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

Chapter 3: Introduction to Graphs and Network Coding 29
Definition 3.15 Let the ω-dimensional row vector x represent the message generated
by the source node S, a global encoding mapping ˜f e (x) is called linear if there exists an
ω-dimensional column vector f e such that
˜f e (x) = x · f e . (3.12)
Definition 3.16 Let the |I(U)|-dimensional row vector y represent the data units received
at the node U, a local encoding mapping ˜k e (y) is called linear if there exists an
|I(U)|-dimensional column vector ˆk e = [k 1,e k 2,e ... k |I(U)|,e ] T such that
˜k e (y) = y · ˆk e . (3.13)
In Definition 3.17, the phrase “adjacent pair” is defined to facilitate definitions 3.18
and 3.19. Definition 3.18 defines the local encoding kernel matrix K U at the node U as
a concatenation of all the linear local encoding mappings ˆk e at U. Definition 3.19 shows
that the vector f e in Definition 3.15, which is used for the global encoding mapping at
the node U to the outgoing edge e, can be derived from the global encoding mappings
f d of all the incoming edges d and the local encoding kernel K U . f e is now called global
encoding kernel.
Definition 3.17 A pair of edges (d, e) is called an adjacent pair if and only if there exists
a node U such that d ∈ I(U) and e ∈ O(U) [61].
Definition 3.18 Let F be a finite field and ω a positive integer. An ω-dimensional F-
valued linear network code on an acyclic communication network consists of a scalar k d,e ,
called the local encoding kernel, for every adjacent pair (d, e). The local encoding kernel
at the node U means the |I(U)| × |O(U)| matrix K U , of which element at the d th row and
e th column is k d,e .
K U = [k d,e ] d∈I(U),e∈O(U) (3.14)
Note that K U is a concatenation of all vectors ˆk e , e ∈ O(U) in Definition 3.16 [61].