New Varieties of Hadamard-Type Matrices over Finite Fields and Their Properties

Hadamard matrix is defined as a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. On the other hand, Hadamard-type matrix on finite fields has been proposed. This matrix is a similar one as a binary Hadamard matrix, but has multi-valued components on finite fields. To be more specific, we consider n×n matrices that have their elements on the given finite fields GF(p), and satisfy HHT ≡ nI mod p, where I is an identity matrix. Any additions and multiplications should be executed under modulo p. In this paper, the authors introduce some new Hadamard-type matrices found in computer searches as well as their properties. Specifically, we define special types of Hadamard-type matrices called cyclic Hadamard-type matrices on finite fields, and propose the methods to generate them. In addition, it is shown that the order of an arbitrary Hadamard-type matrix of odd order is limited to quadratic residues of the given prime p. Some methods to extend the order of Hadamard-type matrices are also discussed.