# zbMATH — the first resource for mathematics

Constructing cocyclic Hadamard matrices of order $$4p$$. (English) Zbl 1451.05032
Summary: Cocyclic Hadamard matrices (CHMs) were introduced by W. de Launey and K. J. Horadam [Des. Codes Cryptography 3, No. 1, 75–87 (1993; Zbl 0838.05019)] as a class of Hadamard matrices (HMs) with interesting algebraic properties. P. Ó Catháin and M. Röder [Des. Codes Cryptography 58, No. 1, 73–88 (2011; Zbl 1246.05033)] 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. On the basis of work of W. de Launey and D. Flannery [Algebraic design theory. Providence, RI: American Mathematical Society (AMS) (2011; Zbl 1235.05001)], we describe a classification algorithm for CHMs of order $$4p$$ with $$p$$ a prime; we prove refined structure results and provide a classification for $$p \leq 13$$. Our analysis shows that every CHM of order $$4p$$ with $$p \equiv 1 \pmod 4$$ is equivalent to a HM with one of five distinct block structures, including Williamson-type and (transposed) Ito matrices. If $$p \equiv 3 \pmod 4$$, then every CHM of order $$4 p$$ is equivalent to a Williamson-type or (transposed) Ito matrix.
##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.) 05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
##### Keywords:
cocyclic development; Hadamard matrix; Ito type; Williamson type
Full Text: