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Zbl 0943.05026
Shinoda, Koichi; Yamada, Mieko
A family of Hadamard matrices of dihedral group type.
(English)
[J] Discrete Appl. Math. 102, No.1-2, 141-150 (2000). ISSN 0166-218X

Summary: Let $D_{2n}$ be a dihedral group of order $2n$ and $\bbfZ$ 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 $\bbfZ[D_{2n}]$ 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 $\bbfZ[D_{2p}]$ 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 2000:
*05B20 (0,1)-matrices (combinatorics)
11L05 Gauss and Kloosterman sums

Keywords: dihedral group; group ring; Hadmard matrix; Gauss sum

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