A system of equations for describing cocyclic Hadamard matrices

• Published in: Journal of combinatorial designs 2008-07, Vol.16 (4), p.276-290
• Main Authors: Álvarez, V., Armario, J. A., Frau, M. D., Real, P.
• Format: Article
• Language: English
• Subjects: coboundary matrix; cocyclic matrix; Hadamard matrix
• ISSN: 1063-8539; 1520-6610
• DOI: 10.1002/jcd.20191

Given a basis ${\cal B} = \{f_1,\ldots, f_k\}$ for 2‐cocycles $f:G \times G \rightarrow \{\pm 1\}$ over a group G of order $\vert G\vert=4t$, we describe a nonlinear system of 4t‐1 equations and k indeterminates $x_i$ over ${\cal Z}_2, 1\leq i \leq k$, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that ($x_1,\ldots ,x_k$) is a solution of the system if and only if the 2‐cocycle $f=f_1^{x_1}\cdots f_k^{x_k}$ gives rise to a cocyclic Hadamard matrix $M_f=(f(g_i,g_j))$. Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in ${\cal B}$ which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups ${\cal Z}_2^2 \times {\cal Z}_t$ and $D_{4t}$. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups $D_{4t}$, we have found that it suffices to check t instead of the 4t rows of $M_f$, to decide the Hadamard character of the matrix (for a special class of cocycles f). © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 276–290, 2008