Classification and Generation of the Hadamard-Type Matrices of Third Order over Finite Fields

Hadamard matrix is 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 is a similar one, but has multi-valued components on finite fields. To be more specific, we consider n × n matrices H that have their elements on the given finite fields GF(p), and satisfy HHT = nI under modulo p, where I is an identity matrix. Any additions and multiplications should be executed under modulo p. In the authors' previous studies, some properties of Hadamard-type matrices on finite fields have been proven. For example, it has been shown that the order of a Hadamard-type matrix of odd order on GF(p) is limited to a quadratic residue of a given prime p. On the other hand, it is not clear how many and how various Hadamard-type matrices on GF(p) exist in general. In this paper, we count all possible Hadamard-type matrices on finite fields when the order n of a matrix is small. We also categorize Hadamard-type matrices into six different types when n = 3 and p = 11. In addition, we prove that for any prime p, a Hadamard-type matrix over GF(p) in each of the six types always exists if and only if the order ‘3’ is a quadratic residue of a given p.