Classifying cocyclic Butson Hadamard matrices. (English) Zbl 1329.05040

Egan, Ronan; Flannery, Dane; Ó Catháin, Padraig

Colbourn, Charles J. (ed.), Algebraic design theory and Hadamard matrices. ADTHM, Lethbridge, Alberta, Canada, July 8–11, 2014. Selected papers based on the presentations at the workshop and at the workshop on algebraic design theory with Hadamard matrices: applications, current trends and future directions, Banff International Research Station, Alberta, Canada, July 11–13, 2014. Cham: Springer (ISBN 978-3-319-17728-1/hbk; 978-3-319-17729-8/ebook). Springer Proceedings in Mathematics & Statistics 133, 93-106 (2015).

Summary: We classify all the cocyclic Butson Hadamard matrices \(\mathrm{BH}(n,p)\) of order \(n\) over the \(p\)th roots of unity for an odd prime \(p\) and \(np\leq 100\). That is, we compile a list of matrices such that any cocyclic \(\mathrm{BH}(n,p)\) for these \(n\), \(p\) is equivalent to exactly one element in the list. Our approach encompasses non-existence results and computational machinery for Butson and generalized Hadamard matrices that are of independent interest.
For the entire collection see [Zbl 1329.05003].

MSC:

05B20	Combinatorial aspects of matrices (incidence, Hadamard, etc.)
20B25	Finite automorphism groups of algebraic, geometric, or combinatorial structures
20J06	Cohomology of groups
15B34	Boolean and Hadamard matrices

Keywords:

automorphism group; Butson Hadamard matrix; cocyclic matrices; relative difference set

Software:

GAP; Magma; RDS