Network Coding and Wireless Physical-layer ... - Jacobs University

36 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
and concludes the topic.
4.2 The Edge-Disjoint Path Model with Binary Erasure
Channels for Network Coding
Figure 4.2 shows network coding in a network that is usually called the butterfly network,
where A aims to multicast two binary symbols b 1 and b 2 to D and E. In Fig. 4.2, we
can see that the node F encodes b 1 and b 2 together to achieve the multicast rate of 2 bits
per unit time, if each edge represents the capacity of one bit per unit time, D receives b 1
from the path ABD and can recover b 2 from the symbol b 1 ⊕ b 2 from ACF GD, whereas
E receives b 2 from the path ACE and recovers b 1 from the symbol b 1 ⊕ b 2 from ABF GD.
Had network coding not been there, only either b 1 or b 2 would have been able to pass the
bottleneck F G in one unit time, i.e., one receiver would have been unable to use one of
its possible transmission paths [55].
A
b 1 b 2
B
C
b 1 b 2
F
b 1 b 1 ⊕ b 2
b 2
G
b 1 ⊕ b 2 b 1 ⊕ b 2
D
E
Figure 4.2: Network coding in a butterfly network
By means of network coding, all receivers can use all of their possible paths at the
same time. In general, the multicast rate of ω suggests the existence of ω non-intersecting
paths from the source to each sink, which are called edge-disjoint paths by Jaggi et al. [63],