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Multicast Network Coding and Field Sizes
In an acyclic multicast network, it is well known that a linear network
coding solution over GF() exists when is sufficiently large. In
particular, for each prime power no smaller than the number of receivers, a
linear solution over GF() can be efficiently constructed. In this work, we
reveal that a linear solution over a given finite field does \emph{not}
necessarily imply the existence of a linear solution over all larger finite
fields. Specifically, we prove by construction that: (i) For every source
dimension no smaller than 3, there is a multicast network linearly solvable
over GF(7) but not over GF(8), and another multicast network linearly solvable
over GF(16) but not over GF(17); (ii) There is a multicast network linearly
solvable over GF(5) but not over such GF() that is a Mersenne prime
plus 1, which can be extremely large; (iii) A multicast network linearly
solvable over GF() and over GF() is \emph{not} necessarily
linearly solvable over GF(); (iv) There exists a class of
multicast networks with a set of receivers such that the minimum field size
for a linear solution over GF() is lower bounded by
, but not every larger field than GF() suffices to
yield a linear solution. The insight brought from this work is that not only
the field size, but also the order of subgroups in the multiplicative group of
a finite field affects the linear solvability of a multicast network
On Network Coding Capacity - Matroidal Networks and Network Capacity Regions
One fundamental problem in the field of network coding is to determine the
network coding capacity of networks under various network coding schemes. In
this thesis, we address the problem with two approaches: matroidal networks and
capacity regions.
In our matroidal approach, we prove the converse of the theorem which states
that, if a network is scalar-linearly solvable then it is a matroidal network
associated with a representable matroid over a finite field. As a consequence,
we obtain a correspondence between scalar-linearly solvable networks and
representable matroids over finite fields in the framework of matroidal
networks. We prove a theorem about the scalar-linear solvability of networks
and field characteristics. We provide a method for generating scalar-linearly
solvable networks that are potentially different from the networks that we
already know are scalar-linearly solvable.
In our capacity region approach, we define a multi-dimensional object, called
the network capacity region, associated with networks that is analogous to the
rate regions in information theory. For the network routing capacity region, we
show that the region is a computable rational polytope and provide exact
algorithms and approximation heuristics for computing the region. For the
network linear coding capacity region, we construct a computable rational
polytope, with respect to a given finite field, that inner bounds the linear
coding capacity region and provide exact algorithms and approximation
heuristics for computing the polytope. The exact algorithms and approximation
heuristics we present are not polynomial time schemes and may depend on the
output size.Comment: Master of Engineering Thesis, MIT, September 2010, 70 pages, 10
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