ICs with more than two transmitter-receiver pairs. The results of the paper in fact suggest that the optimal strategy might in fact be interference alignment through lattice coding. Lattice coding while allowing the interference to be decoded without decoding the interfering messages places fewer constraints on the rates of the interfering users. In particular, it eliminates the MAC type constraints that arise when decoding the interfering messages separately.

Generalizations to non-symmetric channels
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In this section, we generalize the results of Section IV to a class of non-symmetric three user Gaussian ICs. We consider channel whose channel matrix $H \in \mathcal{H}_1$. As $H \in \mathcal{H}_1$, there exists integers $p$ and $q$ such that $GCD(p, q) = 1$ and $$\label{eqn : channel matrix condition 2}
\frac{h_{12}}{h_{21}}\times \frac{h_{23}}{h_{32}}\times \frac{h_{31}}{h_{13}} = \frac{p}{q}.$$ In the next Lemma, we provide a generalization of the “very strong" interference result in [@Sridharan-Globecom] to non-symmetric channels.

Consider a three user Gaussian IC, whose channel matrix $H \in \mathcal{H}_1$ and whose channel gains satisfy (\[eqn : channel matrix condition 2\]). The power constraints at the transmitters are $P_1, P_2, P_3$ and the noise variances at the receivers are $\sigma_1^2, \sigma_2^2$ and $\sigma_3^2$. If the channel gains satisfy one of 