  We present a family of automata networks that solve the k-parity problem when
run in parallel. These solutions are constructed by connecting cliques in a
non-cyclical fashion. The size of the local neighbourhood is linear in the size
of the alphabet, and the convergence time is proven to always be the diameter
of the interaction graph. We show that this family of solutions can be slightly
altered to obtain an equivalent family of solutions to the k-synchronisation
problem, which means that these solutions converge from any initial
configuration to the cycle which contains all the uniform configurations over
the alphabet, in order.
