Network-Coding Solutions for Minimal Combination Networks and Their Sub-Networks

Han Cai, Johan Chrisnata, Tuvi Etzion, Moshe Schwartz, Antonia Wachter-Zeh

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Minimal multicast networks are fascinating and efficient combinatorial objects, where the removal of a single link makes it impossible for all receivers to obtain all messages. We study the structure of such networks, and prove some constraints on their possible solutions. We then focus on the combination network, which is one of the simplest and most insightful network in network-coding theory. Of particular interest are minimal combination networks. We study the gap in alphabet size between vector-linear and scalar-linear network-coding solutions for such minimal combination networks and some of their sub-networks. For minimal multicast networks with two source messages we find the maximum possible gap. We define and study sub-networks of the combination network, which we call Kneser networks, and prove that they attain the upper bound on the gap with equality. We also prove that the study of this gap may be limited to the study of sub-networks of minimal combination networks, by using graph homomorphisms connected with the q-analog of Kneser graphs. Additionally, we prove a gap for minimal multicast networks with three or more source messages by studying Kneser networks. Finally, an upper bound on the gap for full minimal combination networks shows nearly no gap, or none in some cases. This is obtained using an MDS-like bound for subspaces over a finite field.

Original languageEnglish
Article number9097246
Pages (from-to)6786-6798
Number of pages13
JournalIEEE Transactions on Information Theory
Issue number11
StatePublished - 1 Nov 2020


  • Linear network coding
  • combination network
  • graph coloring
  • minimal networks
  • q-Kneser graphs

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


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