user Gaussian IC is more than $1$ for a wide range of channel parameter $a$.

\[thm : degree of freedom of symmetric channel\] Consider a symmetric three user Gaussian IC with channel parameter $a$, and noise variance equal to $1$. The total degrees of freedom of the channel satisfies $$\label{eqn : Theorem 1}
\mathcal{D}_{sum} \geq \left\{\begin{array}{ll} \max\left(1, 3\times\frac{\log(a^2-1)}{\log(2a^4-a^2)}\right), &\ a^2 \geq 2\\
1, & \frac{1}{3} \leq a^2 \leq 2\\
\max\left(1, 3\times\frac{\log\left(\frac{1-a^2}{2a^2}\right)}{\log\left(\frac{1+a^2}{2a^4}\right)}\right), &\ a^2 \leq \frac{1}{3}
\end{array}\right.$$

*:* The proof of the Theorem for $\frac{1}{3} \leq a^2 \leq 2$ is obvious, because a simple time sharing scheme will achieve one degree of freedom for any $a$. Hence, we focus on the other two cases.

First, we consider the case $a^2 \geq 2$. As the channel is symmetric and we are analyzing the total degrees of freedom, we look at only symmetric rate points. For $j \in \{1,2,3\}$, transmitter $j$ communicates message $m_j \in \{1, \ldots, 2^{nR}\}$ to receiver $j$. Transmitter $j$ splits its message $m_j$ into $N$ parts $m_{j1}, \ldots, m_{jN}$, such that a rate $R_i$ is associated with the $i^{th}$ sub-message of each message. For $i \in \{1,\ldots,N\}$, the $i^{th}$ sub-message is encoded to codeword $X_{ji}^n$ by the $j^{th}$ transmitter, 