Construction of Goethals-Seidel sequences, 2

We make use of the Gauss-Legendre three squares theorem to formulate a plausible construction of Goethal-Seidel sequences. Empirical evidence suggests that such four sequences (one of which is repeated) are likely to always exist. There is a close interplay between such sequences, Gauss sums and the Discrete Fourier Transform. We explore and try to better understand these connections.