Hadamard-type Matrices on Finite Fields and Their Applications

摘要

Hadamard matrix is defined as a square matrix where any components are -1 or +1, and where any pairs of rows are mutually orthogonal. In this study, we consider the similar matrix on finite field GF(p) where p is an odd prime. In such a matrix, every component is one of the integers on GF(p)／{0}, that is, (1,2,... ,p - 1}. Any addition and multiplication should be executed under modulo p. In this paper, we classify such matrices into some typical types, propose methods to generate them, and introduce some simple examples on GF(3), GF(5) and GF(7), and so on. The applications to the sequence generation are also discussed.