A family of Hadamard matrices of dihedral group type

Abstract

Let D 2 n be a dihedral group of order 2 n 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 8 n+4 obtained from the elements of the group ring Z [ D 2 n ] becomes a Hadamard matrix. We show that if p≡1 ( mod 4) is an odd prime and q=2 p−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 [ D 2 p ] concretely. The Gauss sum over GF( p) and the relative Gauss sum over GF( q 2) are important to prove the theorem.