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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 j∈{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.

Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks, without revealing the privacy of the specific item to any single server. Classical models of PIR protocols require that each server stores a whole copy of the database. Recently new PIR models are proposed with coding techniques arising from distributed storage system. In these new models each server only stores a fraction 1/s of the whole database, where s > 1 is a given rational number. PIR array codes are recently proposed by Fazeli, Vardy and Yaakobi to characterize the new models. Consider a PIR array code with m servers and the k-PIR property (which indicates that these m servers may emulate any efficient k-PIR protocol). The central problem is to design PIR array codes with optimal rate k/m. Our contribution to this problem is three-fold. First, for the case 1 < s ≤ 2, although PIR array codes with optimal rate have been constructed recently by Blackburn and Etzion, the number of servers in their construction is impractically large. We determine the minimum number of servers admitting the existence of a PIR array code with optimal rate for a certain range of parameters. Second, for the case s > 2, we derive a new upper bound on the rate of a PIR array code. Finally, for the case s > 2, we analyze a new construction by Blackburn and Etzion and show that its rate is better than all the other existing constructions.

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