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A family of Hadamard matrices of dihedral group type. (English) Zbl 0943.05026
Summary: Let D 2n be a dihedral group of order 2n and be the rational integer ring where n is an odd integer. Kimura gave the necessary and sufficient conditions such that a matrix of order 8n+4 obtained from the elements of the group ring [D 2n ] becomes a Hadmard matrix. We show that if p1(mod4) is an odd prime and q=2p-1 is a prime power, then there exists a family of Hadamard matrices of dihedral group type. We prove this theorem by giving the elements of [D 2p ] concretely. The Gauss sum over GF(p) and the relative Gauss sum over GF(q 2 ) are important to prove the theorem.
MSC:
05B20Matrices (incidence, Hadamard, etc.)
11L05Gauss and Kloosterman sums