approach for a symmetric three user Gaussian IC from degree of freedom perspective.

Symmetric IC: Degrees of Freedom
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In this section, we analyze the degrees of freedom of three user symmetric Gaussian IC. For this channel, the channel gain matrix is of the form $$H = \left(\begin{array}{ccc}1 & a & a\\ a & 1 & a\\ a & a & 1\end{array}\right).$$ We analyze the degrees of freedom of such a channel using a layered lattice coding approach. First, we briefly review the “very strong" interference result of [@Sridharan-Globecom] for a symmetric three user Gaussian IC in the next lemma.

[@Sridharan-Globecom Theorem 2]\[lem : very strong interference\] Consider a symmetric three user Gaussian IC with channel gain $a$, power constraint $P$ at the transmitters and noise variance $\sigma^2$ at the receivers. If the channel gain satisfies $a^2 \geq \frac{P}{\sigma^2} + 1$, then each user can achieve a symmetric rate of $\frac{1}{2}\log\left(\frac{P}{\sigma^2}\right)$.

The proof of Lemma \[lem : very strong interference\] for $\sigma^2 = 1$ is given in [@Sridharan-Globecom Theorem 2]. The proof for any $\sigma^2$ follows with little modifications. We use this lemma frequently in the next Theorem which shows that the total degrees of freedom of a symmetric three 