# A New Coding Paradigm for the Primitive Relay Channel +
## 3 Main Result
By using Wyner-Ziv coding, the destination needs from the relay a number of bits approaching from above \(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}\mid Y_{\text{\tiny SD}})\cdot n ^{\prime}_{1}\), in order to decode the message sent by the source. As \(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{\tiny SD}})\geq C_{ \text{\tiny ED}}\), the relay transmits right away a number of these bits approaching from below \(C_{\text{\tiny ED}}\cdot n^{\prime}_{1}\). The number of remaining bits approaches from above \(\left(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{\tiny SD}})-C_{ \text{\tiny ED}}\right)\cdot n^{\prime}_{1}\) and it is stored by the relay. The destination stores the message received from the relay and the observation \(Y_{\text{\tiny SD}}\) obtained from the source. As concerns the transmission of the second block, define  
\[\alpha^{\prime}=\frac{I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text {\tiny SD}})-C_{\text{\tiny ED}}}{C_{\text{\tiny ED}}}, \tag{43}\]  
and denote by \(n^{\prime}_{2}\) and \(R^{\prime}_{2}\) the block length and the rate of the message transmitted by the source. Let \(n^{\prime}_{2}=n^{\prime}_{1}\cdot\alpha^{\prime}\) and let \(R^{\prime}_{2}\) approach from below \(I(X_{\text{\tiny SF}};Y_{\text{\tiny SD}})\). The relay discards the received message and transmits to the destination the \(\left(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{\tiny SD}})-C_{ \text{\tiny ED}}\right)\cdot n^{\prime}_{1}\) bits remaining from the previous block. This transmission is reliable as (43) implies that  
\[\left(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{\tiny SD}})-C_{ \text{\tiny ED}}\right)\cdot n^{\prime}_{1}=C_{\text{\tiny ED}}\cdot n^{\prime} _{2}. \tag{44}\]  
At this point, the destination can reconstruct the second block by using the message received from the source. Furthermore, it can also reconstruct the first block by using the side information previously received from the source and the extra \(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}\mid Y_{\text{\tiny SD}})\cdot n _{1}\) bits received from the relay (partly in the first and partly in the second block). The overall block length is \(n^{\prime}=n^{\prime}_{1}+n^{\prime}_{2}=(1+\alpha^{\prime})n^{\prime}_{1}\) and the achievable rate is  
\[R^{\prime}=\frac{R^{\prime}_{1}+\alpha^{\prime}R^{\prime}_{2}}{1+\alpha^{ \prime}}, \tag{45}\]  
which approaches from below  
\[\frac{C_{\text{\tiny ED}}\cdot I(X_{\text{\tiny SF}};\hat{Y}_{\text{\tiny SR}},Y_{\text{\tiny SD}})+\left(I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{ \text{\tiny SD}})-C_{\text{\tiny ED}}\right)I(X_{\text{\tiny SF}};\hat{Y}_{ \text{\tiny SD}})}{I(Y_{\text{\tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{ \tiny SD}})}. \tag{46}\]  
Note that the expression (46) coincides with (33) when \(I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}})<I(X_{\text{\tiny SF}};Y_{\text{ \tiny SD}})\). Clearly, the coding scheme described so far can be implemented with codes that are suitable for compress-and-forward and for decode-and-forward. Hence, we can employ the polar coding schemes for compress-and-forward and for decode-and-forward presented in [24]. However, polar codes require block lengths \(n_{1}\) and \(n_{2}\) (or \(n^{\prime}_{1}\) and \(n^{\prime}_{2}\)) that are powers of two, which puts a constraint on the possible values for \(\alpha=n_{2}/n_{1}\) (or \(\alpha^{\prime}=n^{\prime}_{2}/n^{\prime}_{1}\)). To remove this constraint and achieve the rate (33) for any \(\alpha\), it suffices to use the punctured polar codes described in ([36] Theorem 1).