Hadamard matrices of generalized quaternion type

✍ Scribed by Mieko Yamada

• Publisher: Elsevier Science
• Year: 1991
• Tongue: English
• Weight: 525 KB
• Volume: 87
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(91)90048-7

✦ Synopsis

Yamada, M., Hadamard matrices of generalized quaternion type, Discrete Mathematics 87 (1991) 187-196. Let G be a semi-direct product of a cyclic group of an odd order by a generalized quaternion group Q,. We consider the ring % obtained from the group ring ZG by identifying the elements f 1 in the center of Q, with f 1 of the rational integer ring Z. If the right regular representation of matrix of an element in 8 is an Hadamard matrix, we call this an Hadamard matrix of generalized quatemion type.

An Hadamard matrix generated by the Paley type 1 matrix is Seidel-equivalent to an Hadamard matrix of generalized quaternion type, bound by some conditions. When the order of generalized quatemion group is minimum, i.e. when Q, is the quaternion group, then an Hadamard matrix of generalized quatemion type is exactly an Hadamard matrix of type Q. See Ito . Moreover if the four component matrices of an Hadamard matrix of type Q are symmetric, then this becomes an Hadamard matrix of Williamson type.

The purpose of this paper is to prove the existence of some infinite series of Hadamard matrices of generalized quaternion type. The theory of the relative Gauss sum is very important for the construction of our infinite series.

In the last section, we give examples of Hadamard matrices of generalized quaternion type of order 24 in detail.