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Generalized Index Coding Problem and Discrete Polymatroids.

Anoop Thomas1, Balaji Sundar Rajan2

  • 1School of Electrical Sciences, Indian Institute of Technology Bhubaneswar, Odisha 752050, India.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary
This summary is machine-generated.

This study explores generalized index coding and discrete polymatroids. We establish conditions for vector linear solutions and link representable polymatroids to optimal index coding solutions.

Keywords:
generalized index coding problemsindex codingmatroidsrepresentable discrete polymatroids

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Area of Science:

  • Information Theory
  • Algebraic Combinatorics

Background:

  • Index coding connects to matroid theory via multilinear representations.
  • Discrete polymatroids generalize matroids for vector linear index codes.
  • Recent generalizations of index coding involve receivers with functional demands and capabilities.

Purpose of the Study:

  • To explore connections between generalized index coding and discrete polymatroids.
  • To establish conditions for vector linear solutions in generalized index coding problems based on representable discrete polymatroids.

Main Methods:

  • Investigating the relationship between generalized index coding and discrete polymatroids.
  • Constructing index coding problems with coded side information from discrete polymatroids.
  • Analyzing conditions for optimal length solutions and binary scalar linear solutions.

Main Results:

  • Conditions for a representable discrete polymatroid to yield a vector linear solution for generalized index coding are established.
  • A discrete polymatroid is shown to be representable if an associated index coding problem has an optimal length solution.
  • For matroids, a binary scalar linear solution of optimal length exists if and only if the matroid is binary representable.

Conclusions:

  • The work establishes a significant link between generalized index coding and discrete polymatroids.
  • Findings provide criteria for the existence of efficient solutions in generalized index coding problems.
  • The study contributes to understanding the interplay between coding theory and combinatorial structures.