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Entropy of Closure Operators and Network Coding Solvability

School of Engineering and Computing Sciences, Durham University, South Road, DH1 3LE, Durham, UK
Entropy 2014, 16(9), 5122-5143; https://doi.org/10.3390/e16095122
Received: 17 June 2014 / Revised: 16 August 2014 / Accepted: 11 September 2014 / Published: 25 September 2014
(This article belongs to the Section Information Theory, Probability and Statistics)
The entropy of a closure operator has been recently proposed for the study of network coding and secret sharing. In this paper, we study closure operators in relation to their entropy. We first introduce four different kinds of rank functions for a given closure operator, which determine bounds on the entropy of that operator. This yields new axioms for matroids based on their closure operators. We also determine necessary conditions for a large class of closure operators to be solvable. We then define the Shannon entropy of a closure operator and use it to prove that the set of closure entropies is dense. Finally, we justify why we focus on the solvability of closure operators only. View Full-Text
Keywords: closure operators; entropy; network coding; information inequalities closure operators; entropy; network coding; information inequalities
MDPI and ACS Style

Gadouleau, M. Entropy of Closure Operators and Network Coding Solvability. Entropy 2014, 16, 5122-5143.

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