the sources in block $3f+1$ is done as follows. Users 1 and 2 map $m_{12}$ and $m_{21}$ to codewords (lattice points) $x_1^n(m_{12})=v_{12}$ and $x_2^n(m_{21})=v_{21}$ respectively, with $v_{12}\in\Lambda_{12}$ and $v_{21}\in\Lambda_{21}$. Then they transmit these codewords. Users 2 and 3 transmit $x_2^n(m_{23})$ and $x_3^n(m_{32})$ respectively in block $3f+2$, and users 3 and 1 transmit $x_3^n(m_{31})$ and $x_1^n(m_{13})$ respectively in block $3f+3$.

Processing at the relay
-----------------------

The received signal at the relay in block $3f+1$ is $$\begin{aligned}
y_r^n&=h_1x_1^n+h_2x_2^n+z_r^n\nonumber\\
&=h_1v_{12}+h_2v_{21}+z_r^n.\end{aligned}$$ Notice that $h_1v_{12}+h_2v_{21}$ is also a lattice point $h_1v_{12}+h_2v_{21}\in h_1\Lambda_{12}$. The relay can decode the superposition $h_1v_{12}+h_2v_{21}$ with arbitrarily small probability of error if [@NarayananPravinSprintson; @NamChungLee] $$\begin{aligned}
R_{12}&=R_{21}\leq\left[C\left(h_1^2P'-\frac{1}{2}\right)\right]^+\\
R_{12}&=R_{21}\leq\left[C\left(h_2^2P'-\frac{1}{2}\right)\right]^+,\end{aligned}$$ where $P'$ is the transmit power. Since each user transmits in 2 blocks out of 3, we can set $P'=3P/2$ without violating the power constraint of the users. Thus, the following rates are achievable $$\begin{aligned}
R_{12}=R_{21}\leq\left[C\left(\frac{3h_2^2P}{2}-\frac{1}{2}\right)\right]^+,\end{aligned}$$ since $h_2\leq h_1$. At the end of block $3f+1$, the relay knows $h_1v_{12}+h_2v_{21}\in h_1\Lambda_{12}$, and maps it to an index $u_{12}\in\mathcal{U}_{12}$. Then, it maps $u_{12}$ into a codeword $x_r^n(u_{12})$, and transmits $x_r^n(u_{12})$ in the next block, block $3f+2$[^2]. Keep in mind that this message $u_{12}$ is meant for users 1 and 2.

In block $3f+2$, the relay decodes 