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

Chapter 3: Introduction to Graphs and Network Coding 27
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Figure 3.8: The butterfly network with imaginary edges
Coding” [51] that each intermediate node combines the network coding from the previous
nodes along the path with the coding of its own before writing the information about the
combined code on data packets’ headers and sending the packets downstream. Here, the
function of the combined code is called “global encoding mapping” whereas that of the
local code is called “local encoding mapping.” They are defined as follows.
Definition 3.13 Let F be a finite field and ω a positive integer. An ω-dimensional F-
valued network code on an acyclic communication network consists of a local encoding
mapping
˜k e : F |I(U)| → F (3.10)
for each node U in the network and each edge e ∈ O(U) [61].
Definition 3.13 tells us that the local encoding mapping ˜k e for a specific outgoing
edge e of the node U maps the |I(U)|-dimensional vector F |I(U)| , each element of which
represents the data from a distinct incoming edge in the set I(U), into an outgoing data.
Definition 3.14 Let F be a finite field and ω a positive integer. An ω-dimensional F-