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On Cocyclic Hadamard Matrices over Goethals-Seidel Loops

Author

Listed:
  • Víctor Álvarez

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • José Andrés Armario

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • Raúl M. Falcón

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • María Dolores Frau

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • Félix Gudiel

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • María Belén Güemes

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

  • Amparo Osuna

    (Department of Applied Mathematics I, University of Seville, 41004 Seville, Spain)

Abstract

About twenty-five years ago, Horadam and de Launey introduced the cocyclic development of designs, from which the notion of cocyclic Hadamard matrices developed over a group was readily derived. Much more recently, it has been proved that this notion may naturally be extended to define cocyclic Hadamard matrices developed over a loop. This paper delves into this last topic by introducing the concepts of coboundary, pseudocoboundary and pseudococycle over a quasigroup, and also the notion of the pseudococyclic Hadamard matrix. Furthermore, Goethals-Seidel loops are introduced as a family of Moufang loops so that every Hadamard matrix of Goethals-Seidel type (which is known not to be cocyclically developed over any group) is actually pseudococyclically developed over them. Finally, we also prove that, no matter if they are pseudococyclic matrices, the usual cocyclic Hadamard test is unexpectedly applicable.

Suggested Citation

  • Víctor Álvarez & José Andrés Armario & Raúl M. Falcón & María Dolores Frau & Félix Gudiel & María Belén Güemes & Amparo Osuna, 2019. "On Cocyclic Hadamard Matrices over Goethals-Seidel Loops," Mathematics, MDPI, vol. 8(1), pages 1-23, December.
    • Handle: RePEc:gam:jmathe:v:8:y:2019:i:1:p:24-:d:300421
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