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42 Chapter 4: Unequal Erasure Protection (UEP) in Network Coding
where ϱ il and ρ i,j represent the probabilities that the symbol m l and the prefix P j are
recovered at the sink i, respectively. From (4.8), the quality improves if the term ρ i,j
becomes larger, especially for a small j implying a large ∆ j . This reaffirms the essence
of UEP, which is to better protect the high-priority preamble. ρ i,j depends on the transmission
channels and our network codes, which will be investigated in the next section.
The equation (4.8) satisfies the paragraph 2 of Law 4.2 generalized by linearity assumption,
such that any two probabilistic objects having the same expected value, despite
being linearly-combined by different probabilistic proportion of data layers, are considered
economically equivalent.
4.6 Utility of Global Encoding Kernels (GEKs) for
Linear Network Codes
In the previous section, we consider scalable data as probabilistic objects and derive its
utility. In this section, after having a quick review of the meaning of global encoding
kernels (GEKs), we will see that the assignment of GEKs to the edges in the network is
analogous to assigning received symbols to the sinks in an erasure-free network. In case
there are erasures, we can identify the utility of GEKs based on erasure probabilities of
transmission channels just as we can identify the utility of scalable data.
Let us now revisit the global encoding kernel (GEK) defined in Chapter 3.
For a
network that employs linear network coding, each of its edges in the graphical model,
such as Fig. 4.2, is used to transmit the linear combination of the source symbols. This
linear combination can either be represented locally as a linear function of symbols from
adjacent edges, which is called “a local encoding mapping,” or globally as a linear function
of source symbols, which is called “a global encoding mapping” [61]. The global encoding
mapping is described by a vector called ”a global encoding kernel (GEK),” which is
introduced in Definition 4.6.