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On Z Z t × Z Z 2 2 ‐Cocyclic Hadamard Matrices
Author(s) -
Álvarez Víctor,
Gudiel Félix,
Belén Güemes Maria
Publication year - 2015
Publication title -
journal of combinatorial designs
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.618
H-Index - 34
eISSN - 1520-6610
pISSN - 1063-8539
DOI - 10.1002/jcd.21406
Subject(s) - hadamard transform , mathematics , hadamard matrix , complex hadamard matrix , combinatorics , disjoint sets , hadamard product , matrix (chemical analysis) , hadamard's maximal determinant problem , set (abstract data type) , orthogonality , hadamard three lines theorem , hadamard's inequality , order (exchange) , discrete mathematics , computer science , mathematical analysis , materials science , geometry , finance , economics , composite material , programming language
A characterization ofZ Z t × Z Z 2 2‐cocyclic Hadamard matrices is described, depending on the notions of distributions , ingredients , and recipes . In particular, these notions lead to the establishment of some bounds on the number and distribution of 2‐coboundaries overZ Z t × Z Z 2 2to use and the way in which they have to be combined in order to obtain aZ Z t × Z Z 2 2‐cocyclic Hadamard matrix. Exhaustive searches have been performed, so that the table in p. 132 in A. Baliga, K. J. Horadam, Australas. J. Combin., 11 (1995), 123–134 is corrected and completed. Furthermore, we identify four different operations on the set of coboundaries definingZ Z t × Z Z 2 2‐cocyclic matrices, which preserve orthogonality. We split the set of Hadamard matrices into disjoint orbits, define representatives for them, and take advantage of this fact to compute them in an easier way than the usual purely exhaustive way, in terms of diagrams . Let H be the set of cocyclic Hadamard matrices overZ Z t × Z Z 2 2having a symmetric diagram. We also prove that the set of Williamson‐type matrices is a subset of H of size| H | t .