Some Properties of Hadamard-type Matrices on Finite Fields

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 work, 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 additions and multiplications should be executed under modulo p. The author has proposed a method to generate such matrices, and applied them to generate n-shift orthogonal sequences and complete complementary codes. The generated complete complementary code is a family of multi-valued sequences on GF(p)\{0}, where the number of sequence sets, the number of sequences in each sequence set and the sequence length depend on the various divisors of p – 1. Such complete complementary codes with various parameters have not been proposed in previous studies. In this paper, some properties of those matrices are shown to give various construction methods.