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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 \(\mathbb{Z}\) 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 \(\mathbb{Z}[D_{2n}]\) becomes a Hadmard matrix. We show that if \(p\equiv 1\pmod 4\) 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 \(\mathbb{Z}[D_{2p}]\) concretely. The Gauss sum over \(\text{GF}(p)\) and the relative Gauss sum over \(\text{GF}(q^2)\) are important to prove the theorem.

MSC:

05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
11L05 Gauss and Kloosterman sums; generalizations
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References:

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