  Many code families such as low-density parity-check codes, fractional
repetition codes, batch codes and private information retrieval codes with low
storage overhead rely on the use of combinatorial block designs or derivatives
thereof. In the context of distributed storage applications, one is often faced
with system design issues that impose additional constraints on the coding
schemes, and therefore on the underlying block designs. Here, we address one
such problem, pertaining to server access frequency balancing, by introducing a
new form of Steiner systems, termed MaxMinSum Steiner systems. MaxMinSum
Steiner systems are characterized by the property that the minimum value of the
sum of points (elements) within a block is maximized, or that the minimum sum
of block indices containing some fixed point is maximized. We show that proper
relabelings of points in the Bose and Skolem constructions for Steiner triple
systems lead to optimal MaxMin values for the sums of interest; for the duals
of the designs, we exhibit block labelings that are within a 3/4 multiplicative
factor from the optimum.
