Some New Results on Z ‐Cyclic Patterned Starter Whist Tournaments and Related Frames

The whist tournament problem was introduced by Moore in 1896. To construct Z ‐cyclic whist tournaments, Z ‐cyclic patterned starter whist tournaments were introduced. The systematic research on Z ‐cyclic patterned starter whist tournaments dates back to 1954. Recently, Z ‐cyclic patterned starter whist tournament frames were shown to be instrumental in the constructions of Z ‐cyclic patterned starter whist tournaments. In this paper, we develop some necessary conditions for the existence of Z ‐cyclic patterned starter whist tournament frames and some new necessary conditions for the existence of Z ‐cyclic patterned starter whist tournaments. An infinite family of Z ‐cyclic patterned starter whist tournament frames is constructed through Weil's theorem on character sums. As a consequence, a new infinite family of Z ‐cyclic patterned starter whist tournaments is obtained. In addition, we also give a near complete solution to the existence of Z ‐cyclic patterned starter whist tournaments on no greater than 300 players.