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 figure

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

Computing Bounds on Network Capacity Regions as a Polytope Reconstruction Problem

We define a notion of network capacity region of networks that generalizes the notion of network capacity defined by Cannons et al. and prove its notable properties such as closedness, boundedness and convexity when the finite field is fixed. We show that the network routing capacity region is a computable rational polytope and provide exact algorithms and approximation heuristics for computing the region. We define the semi-network linear coding capacity region, with respect to a fixed finite field, that inner bounds the corresponding network linear coding capacity region, show that it is a computable rational polytope, and provide exact algorithms and approximation heuristics. We show connections between computing these regions and a polytope reconstruction problem and some combinatorial optimization problems, such as the minimum cost directed Steiner tree problem. We provide an example to illustrate our results. The algorithms are not necessarily polynomial-time.Comment: Appeared in the 2011 IEEE International Symposium on Information Theory, 5 pages, 1 figur

Childcare quality improvement and assurance practices

This study set out to examine how quality assurance and quality improvement schemes are being used by Early Years Development and Childcare Partnerships and childcare providers in England to improve the quality of services beyond minimum standards. Three types of early years providers took part: day nurseries, out-of-school clubs and childminding networks

Financial Disintermediation in the 1990s : Implications on Monetary Policy in Malaysia

The increased financial disintermediation that characterizes the Malaysia's financial system since the early 1990s has contributed towards changes in the dynamics of monetary transmission mechanism. Using quarterly data from 1980: 1 to 2005: 4, we found a greater effectiveness of monetary policy during the pre-1990:3 period, but the post-1990:3 period poses much difficulty for the conduct of monetary policy. Innovations in the financial market appeared to have led to lower output variability. Further, when the real interest rate is made a function of financial disintermediation, the real interest rate appeared to have lost its significance in influencing real variables in the post-1990: 3 period. This study did not, however, find evidence in support of the significance of the real interest rate in affecting real variables through the direct financing channel via the capital market.bank lending channel, capital market, cointegration, VAR