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Hadamard matrices and their applications: Progress 2007–2010

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Abstract

We survey research progress in Hadamard matrices, especially cocyclic Hadamard matrices, their generalisations and applications, made over the past three years. Advances in 20 specific problems and several new research directions are outlined. Two new problems are presented.

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Notes

  1. Also denoted GH(v, N), or GH(v, m) when N is a group of m th roots of unity.

  2. The “opposite” multiplication in Aut(N) is needed: σ 1 σ 2 = σ 1 ∘ σ 2.

  3. The formula in [34, Def. 6.2] is \(\partial\phi(g, h) = \phi(g)^{-1} (\phi(h)^{\varepsilon(g)})^{-1} \phi(g h)\); that is, the coboundary \(\partial(\phi^{-1})\) according to (3). Both definitions of \(\partial\phi\) are in use. Each is correct, if applied consistently.

  4. A different definition of Hadamard graph (due to Ito [39]) is the graph with vertex set \(V_{4n} = \{0, 1\}^{4n}\) and edge set \(E_{4n} = \{(u, v) \in V_{4n}^2 ~|~ d_H(u, v) = 2n\}\), where d H (u, v) is the Hamming distance between u and v.

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Acknowledgements

I am most grateful to the referees for comprehensive and expert reviews which greatly assisted me to clarify and polish this survey.

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    K. J. Horadam

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Correspondence to K. J. Horadam.

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Dedicated to Warwick de Launey for his 50th birthday.

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Horadam, K.J. Hadamard matrices and their applications: Progress 2007–2010. Cryptogr. Commun. 2, 129–154 (2010). https://doi.org/10.1007/s12095-010-0032-0

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