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Williamson Hadamard matrices and Gauss sums. (English) Zbl 0584.05016
In this paper, the study of the well-known Williamson equation is continued for constructing Hadamard matrices of Williamson type. Some important special cases of the equation are investigated. The main result is contained in the following theorem: Let n be odd and \(q=2n-1\) be a power of prime. Let K be a quadratic extension of the finite field \(F=GF(g)\), \(\psi\) be the quadratic residue character of F and \(\alpha\) be an element of K such that \(2F^*\) generates the quotient group \(K^*/F^*\). Let’s define \(z_ m\) \((m=1,2,...,n-1):\) \[ z_ m=(\psi (2S_{K/F}\alpha^{4m})-i\psi (2S_{K/F}\alpha^{4m-n}))/(1-i), \] where \(S_{K/F}\) is the relative trace in K/F. Then \(H=wI+\sum^{n- 1}_{m=1}z_ mT^ m\) is a symmetric circulant quaternion Hadamard matrix of order n where I is the unit matrix and T is the basic circulant matrix \((T^ n=I)\), \(w=(1+i+j+k)/2\), 1,i,j,k are the quaternion units.
Reviewer: S.S.Agayan

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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