R}{\frac{1}{2}\log(P)} \geq 3\times\frac{\log(a^2-1)}{\log(2a^4-a^2)}.$$ Next, we consider the case $a^2 \leq \frac{1}{3}$. The proof for this case is very similar to that of $a^2 \geq 2$ with very few modifications. We again focus only on symmetric rates. For $j \in \{1,2,3\}$, transmitter $j$ splits its message $m \in \{1,\ldots,2^{nR}\}$ into $N$ sub-parts $m_{j1}, \ldots, m_{jN}$ such that rate $R_i$ and power $P_i$ is associated with the $i^{th}$ sub-message. Power $P_i$ is chosen as $$\label{eqn : Power assignment 2}
P_i = \frac{1-a^2}{2a^4}\left(\frac{1+a^2}{2a^4}\right)^{N-i}.$$ The encoding strategy is similar to the one described for the case $a^2 \geq 2$ in that each transmitter uses lattice coding to encode all its sub-messages. However, the decoding strategy is slightly different. The decoding process again proceeds through $N$ stages. In stage $i$, receiver $j$ first decodes message $m_{ji}$ and then decodes interference $I_{ji}^n$. This is because decoding interference first will lead to rate constraints that are more binding than the constraints due to decoding the message.

Choice of Lattices, Shifts and Shaping Regions: For $i \in \{1,\ldots, N\}$, each transmitter chooses shaping region $S_i$ to be a $n$ dimensional sphere of radius $\sqrt{nP_i}$. Lattice $\Lambda_i$ is generated using construction A such that

-   the volume of the 