scheme that we used for the symmetric case. To present the main ideas, we consider an example three user Gaussian IC with channel matrix given by $$\label{eqn : toy example channel}
H = \left(\begin{array}{ccc}1 & a_1 & a_1\\ a_2 & 1 & a_2 \\ a_3 & a_3 & 1\end{array}\right),$$ where $a_1^2, a_2^2, a_3^2 \geq 2$. Without loss of generality we assume $a_1 \leq a_2 \leq a_3$. The analysis for other channel matrices in $H \in \mathcal{H}_1$ are similar to the one presented and is omitted here. We describe the encoding and decoding strategy below:

For $j \in \{1,2,3\}$, transmitter $j$ communicates message $m_j \in \{1, \ldots, 2^{nR_j}\}$ to receiver $j$. Transmitter $j$ splits its message into $N$ parts - $m_{j_1}, m_{j_2}, \ldots, m_{jN}$ such that rate $R_{ji}$ is associated with the $i^{th}$ sub-message. For $i \in \{1, \ldots, N\}$, transmitter $j$ encodes message $m_{ji}$ into codeword $X_{ji}^n$ and transmits $X_j^n = \sum_{i=1}^N X_{ji}^N$. Also, transmitter $j$ assigns power $P_{ji}$ to encode its $i^{th}$ sub-message.

Encoding Strategy: Each transmitter encodes all its sub-messages using lattice coding, and chooses lattices $\Lambda_1, \ldots, \Lambda_N$. Transmitter $j$ chooses shifts $s_{j1}, \ldots, s_{jN}$ and spherical shaping regions $S_{j1}, \ldots, S_{jN}$. The codebook for the $i^{th}$ 