On the Gap between Scalar and Vector Solutions of Generalized Combination Networks
Abstract
We study scalarlinear and vectorlinear solutions of the generalized combination network. We derive new upper and lower bounds on the maximum number of nodes in the middle layer, depending on the network parameters and the alphabet size. These bounds improve and extend the parameter range of known bounds. Using these new bounds we present a lower bound and an upper bound on the gap in the alphabet size between optimal scalarlinear and optimal vectorlinear network coding solutions. For a fixed network structure, while varying the number of middlelayer nodes $r$, the asymptotic behavior of the upper and lower bounds shows that the gap is in $\Theta(\log(r))$.
 Publication:

arXiv eprints
 Pub Date:
 June 2020
 arXiv:
 arXiv:2006.04870
 Bibcode:
 2020arXiv200604870L
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Networking and Internet Architecture;
 Computer Science  Social and Information Networks;
 Mathematics  Combinatorics
 EPrint:
 extended version of arXiv:2001.04150v2