  This work studies consensus strategies for networks of agents with limited
memory, computation, and communication capabilities. We assume that agents can
process only values from a finite alphabet, and we adopt the framework of
finite fields, where the alphabet consists of the integers {0,...,p-1}, for
some prime number p, and operations are performed modulo p. Thus, we define a
new class of consensus dynamics, which can be exploited in certain applications
such as pose estimation in capacity and memory constrained sensor networks. For
consensus networks over finite fields, we provide necessary and sufficient
conditions on the network topology and weights to ensure convergence. We show
that consensus networks over finite fields converge in finite time, a feature
that can be hardly achieved over the field of real numbers. For the design of
finite-field consensus networks, we propose a general design method, with high
computational complexity, and a network composition rule to generate large
consensus networks from smaller components. Finally, we discuss the application
of finite-field consensus networks to distributed averaging and pose estimation
in sensor networks.
