$h_2v_{23}+h_3v_{32}$, maps it to $u_{23}\in\mathcal{U}_{23}$ and sends $x_r^n(u_{23})$, and in block $3f+3$ the relay decodes $h_1v_{13}+h_3v_{31}$, maps it to $u_{31}\in\mathcal{U}_{31}$ and sends $x_r^n(u_{31})$ (cf. Table \[Cycle\]).

Decoding at the destinations
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At the end of the block $3f+1$, the first and third users have $y_1^n=h_1x_r^n+z_1^n$ and $y_3^n=h_3x_r^n+z_3^n$ and aim to decode $u_{31}$. This can be done with an arbitrarily small probability of error if $$\begin{aligned}
R_{31}=R_{13}&\leq C\left(h_1^2P_r\right)\\
R_{13}=R_{13}&\leq C\left(h_3^2P_r\right).\end{aligned}$$ Knowing $u_{31}$, users 1 and 3 are able to calculate $h_1v_{13}+h_3v_{31}$ and since each knows his own message $m_{13}$ and $m_{31}$ respectively, they can obtain $m_{31}$ and $m_{13}$. Similarly, users 1 and 2 decode $m_{21}$ and $m_{12}$ in block $3f+2$, and users 2 and 3 decode $m_{32}$ and $m_{23}$ in block $3f+3$.

As a result, the achievable rate using this scheme is bounded by $$\begin{aligned}
R_{12}&\leq\min\left\{\left[C\left(\frac{3h_2^2P}{2}-\frac{1}{2}\right)\right]^+,C\left(h_2^2P_r\right)\right\}\\
R_{13}&\leq\min\left\{\left[C\left(\frac{3h_3^2P}{2}-\frac{1}{2}\right)\right]^+,C\left(h_3^2P_r\right)\right\}\\
R_{23}&\leq\min\left\{\left[C\left(\frac{3h_3^2P}{2}-\frac{1}{2}\right)\right]^+,C\left(h_3^2P_r\right)\right\},\end{aligned}$$ and $R_{12}=R_{21}$, $R_{13}=R_{31}$, $R_{23}=R_{32}$. Since we have used 3 blocks to transmit all messages, we obtain the following theorem.

\[LowerBound:FDF\] The sum-capacity of the Y-channel satisfies $C_g\geq\underline{C}^{II}$ where $$\begin{aligned}
\underline{C}^{II}&=\frac{2}{3}\min\left\{\left[C\left(\frac{3h_2^2P}{2}-\frac{1}{2}\right)\right]^+,C\left(h_2^2P_r\right)\right\}\\
&+\frac{4}{3}\min\left\{\left[C\left(\frac{3h_3^2P}{2}-\frac{1}{2}\right)\right]^+,C\left(h_3^2P_r\right)\right\}.\end{aligned}$$

Functional decode and forward with two active users
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We can also obtain a sum-capacity lower bound by letting two out of three users communicate all the time as in a two-way 