encoding and decoding scheme are similar to those used in [@Sridharan-Globecom Theorem 2]. Each transmitter uses lattice codes to transmit their messages. For $i \in \{1,2,3\}$, transmitter $i$ chooses lattice $\Lambda_i$ generated using construction A described in (\[eqn : construction A\]) or in [@Loeliger]. To ensure that the interference lattices are aligned at each receiver, we choose the lattices so that $$h_{12}\Lambda_2 = ph_{13}\Lambda_3,\  h_{21}\Lambda_1 = qh_{23}\Lambda_3, \ h_{31}\Lambda_1 = h_{32}\Lambda_2.$$ Let $V_i$ denote the volume of the Voronoi region of lattice $\Lambda_i$. The shaping region chosen by user $i$ is a $n$ dimensional sphere of radius $\sqrt{nP_i}$ denoted by $S_i$. We denote the volume of $S_i$ by $V_{S_i}$. Then, we choose the following relationship between $V_i, V_{S_i}$ and $R_i$ $$V_i = \frac{V_{S_i}}{2^{nR_i}}, \quad i \in \{1,2,3\}.$$ The shifts for transmitter $i$ denoted by $t_i$ is chosen such that the codebook $\mathcal{C}_i$ given by $\mathcal{C}_i = (\Lambda_i + t_i) \cap S_i$ has at least $2^{nR_i}$ codewords. For decoding, receiver $i$ first decodes the total interference it sees and then decodes its own message. We describe the rate constraints involved in successful decoding at receiver $1$.

The total interference seen by receiver $1$ is given by $$I_1 = h_{12} X_2 + 