Representations of the Multicast Network Problem
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.
© 2017 The Author(s) and the Association for Women in Mathematics. First Online: 16 November 2017. The authors would like to acknowledge the hospitality of IPAM and the organizers of its Algebraic Geometry for Coding Theory and Cryptography Workshop: Everett Howe, Kristin Lauter, and Judy Walker. The third author was supported by NSA Young Investigator Grant H98230-16-10305 and by an AMS–Simons Travel Grant. The fourth author was partially supported by CONACyT, CVU No. 268999 project "Network Codes," and Universidad Autónoma de Aguascalientes. The fifth author was partially supported by the National Science Foundation under grant DMS-1547399.