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arXiv · 1904.11460

Constructing cocyclic Hadamard matrices of order 4p

Santiago Barrera Acevedo·Heiko Dietrich·Padraig O Cathain
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Abstract

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. \'O Cath\'ain and R\"oder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.

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Santiago Barrera Acevedo, Heiko Dietrich, Padraig O Cathain. 2019-04-25. Constructing cocyclic Hadamard matrices of order 4p. https://arxiv.org/abs/1904.11460

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