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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

##### MSC:
 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
##### Keywords:
Gauss sums; Williamson equation; Hadamard matrices
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