Parameterization of high-dimensional perfect sequences over a composition algebra over ℝ

To analyze the structure of a set of high-dimensional perfect sequences over a composition algebra over ℝ, we developed the theory of Fourier transforms of such sequences. Transforms that are similar to discrete Fourier transforms (DFTs) are introduced for a set of sequences. We define the discrete cosine transform, the discrete sine transform, and the generalized discrete Fourier transform (GDFT) of the sequences, and we prove the fundamental properties of these transforms. We show that the GDFT is bijective and that there exists a relationship between these transforms and a convolution of sequences. By applying these properties to a set of perfect sequences, we obtain a parameterization theorem for the sequences. Using this theorem, we show the equivalence of the left and right perfectness.