The cocyclic Hadamard matrices of order less than 40 |
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Authors: | Padraig Ó Catháin Marc Röder |
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Institution: | (1) School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland;(2) National Security Agency, 9800 Savage Road, Fort George G. Meade, MD 20755-6565, USA |
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Abstract: | In this paper all cocyclic Hadamard matrices of order less than 40 are classified. That is, all such Hadamard matrices are
explicitly constructed, up to Hadamard equivalence. This represents a significant extension and completion of work by de Launey
and Ito. The theory of cocyclic development is discussed, and an algorithm for determining whether a given Hadamard matrix
is cocyclic is described. Since all Hadamard matrices of order at most 28 have been classified, this algorithm suffices to
classify cocyclic Hadamard matrices of order at most 28. Not even the total numbers of Hadamard matrices of orders 32 and
36 are known. Thus we use a different method to construct all cocyclic Hadamard matrices at these orders. A result of de Launey,
Flannery and Horadam on the relationship between cocyclic Hadamard matrices and relative difference sets is used in the classification
of cocyclic Hadamard matrices of orders 32 and 36. This is achieved through a complete enumeration and construction of (4t, 2, 4t, 2t)-relative difference sets in the groups of orders 64 and 72. |
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