On the Gap between Scalar and Vector Solutions of Generalized Combination Networks

TL;DR

This paper investigates the differences between scalar-linear and vector-linear solutions in generalized combination networks, providing new bounds on network parameters and analyzing the gap in alphabet size between optimal solutions.

Contribution

It introduces improved bounds on the maximum number of middle-layer nodes and characterizes the asymptotic gap in alphabet size between scalar and vector solutions.

Findings

Bounds on the maximum number of middle-layer nodes are improved.

The gap in alphabet size between scalar and vector solutions grows logarithmically with network size.

Asymptotic analysis reveals the gap is in Θ(log(r)).

Abstract

We study scalar-linear and vector-linear 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 scalar-linear and optimal vector-linear network coding solutions. For a fixed network structure, while varying the number of middle-layer nodes $r$, the asymptotic behavior of the upper and lower bounds shows that the gap is in $\Theta(\log(r))$.