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

28 Chapter 3: Introduction to Graphs and Network Coding
valued network code on an acyclic communication network consists of a local encoding
mapping ˜k e : F |I(U)| → F and a global encoding mapping ˜f e : F ω → F for each edge e in
the network such that:
• For every node U and every channel e ∈ O(U), ˜fe (x) is uniquely determined by
the global encoding mappings from incoming edges ( ˜f d (x), d ∈ I(U)), and ˜k e is the
mapping via
( ˜f d (x), d ∈ I(U)) → ˜f e (x) (3.11)
• For the ω imaginary edges e, the mappings ˜f e are the projections from the space F ω
to the ω different coordinates [61].
Definition 3.14 shows formally the difference between the local encoding mapping ˜k e
and the global one ˜f e . The former, on the one hand, relates the data from incoming edges
in the set I(U) to the outgoing data. The latter, on the other hand, maps the source
message x ∈ F ω to the data output at e. Moreover, (3.11) tells us that each node can
derive ˜f e from its local encoding mapping ˜k e and the global encoding mapping ˜f d of the
incoming edges d ∈ I(U). This means any global encoding mapping computed by a node
and written on data packets’ headers will be used by successive nodes, together with their
local encoding mapping, to successively compute their global encoding mapping to be
further transmitted as headers. The process repeats until data packets reach the sinks,
which decodes them according to headers.
This work focuses on a class of network codes called linear network codes, which is
described in the next subsection.
3.4.1 Linear Network Codes
Linear network codes are those of which all local encoding mapping and global encoding
mapping are nothing more than dot-product of input vectors and mapping vectors. Linear
local and global encoding mapping are defined formally in definitions 3.15 and 3.16 [61].