192 research outputs found
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
Network coding for non-uniform demands
Non-uniform demand networks are defined as a useful connection model, in between multicasts and general connections. In these networks, each sink demands a certain number of messages, without specifying their identities. We study the solvability of such networks and give a tight bound on the number of sinks for which the min cut condition is sufficient. This sufficiency result is unique to the non-uniform demand model and does not apply to general connection networks. We propose constructions to solve networks at, or slightly below capacity, and investigate the effect large alphabets have on the solvability of such networks. We also show that our efficient constructions are suboptimal when used in networks with more sinks, yet this comes with little surprise considering the fact that the general problem is shown to be NP-hard
Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids
We study matroidal networks introduced by Dougherty et al. We prove the
converse of the following theorem: If a network is scalar-linearly solvable
over some finite field, then the network is a matroidal network associated with
a representable matroid over a finite field. It follows that a network is
scalar-linearly solvable if and only if the network is a matroidal network
associated with a representable matroid over a finite field. We note that this
result combined with the construction method due to Dougherty et al. gives a
method for generating scalar-linearly solvable networks. Using the converse
implicitly, we demonstrate scalar-linear solvability of two classes of
matroidal networks: networks constructed from uniform matroids and those
constructed from graphic matroids.Comment: 5 pages, submitted to IEEE ISIT 201
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