Finite discrete memoryless relay channel and causal code

A finite discrete memoryless relay channel is the three-node instance of Finite discrete memoryless networks and causal codes with source input $X\in \mathcal{X}$, relay input $X_r\in {\mathcal{X}}_r$, relay observation $Y_r\in {\mathcal{Y}}_r$, destination observation $Y\in \mathcal{Y}$, and transition law $W(y_r,y\mid x,x_r)$. For a positive integer message-set size $M$, the one-way message is uniform on $[M]$ and the destination estimates it from $Y^n$.

In the relay-only causal model, $X_i=f_i(M)$ and $X_{r,i}=g_i(Y_r^{i-1})$. In the full output-feedback model, both encoders know the past pair $(Y^{i-1},Y_r^{i-1})$, so $X_i=f_i(M,Y^{i-1},Y_r^{i-1})$ and $X_{r,i}=g_i(Y^{i-1},Y_r^{i-1})$. In either model the channel outputs at time $i$ are drawn from $W(\cdot ,\cdot \mid X_i,X_{r,i})$; no encoder sees a current-time output before choosing its current input. The rate is $n^{-1}\ln M$, and achievability uses average error under the uniform message law.