Some difference matrix constructions and an almost completion for the existence of triplewhist tournaments

A necessary condition for the existence of a triplewhist tournament TWh ( v ) is v ≡ 0 or 1 ( mod 4 ) ; this condition is known to be sufficient except for v = 5 , 9 , 12 , 13 and possibly v = 17 , 57 , 65 , 69 , 77 , 85 , 93 , 117 , 129 , 153 . In this paper, we remove all the possible exceptions except v = 17 . This provides an almost complete solution for the more than 100 year old problem on the existence of triplewhist tournaments TWh ( v ) . By applying frame constructions and product constructions, several new infinite classes of Z -cyclic triplewhist tournaments are also obtained. A couple of new cyclic difference matrices are also obtained.