# A New Coding Paradigm for the Primitive Relay Channel +
## 3 Main Result
As \(I(Y_{\text{ss}};\hat{Y}_{\text{ss}}|Y_{\text{sD}})\geq C_{\text{sD}}\), the relay transmits right away a number of these bits approaching from below \(C_{\text{sD}}\cdot n_{1}\). The number of remaining bits approaches from above \((I(Y_{\text{ss}};\hat{Y}_{\text{ss}}|Y_{\text{sD}})-C_{\text{sD}})\cdot n_{1}\), and it is stored by the relay. The destination stores the message received from the relay and the observation \(Y_{\text{sD}}\) obtained from the source. Consider the transmission of the second block and define  
\[\alpha=\frac{I(Y_{\text{ss}};\hat{Y}_{\text{ss}}|Y_{\text{sD}})-C_{\text{sD}} }{C_{\text{sD}}-\big{(}I(X_{\text{s}};Y_{\text{ss}})-I(X_{\text{s}};Y_{\text {sD}})\big{)}}. \tag{39}\]  
Denote by \(n_{2}\) and \(R_{2}\) the block length and the rate of the message transmitted by the source. Let \(n_{2}=n_{1}\cdot\alpha\) and let \(R_{2}\) approach from below \(I(X_{\text{s}};Y_{\text{ss}})\). The relay receives \(Y_{\text{ss}}\) and successfully decodes the message. Again, the destination receives the side information \(Y_{\text{sD}}\) from the source. Hence, it needs from the relay a number of bits approaching from above \((I(X_{\text{s}};Y_{\text{ss}})-I(X_{\text{s}};Y_{\text{sD}}))\cdot n_{1}\cdot\alpha\), in order to decode the message sent by the source. The relay transmits to the destination these \((I(X_{\text{s}};Y_{\text{ss}})-I(X_{\text{s}};Y_{\text{sD}}))\cdot n_{1}\cdot\alpha\) information bits plus the \((I(Y_{\text{ss}};\hat{Y}_{\text{ss}}|Y_{\text{sD}})-C_{\text{sD}})\cdot n_{1}\) bits remaining from the previous block. This transmission is reliable as (39) implies that  
\[\big{(}I(X_{\text{s}};Y_{\text{ss}})-I(X_{\text{s}};Y_{\text{sD}})\big{)}\cdot n _{1}\cdot\alpha+\big{(}I(Y_{\text{ss}};\hat{Y}_{\text{ss}}|Y_{\text{sD}})-C_ {\text{sD}}\big{)}\cdot n_{1}=C_{\text{sD}}\cdot n_{2}. \tag{40}\]  
At this point, the destination can reconstruct the second block by using the side information received from the source and the extra \((I(X_{\text{s}};Y_{\text{ss}})-I(X_{\text{s}};Y_{\text{sD}}))\cdot n_{1}\cdot\alpha\) bits received from the relay. Furthermore, it can also reconstruct the first block by using the side information previously received from the source and the extra \(I(Y_{\text{ss}};\hat{Y}_{\text{ss}}\mid Y_{\text{sD}})\cdot n_{1}\) bits received from the relay (partly in the first and partly in the second block). The overall block length is \(n=n_{1}+n_{2}=(1+\alpha)n_{1}\), and the achievable rate is  
\[R=\frac{R_{1}+\alpha R_{2}}{1+\alpha}, \tag{41}\]which approaches from below  
\[\frac{\left(C_{\text{\tiny ED}}-\left(I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}})-I(X _{\text{\tiny SF}};Y_{\text{\tiny SD}})\right)\right)I(X_{\text{\tiny SF}};\hat{Y} _{\text{\tiny SR}},Y_{\text{\tiny SD}})}{I(Y_{\text{\tiny SR}};\hat{Y}_{\text{ \tiny SR}}|Y_{\text{\tiny SD}})-\left(I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}} )-I(X_{\text{\tiny SF}};Y_{\text{\tiny SD}})\right)}+\frac{\left(I(Y_{\text{ \tiny SR}};\hat{Y}_{\text{\tiny SR}}|Y_{\text{\tiny SD}})-C_{\text{\tiny ED}} \right)I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}})}{I(Y_{\text{\tiny SR}};\hat{Y} _{\text{\tiny SR}}|Y_{\text{\tiny SD}})-\left(I(X_{\text{\tiny SF}};Y_{\text{ \tiny SR}})-I(X_{\text{\tiny SF}};Y_{\text{\tiny SD}})\right)}. \tag{42}\]  
Note that the expression (42) coincides with (33) when \(I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}})\geq I(X_{\text{\tiny SF}};Y_{\text{ \tiny SD}})\). The case \(I(X_{\text{\tiny SF}};Y_{\text{\tiny SR}})<I(X_{\text{\tiny SF}};Y_{\text{ \tiny SD}})\) is handled in a similar way. As concerns the transmission of the first block, nothing changes. Denote by \(n^{\prime}_{1}\) and \(R^{\prime}_{1}\) the block length and the rate of the message transmitted by the source, and let \(R^{\prime}_{1}\) approach from below \(I(X_{\text{\tiny SF}};\hat{Y}_{\text{\tiny SR}},Y_{\text{\tiny SD}})\). The relay receives \(Y_{\text{\tiny SR}}\) and constructs the compressed description \(\hat{Y}_{\text{\tiny SR}}\).