common part which is also decoded by receiver $2$, 3) $m_{13}$, the common part which is also decoded by receiver $3$ and 4) $m_{123}$, the common part which is decoded by receivers $2$ and $3$. We restrict ourselves to Gaussian codebooks when considering the Han-Kobayashi coding strategy. Finally, as the channel is symmetric, we compare the maximum symmetric rate that can be achieved using the two approaches. In the next Lemma, we derive a symmetric rate point that can be achieved using the layered lattice coding approach for the three user symmetric Gaussian IC with cross channel gain $a$ and power constraint $P$. We define $P_a^N$ as follows $$P_a^N \triangleq \left\{\begin{array}{l}(a^2-1)\frac{(2a^4-a^2)^N-1}{2a^4-a^2-1}, \textrm{ if } a^2 \geq 2 \vspace{0.1cm}\\
\frac{1-a^2}{2a^4}\frac{\left(\frac{1+a^2}{2a^4}\right)^{N_2}-1}{\left(\frac{1+a^2}{2a^4}\right)-1}, \textrm{ if } a^2 \leq \frac{1}{3}\end{array}\right..$$ Let $\mathcal{R}_{HK}(P, \sigma^2, a)$ denote the maximum symmetric rate that can be obtained by Han-Kobayashi coding scheme in a three user symmetric Gaussian IC with power constraint $P$, noise at the receiver $\sigma^2$ and the cross channel gain equal to $a$.

\[lem : symmetric rate\] Consider a symmetric three user Gaussian IC with cross channel gain $a$ and power constraint $P$ at each transmitter. Then\
a) if $a^2 \geq 2$ and $P \leq a^2 - 