Z-cyclic (t,8) GWhD(v), t = 2,4

Abstract

For v = 0, 1 (mod 8), (t, 8) Generalized Whist Tournaments on v players, denoted (t, 8) GWhD(v), for t = 2, 4 are (v, 8, 7)-(N)RBIBDs that satisfy additional balance conditions. For both v = 8n and v = 8n+1, considerable information is known about the existence of (4, 8) GWhD(w). Here we focus on a specialization of these designs for which very little is known, namely Z-cyclic (t, 8) GWhD(v)s. The generality of the methodology employed and the tools utilized allow us to study t = 2 as well as t = 4. In particular, we establish, for each of t = 2,4, the existence of Z-cyclic (t, 8) GWhD(7p+1) for all primes p = 8m + 1, except for p = 17, 89. It is further demonstrated that these designs possess a structure described as the Moore property. This terminology is introduced so as to emphasize a parallel between our work for designs with block size 8 with that of E.M. Moore for analogous designs with block size 4 [22]. Several additional infinite families of Z-cyclic (t,8) GWhD(v) are also obtained.