# Representations of the Multicast Network Problem

## TL;DR

This paper explores the structure of multicast networks in linear network coding, introducing concepts like coding points and code graphs, and connects network capacity problems to algebraic geometry.

## Contribution

It introduces the notions of coding points and code graphs, and links network capacity to rational points on algebraic varieties, providing new perspectives.

## Key findings

- Defined coding points as edges where messages combine
- Introduced the code graph as a simplified network representation
- Connected network capacity to algebraic geometry concepts

## Abstract

We approach the problem of linear network coding for multicast networks from different perspectives. We introduce the notion of the coding points of a network, which are edges of the network where messages combine and coding occurs. We give an integer linear program that leads to choices of paths through the network that minimize the number of coding points. We introduce the code graph of a network, a simplified directed graph that maintains the information essential to understanding the coding properties of the network. One of the main problems in network coding is to understand when the capacity of a multicast network is achieved with linear network coding over a finite field of size q. We explain how this problem can be interpreted in terms of rational points on certain algebraic varieties.