2 research outputs found
Linear Fractional Network Coding and Representable Discrete Polymatroids
A linear Fractional Network Coding (FNC) solution over is a
linear network coding solution over in which the message
dimensions need not necessarily be the same and need not be the same as the
edge vector dimension. Scalar linear network coding, vector linear network
coding are special cases of linear FNC. In this paper, we establish the
connection between the existence of a linear FNC solution for a network over
and the representability over of discrete
polymatroids, which are the multi-set analogue of matroids. All previously
known results on the connection between the scalar and vector linear
solvability of networks and representations of matroids and discrete
polymatroids follow as special cases. An algorithm is provided to construct
networks which admit FNC solution over from discrete
polymatroids representable over Example networks constructed
from discrete polymatroids using the algorithm are provided, which do not admit
any scalar and vector solution, and for which FNC solutions with the message
dimensions being different provide a larger throughput than FNC solutions with
the message dimensions being equal.Comment: 8 pages, 5 figures, 2 tables. arXiv admin note: substantial text
overlap with arXiv:1301.300
Linear Network Coding, Linear Index Coding and Representable Discrete Polymatroids
Discrete polymatroids are the multi-set analogue of matroids. In this paper,
we explore the connections among linear network coding, linear index coding and
representable discrete polymatroids. We consider vector linear solutions of
networks over a field with possibly different message and edge
vector dimensions, which are referred to as linear fractional solutions. We
define a \textit{discrete polymatroidal} network and show that a linear
fractional solution over a field exists for a network if and
only if the network is discrete polymatroidal with respect to a discrete
polymatroid representable over An algorithm to construct
networks starting from certain class of discrete polymatroids is provided.
Every representation over for the discrete polymatroid, results
in a linear fractional solution over for the constructed
network. Next, we consider the index coding problem and show that a linear
solution to an index coding problem exists if and only if there exists a
representable discrete polymatroid satisfying certain conditions which are
determined by the index coding problem considered. El Rouayheb et. al. showed
that the problem of finding a multi-linear representation for a matroid can be
reduced to finding a \textit{perfect linear index coding solution} for an index
coding problem obtained from that matroid. We generalize the result of El
Rouayheb et. al. by showing that the problem of finding a representation for a
discrete polymatroid can be reduced to finding a perfect linear index coding
solution for an index coding problem obtained from that discrete polymatroid.Comment: 24 pages, 6 figures, 4 tables, some sections reorganized, Section VI
newly added, accepted for publication in IEEE Transactions on Information
Theor