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

68 Chapter 5: Network Coding with LT Codes as Erasure Codes
A
b 2
B
C
b 1 ⊕ b 2
b 2
b 2
b 1 ⊕ b 2
F
b 1
b 1 ⊕ b 2 b 1
b 1
G
D
E
Figure 5.8: Network coding in a butterfly network when only BD is erasure-prone
This solution can be performed by the source alone. When the source A is informed
by B that there are severe erasures at BD, it simply transmits b 1 + b 2 instead of b 1 to B
while other nodes just work as usual. In general, this solution is applicable when erasures
occur along a single edge, with the transmission pattern depending on which edge is
erasure-prone, as shown in Table 5.1.
Table 5.1: Recommended Transmission When An Edge is Erasure-prone
.
Erasure-prone edge
Choices of
recommended transmission to (AB, AC)
AB or BF or GE (b 1 , b 2 ), (b 2 , b 1 ), (b 1 + b 2 , b 1 ), (b 1 + b 2 , b 2 )
AC or CF or GD (b 1 , b 2 ), (b 2 , b 1 ), (b 1 , b 1 + b 2 ), (b 2 , b 1 + b 2 )
BD (b 1 + b 2 , b 1 ), (b 1 + b 2 , b 2 )
CE (b 1 , b 1 + b 2 ), (b 2 , b 1 + b 2 )
F G (b 1 , b 2 ), (b 2 , b 1 )
The next section deals with a more general case in which more than one edge is
erasure-prone.