Existence results for translation nets, II

Abstract

Let D be a translation Bruck net of order s and degree r, with translation group G, and let s = q 1… q n be the prime power factorization of s. We prove that r ⩽ min q i + 1: i = 1,…, n; moreover, equality can be realized provided that G is the direct sum of elementary abelian groups. Similar results are obtained for translation ( s, r; μ)-nets and, more generally, for groups with large systems of large subgroups. As a corollary, we considerably strengthen a known result on symmetric translation nets by showing that any translation ( s, sμ; μ)-net has parameters of the form s = p i , μ = p j for some prime p and also necessarily has elementary abelian translation group.