A system of equations for describing cocyclic Hadamard matrices
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Abstract
Abstract 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 4 t ‐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 4 t 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