A family of Hadamard matrices of dihedral group type Original Research Article

Let D2n be a dihedral group of order 2n and 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 Z[D2n] becomes a Hadamard matrix. We show that if p≡1 (mod 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 Z[D2p] concretely. The Gauss sum over GF(p) and the relative Gauss sum over GF(q2) are important to prove the theorem.