Matroidal Entropy Functions: A Quartet of Theories of Information, Matroid, Design, and Coding

In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classi...

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Bibliographic Details
Published in:Entropy (Basel, Switzerland) Vol. 23; no. 3; p. 323
Main Authors: Chen, Qi, Cheng, Minquan, Bai, Baoming
Format: Journal Article
Language:English
Published: Switzerland MDPI 09.03.2021
MDPI AG
Subjects:
almost affine code
entropy function
matroid
matroidal entropy function
orthogonal array
variable strength orthogonal array
orthogonal array
entropy function
variable strength orthogonal array
polymatroid
MDS code
matroidal entropy function
almost affine code
matroid
ISSN:1099-4300, 1099-4300
Online Access:Get full text
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Summary:In this paper, we study the entropy functions on extreme rays of the polymatroidal region which contain a matroid, i.e., matroidal entropy functions. We introduce variable strength orthogonal arrays indexed by a connected matroid M and positive integer v which can be regarded as expanding the classic combinatorial structure orthogonal arrays. It is interesting that they are equivalent to the partition-representations of the matroid M with degree v and the (M,v) almost affine codes. Thus, a synergy among four fields, i.e., information theory, matroid theory, combinatorial design, and coding theory is developed, which may lead to potential applications in information problems such as network coding and secret-sharing. Leveraging the construction of variable strength orthogonal arrays, we characterize all matroidal entropy functions of order n≤5 with the exception of log10·U2,5 and logv·U3,5 for some v.
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content type line 23
ISSN:1099-4300
1099-4300
DOI:10.3390/e23030323