Cocyclic Development of Designs
Journal of Algebraic Combinatorics: An International Journal
Classification of Hadamard matrices of order 24 and 28
Discrete Mathematics
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Cocyclic Hadamard matrices and difference sets
Discrete Applied Mathematics - Coding, cryptography and computer security
Switching Operations for Hadamard Matrices
SIAM Journal on Discrete Mathematics
On twin prime power Hadamard matrices
Cryptography and Communications
Difference sets and doubly transitive actions on Hadamard matrices
Journal of Combinatorial Theory Series A
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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.