On the Weight Spectra of Conway Matrices Related to the Non-transitive Head-or-Tail Game

摘要

Our paper is devoted to the computation of the weight spectra of Conway matrices related to the non-transitive head-or-tail game. We obtained explicit formulas for the spectra containing partial binomial sums. These sums are rather hard to deal with when the methods of classical algebra and number theory are used; but when we used methods of computer algebra, we were able to handle them quite efficiently and could easily produce visualizations. We suggest a recurrence algorithm for efficient calculation of the weight-spectrum matrices, including, as a special case, integer matrices modulo m. The algorithm is implemented with MATHEMATICA and visualizations for some interesting examples are shown.