  A sum-network is an instance of a network coding problem over a directed
acyclic network in which each terminal node wants to compute the sum over a
finite field of the information observed at all the source nodes. Many
characteristics of the well-studied multiple unicast network communication
problem also hold for sum-networks due to a known reduction between instances
of these two problems. In this work, we describe an algorithm to construct
families of sum-network instances using incidence structures. The computation
capacity of several of these sum-network families is characterized. We
demonstrate that unlike the multiple unicast problem, the computation capacity
of sum-networks depends on the characteristic of the finite field over which
the sum is computed. This dependence is very strong; we show examples of
sum-networks that have a rate-1 solution over one characteristic but a rate
close to zero over a different characteristic. Additionally, a sum-network can
have an arbitrary different number of computation capacities for different
alphabets. This is contrast to the multiple unicast problem where it is known
that the capacity is independent of the network coding alphabet.
