Voronoi region $V_i = 2^{-nR_i}V_{S_i}$,

-   in decoding interference $I_{ji}^n$ at receiver $j$, the probability of error is upper bounded by (\[eqn : Loeliger condition\]) with $\sigma^2 = \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$, and

-   in decoding message $m_{ji}$ at receiver $j$, the probability of error is upper bounded by (\[eqn : Loeliger condition\]) with $\sigma^2 = 2a^2 P_i + \sum_{k=i+1}^N (2a^2 + 1)P_k + 1$.

Finally, shift $s_i$ is chosen such that the cardinality of the codebook $\mathcal{C}_i$ satisfies $|\mathcal{C}_i| = |(\Lambda_i + s_i) \cap S_i| \geq 2^{nR_i}$. The details of the probability of error analysis are similar to the case $a^2 \geq 2$ and are omitted here. Using the power assignments in (\[eqn : Power assignment 2\]), we see that each user achieves a rate $R_i$ for its $i^{th}$ sub message given by $$R_i = \frac{1}{2}\log\left(\frac{1-a^2}{2a^2}\right).$$ Hence, each user achieves a total rate $R$ given by $$R = \frac{1}{2}\log\left(\frac{1-a^2}{2a^2}\right)^N.$$ The total power expended by each transmitter is given by $$\begin{aligned}
P &=& P_1 + \ldots + P_N\nonumber\\
    &\leq & \left(\frac{1+a^2}{2a^4}\right)^N.\end{aligned}$$ Taking $N$ to $\infty$, we get the desired result. This completes the proof of Theorem 1.

In Figure \[fig : dof symmetric\], we plot the degrees of 