On Cocyclic Hadamard Matrices over Goethals-Seidel Loops

Víctor Álvarez, José Andrés Armario, Raúl M. Falcón, María Dolores Frau, Félix Gudiel, María Belén Güemes and Amparo Osuna
Additional contact information
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

Mathematics, 2019, vol. 8, issue 1, 1-23

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.

Keywords: Hadamard matrix; cocyclic matrix; quasigroup; Goethals-Seidel array (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/8/1/24/pdf (application/pdf)
https://www.mdpi.com/2227-7390/8/1/24/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:8:y:2019:i:1:p:24-:d:300421

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().