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Open AccessArticle

Generalized Index Coding Problem and Discrete Polymatroids

1
School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 752050, India
2
Department of Electrical Communication Engineering, Indian Institute of Science Bangalore, Bangalore 560012, India
*
Author to whom correspondence should be addressed.
This paper is an extended version of our paper published in IEEE International Symposium on Information Theory, Aachen, Germany, 2017
Entropy 2020, 22(6), 646; https://doi.org/10.3390/e22060646
Received: 20 April 2020 / Revised: 5 June 2020 / Accepted: 7 June 2020 / Published: 10 June 2020
The connections between index coding and matroid theory have been well studied in the recent past. Index coding solutions were first connected to multi linear representation of matroids. For vector linear index codes, discrete polymatroids, which can be viewed as a generalization of the matroids, were used. The index coding problem has been generalized recently to accommodate receivers that demand functions of messages and possess functions of messages. In this work we explore the connections between generalized index coding and discrete polymatroids. The conditions that need to be satisfied by a representable discrete polymatroid for a generalized index coding problem to have a vector linear solution is established. From a discrete polymatroid, an index coding problem with coded side information is constructed and it is shown that if the index coding problem has a certain optimal length solution then the discrete polymatroid is representable. If the generalized index coding problem is constructed from a matroid, it is shown that the index coding problem has a binary scalar linear solution of optimal length if and only if the matroid is binary representable. View Full-Text
Keywords: index coding; generalized index coding problems; representable discrete polymatroids; matroids index coding; generalized index coding problems; representable discrete polymatroids; matroids
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Thomas, A.; Sundar Rajan, B. Generalized Index Coding Problem and Discrete Polymatroids. Entropy 2020, 22, 646.

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