The Existence of Well-Balanced Triple Systems

A triple system is a collection of b blocks, each of size three, on a set of v points. It is j-balanced when every two j-sets of points appear in numbers of blocks that are as nearly equal as possible, and well balanced when it is j-balanced for each {1,2,3}. Well-balanced systems arise in the minimization of variance in file availability in distributed file systems. It is shown that when a triple system that is 2-balanced and 3-balanced exists, so does one that is well balanced. Using known and new results on variants of group divisible designs, constructions for well-balanced triple systems are developed. Using these, the spectrum of pairs (v,b) for which such a well-balanced triple system exists is determined completely.