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We study a class of combinatorial designs called Kirkman systems, and we showthat infinitely many Kirkman systems are well-distributed in a precise sense.Steiner triple systems of order $n$ can achieve a minimum block sum of $n$.Kirkman triple systems form parallel classes from the blocks of Steiner triplesystems. We prove that there are an infinite number of Kirkman triple systemsthat have a minimum block sum of $n$. We expand this to quadruple systems.These concepts can then be applied to distributed storage to spread data acrossthe servers, and servers across locations, using Kirkman triple systems, whilehaving data well distributed by popularity, measured by the minimum block sum.

Kirkman Systems that Attain the Upper Bound on the Minimum Block Sum, for Access Balancing in Distributed Storage