across time. The channel inputs are subject to the following average power constraints: $$\frac{1}{n}\sum_{i=1}^n X_k(i)^2 \leq P_k, \ \forall\ k \in \{1,2,3\}.$$ Let $H$ denote the $3 \times 3$ matrix of channel gains $$H = \left(\begin{array}{ccc}1 & h_{12} & h_{13} \\ h_{21} & 1 & h_{23}\\ h_{31} & h_{32} & 1\end{array}\right).$$ Let $\mathcal{H}_1$ denote a class of channel matrices such that $$\label{eqn : channel class}
\mathcal{H}_1 = \left\{\begin{array}{c} H \in \mathbb{R}^{3\times 3} : h_{11} = h_{22} = h_{33} = 1,\\
 \textrm{ and } \frac{h_{12}}{h_{21}}\times\frac{h_{23}}{h_{32}}\times\frac{h_{31}}{h_{13}} \in \mathbb{Q} \end{array}\right\},$$ In this work, we will restrict ourselves to three-user Gaussian ICs whose channel matrices are in the set $\mathcal{H}_1$. This is the class of channels to which our coding strategy applies. Note that this is a (highly) non-trivial class of channels which includes the symmetric IC as a special case. Achievable rates and capacity region are defined in the Shannon sense. The total degrees of freedom of the three user Gaussian IC is denoted by $\mathcal{D}_{sum}$ and is defined as $$\label{eqn : total degrees of freedom definition}
\mathcal{D}_{sum} \triangleq \limsup_{P_1 + P_2 + P_3 \rightarrow \infty}\max_{(R_1, R_2, R_3)\in \mathcal{C}_{ap}} \frac{R_1 + R_2 + R_3}{\frac{1}{2}\log(P_1+P_2+P_3)}.$$ The total degrees of freedom represents the rate of 