Entropy of Closure Operators and Network Coding Solvability

Gadouleau, Maximilien

Abstract

The entropy of a closure operator has been recently proposed for the study ofnetwork 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 axiomsfor matroids based on their closure operators. We also determine necessary conditions fora large class of closure operators to be solvable. We then define the Shannon entropy of aclosure 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.

Citation

Gadouleau, M. (2014). Entropy of Closure Operators and Network Coding Solvability. Entropy, 16(9), 5122-5143. https://doi.org/10.3390/e16095122

• Journal Article Type: Article
• Publication Date: Sep 25, 2014
• Deposit Date: Sep 25, 2014
• Publicly Available Date: Sep 25, 2014
• Journal: Entropy
• Electronic ISSN: 1099-4300
• Publisher: MDPI
• Peer Reviewed: Peer Reviewed
• Volume: 16
• Issue: 9
• Pages: 5122-5143
• DOI: https://doi.org/10.3390/e16095122
• Keywords: Closure operators, Entropy, Network coding, Information inequalities.
• Public URL: https://durham-repository.worktribe.com/output/1453680