Conway matrices related to a non-transitive head-or-tail game with a q-sided die and their Hamming weight-spectra via DFT and the MacWilliams duality formula

We consider the well-known problem of string overlapping in connection with the so-called Penney Ante game with a q-sided die. We propose a simple and effective algorithm for calculation and visualization of Conway matrices Cn and the corresponding matrices Pn that give probabilities that player (1) will win for the chosen words w1w1 and w2w2 via the standard technique of the multidimensional discrete Fourier transform (DFT). We consider also the Hamming weight-spectra of these matrices and propose an algorithm for their computation and visualization.