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

## TL;DR

This paper investigates the structure and solutions of minimal multicast networks, especially combination networks, focusing on the gap between vector-linear and scalar-linear solutions, and introduces Kneser networks to analyze this gap.

## Contribution

It characterizes the gap in alphabet size between vector-linear and scalar-linear solutions for minimal combination networks and introduces Kneser networks as a key tool.

## Key findings

- Maximum gap identified for networks with two source messages.
- Kneser networks attain the upper bound on the gap.
- Nearly no gap in full minimal combination networks, with bounds derived.

## Abstract

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.