Existence of (2, 8) GWhD(v) and (4, 8) GWhD(v) with $${v \equiv 0,1 (mod 8)}$$

Abstract

(2, 8) Generalized Whist tournament Designs (GWhD) on v players exist only if $${v \equiv 0,1 (mod 8)}$$. We establish that these necessary conditions are sufficient for all but a relatively small number of (possibly) exceptional cases. For $${v \equiv 1 (mod 8)}$$ there are at most 12 possible exceptions: {177, 249, 305, 377, 385, 465, 473, 489, 497, 537, 553, 897}. For $${v \equiv 0 (mod 8)}$$ there are at most 98 possible exceptions the largest of which is v = 3696. The materials in this paper also enable us to obtain four previously unknown (4, 8)GWhD(8n+1), namely for n = 16,60,191,192 and to reduce the list of unknown (4, 8) GWhD(8n) to 124 values of v the largest of which is v = 3696.