The cocyclic Hadamard matrices of order less than 40
 Padraig Ó Catháin,
 Marc Röder
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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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 Title
 The cocyclic Hadamard matrices of order less than 40
 Journal

Designs, Codes and Cryptography
Volume 58, Issue 1 , pp 7388
 Cover Date
 20110101
 DOI
 10.1007/s1062301093859
 Print ISSN
 09251022
 Online ISSN
 15737586
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Cocyclic Hadamard matrices
 Relative difference sets
 Classification of Hadamard matrices
 05B10
 05B20
 Industry Sectors
 Authors

 Padraig Ó Catháin ^{(1)}
 Marc Röder ^{(1)}
 Author Affiliations

 1. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland