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Chapter 4: Unequal Erasure Protection (UEP) in Network Coding 39
of received data layers.
2. The extra utility that a receiver gains from an increase in the number of received
layers, which is called a marginal utility, is less if the previously received number of layers
is greater.
According to Law 4.1, if any two receivers have exactly the same capability to extract
utility from the same amount of received data, it can be concluded that an unequal amount
of received layers between them will make the marginal utility of the one with privilege
less than the other. Conversely, if a given receiver have more capability to extract utility,
an equal distribution results in the larger marginal utility for that receiver.
The following definitions formally express the utility concept in scalable data. We
formulate, in Definition 4.1, the mapping of scalable data into a scalable message in
which each successive layer adds more details to the data. Therefore, each element in
the scalable message corresponds to the “object” of our interest. After explaining the
term “dependency level” in Definition 4.2, we define the “cumulative utility vector” and
“marginal utility vector” corresponding to the law of diminishing marginal utility in Definition
4.3 and 4.4, respectively. Then, we define an “ordered set of scalable data” that
can be mapped into an “ordered scalable message” in Definition 4.5, which possesses an
interesting practical property obeying Law 4.1: The incremental quality obtained by each
recovered layer in the message is in a decreasing order. The ordered scalable message thus
always prefers more erasure protection in the preceding symbols than the subsequent ones.
Definition 4.1 For any S = {s 1 , s 2 , ..., s ω }, representing a set of scalable data with progressively
increasing quality from s 1 to s ω , the i th element s i can be mapped into a prefix
vector P i = [m 1 , m 2 , ..., m i ] of a scalable message M= [m 1 , m 2 , ..., m ω ], which is an ω-
dimensional row vector of a finite field F. [3, 5]
s i ↦−→ [m 1 , m 2 , ..., m i ] (4.2)