noise at any receiver is independent and identically distributed from symbol to symbol and has variance $\sigma^2$.

An Inner Bound to the Capacity {#subsec:achievable_rate_multirelay_xie_and_kumar}
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For the multi-relay channel, we denote the symbol transmitted by the $i$-th node at time $t$ ($t$ is discrete) by $X_{i}(t)$ for $i=0,1,\cdots,N$. $Z_{k}(t) \sim \mathcal{N}(0,\sigma^{2})$ is the additive white Gaussian noise at node $k$ and time $t$, and is assumed to be [*independent and identically distributed across $k$ and $t$.*]{} Thus, at symbol time $t$, node $k, 1 \leq k \leq N+1$ receives: $$Y_{k}(t)= \sum_{j\in \{0,1,\cdots,N\}, j \neq k} h_{j,k}X_{j}(t)+Z_{k}(t) \label{eqn:network_equation}$$ An inner bound to the capacity of this network, under any path-loss model, is given by (see [@xie-kumar04network-information-theory-scaling-law]): $$R=\min_{1 \leq k \leq N+1} C \bigg(     \frac{1}{\sigma^{2}×} \sum_{j=1}^{k} ( \sum_{i=0}^{j-1} h_{i,k} \sqrt{P_{i,j}}  )^{2}      \bigg)    \label{eqn:achievable_rate_multirelay}$$ where $C(x):=\frac{1}{2}\log_{2}(1+x)$, and node $i$ transmits to node $j$ at power $P_{i,j}$ (expressed in mW).

In Appendix \[section:coding\_scheme\_description\], we provide a descriptive overview of the coding and decoding scheme proposed in [@xie-kumar04network-information-theory-scaling-law]. A sequence of messages are sent from the source to the sink; each message is encoded in a block of symbols and transmitted by using the relay nodes. The scheme involves [*coherent transmission*]{} by the source and 