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Communications in Information and Systems

A class of non-Shannon-type information inequalities and their applications

Raymond W. Yeung, Zhen Zhang

Published Jan 1, 2001• 15 citations
Publisher
DOI
10.4310/cis.2001.v1.n1.a6

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We discuss linear programming techniques that help to deduce corollaries of non-classical inequalities for Shannon's entropy. We focus on direct applications of the copy lemma. These applications involve implicitly some (known or unknown) non-classical universal inequalities for Shannon's entropy, though we do not derive these inequalities explicitly. To reduce the computational complexity of these problems, we extensively use symmetry considerations. We present two examples of use of these techniques: we provide a reduced size formal inference of the best known bound for the Ingleton score (originally proven by Dougherty et al. with explicitly derived non-Shannon type inequalities), and improve the lower bound for the optimal information ratio of the secret sharing scheme for an access structure based on the Vámos matroid.
Many open questions in secret sharing ask about the information rate of a particular access structure or class of access structures. Here we improve the best known upper bounds on the information rates of the access structures induced by the Vamos matroid from 10/11 to 8/9 for V1 and from 9/10 to 17/19 for V6. The method we introduce to obtain the bound for V6 can be generalized and applied to all other known 4-variable non-Shannon information inequalities. We also find the exact information rates for the infinitely many minor-minimal, non-matroid-related access structures whose rates were not previously known. On the topic of information inequalities, we show that the Ingleton inequality holds under certain independence assumptions and use this formulation to get a new proof of the Zhang-Yeung non-Shannon inequality.