Matroidal Entropy Functions: A Quartet of Theories of Information,Matroid, Design,and Coding |
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Authors: | Qi Chen Minquan Cheng Baoming Bai |
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Affiliation: | 1.State Key Laboratory of Integrated Service Networks, Xidian University, Xi’ an 710071, China;2.Guangxi Key Lab of Multi-Source Information Mining & Security, Guangxi Normal University, Guilin 541004, China |
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Abstract: | 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 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 with the exception of and for some v. |
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Keywords: | entropy function matroidal entropy function matroid orthogonal array variable strength orthogonal array almost affine code MDS code polymatroid |
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