  We analyze the asymptotic behavior of compute-and-forward relay networks in
the regime of high signal-to-noise ratios. We consider a section of such a
network consisting of K transmitters and K relays. The aim of the relays is to
reliably decode an invertible function of the messages sent by the
transmitters. An upper bound on the capacity of this system can be obtained by
allowing full cooperation among the transmitters and among the relays,
transforming the network into a K times K multiple-input multiple-output (MIMO)
channel. The number of degrees of freedom of compute-and-forward is hence at
most K. In this paper, we analyze the degrees of freedom achieved by the
lattice coding implementation of compute-and-forward proposed recently by Nazer
and Gastpar. We show that this lattice implementation achieves at most
2/(1+1/K)\leq 2 degrees of freedom, thus exhibiting a very different asymptotic
behavior than the MIMO upper bound. This raises the question if this gap of the
lattice implementation to the MIMO upper bound is inherent to
compute-and-forward in general. We answer this question in the negative by
proposing a novel compute-and-forward implementation achieving K degrees of
freedom.
