range computations. We make use of software package GAP and the library of primitive groups which it contains
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