104 research outputs found

    Classification of Cyclic Steiner Quadruple Systems

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    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

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    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

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    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 d=48d=48 and d=72d=72

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    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 d=48d=48 and d=72d=72. 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

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    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

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    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

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    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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