$x_{ji}$ is a realization of a real random variable $X_{ji}$ such that $$\begin{aligned}
\frac{1}{n}\sum_{i=1}^n\mathbb{E}[X_{ji}^2]\leq P.\end{aligned}$$ The codeword $x_j^n$ can be generated in different ways according to the following cases [@Shannon_TWC]:

-   General encoding: $x_j^n$ is a function of $m_{jk}$, $m_{jl}$, and the previously received symbols at node $j$, thus $$\begin{aligned}
    \label{SourceEncoder}
    x_{ji}=f_j(m_{jk},m_{jl},y_j^{i-1}).\end{aligned}$$

-   Restricted encoding: $x_j^n$ is a function of $m_{jk}$ and $m_{jl}$ only, thus $$\begin{aligned}
    \label{SourceEncoder_R}
    x_{j}^n=f_j(m_{jk},m_{jl}).\end{aligned}$$

In the Y-channel with general encoding, which we call a general Y-channel, the transmit signals of different users are dependent. This is not the case with restricted encoding in what we call a restricted Y-channel.

The received signal at the relay at time instant $i$ can be written as $$\begin{aligned}
y_{ri}=h_1x_{1i}+h_2x_{2i}+h_3x_{3i}+z_{ri},\end{aligned}$$ where $z_{ri}$ is a realization of an i.i.d. Gaussian noise $Z_r\sim\mathcal{N}(0,1)$ and $h_1,h_2,h_3\in\mathbb{R}$ are the channel coefficients from the users to the relay. We assume without loss of generality that $$\begin{aligned}
\label{Ordering}
h_1^2\geq h_2^2\geq h_3^2.\end{aligned}$$ The relay sends a sequence $x_r^n$ of random variables $X_{ri}$ that satisfy $$\begin{aligned}
\frac{1}{n}\sum_{i=1}^n\mathbb{E}[X_{ri}^2]\leq P_r,\end{aligned}$$ which depends on the past received symbols at the relay, i.e. $$\begin{aligned}
\label{RelayEncoder}
X_{ri}=f_r(Y_r^{i-1}).\end{aligned}$$ Then, the received signal at user $j$ and time $i$ can be written as $$\begin{aligned}
\label{ReceivedSignal}
y_{ji}=h_jx_{ri}+z_{ji},\end{aligned}$$ 