$\mathbb{Z}$-Cyclic $DTWh(p)/OTWh(p)$, The Empirical Study Continued For Primes $p\equiv 2^k+1(\mathrm{mod}{2}^{k+1}),k=8$

Abstract

In the past few years, several studies have appeared that relate to the existence of $\mathbb{Z}$-cyclic directed-triplewhist tournaments and $\mathbb{Z}$-cyclic ordered-triplewhist tournaments. In these studies, the number of players in the tournament is taken to be a prime $p$ of the form $p\equiv 2^k+1(\mathrm{mod}{2}^k+1)$, where $k\ge 2$. For the cases $k=2,3,4$ it has been shown [6,4,5,12] that $\mathbb{Z}$-cyclic directed-triplewhist tournaments and $\mathbb{Z}$-cyclic ordered-triplewhist tournaments exist for all such primes except for the impossible cases $p=5,13,17$. For the cases $k=5,6,7$ it has been shown [13] that $\mathbb{Z}$-cyclic directed-triplewhist tournaments exist for all such primes less than $\mathrm{3,200,000}$ and that $\mathbb{Z}$-cyclic ordered-triplewhist tournaments exist for all such primes less than $\mathrm{3,200,000}$ with the exception that existence or non-existence of these designs for $p=97,193,449,577,641,1409$ is an open question. Here the case $k=8$ is considered. It is established that $\mathbb{Z}$-cyclic directed-triplewhist tournaments and $\mathbb{Z}$-cyclic ordered-triplewhist tournaments exist for all primes $p\equiv 257(\mathrm{mod}512)$, $p\le \mathrm{6,944,177}$, except possibly for $p=257,769,3329$. For $p=3329$ we are able to construct a $\mathbb{Z}$-cyclic directed-triplewhist tournament, but the existence of a $\mathbb{Z}$-cyclic ordered-triplewhist tournament remains an open question. Furthermore, for each type of design it is conjectured that our basic constructions will produce these designs whenever $p>\mathrm{5,299,457}$.