Algorithms for computing network coding rate regions via single element extensions of matroids

Abstract

We propose algorithms for finding extreme rays of rate regions achievable with vector linear codes over finite fields Fq, q ∈ {2, 3, 4} for which there are known forbidden minors for matroid representability. We use the idea of single element extensions (SEEs) of matroids and enumeration of nonisomorphic matroids using SEEs, to first propose an algorithm to obtain lists of all non-isomorphic matroids representable over a given finite field. We modify this algorithm to produce only the list of all non-isomorphic connected matroids representable over the given finite field. We then integrate the process of testing which matroids in a list of matroids form valid linear network codes for a given network within matroid enumeration. We name this algorithm, which essentially builds all matroids that form valid network codes for a given network from scratch, as networkconstrained matroid enumeration.Index Terms-Single emement extensions, rate regions of multi-source network coding problems, region of entropic vectors

Computational Techniques for Investigating Information Theoretic Limits of Information Systems

“…Li et al used a similar computational approach to tackle the multilevel diversity coding problem [17] and multi-source network coding problems with simple network topology [21] (also see Reference [22]); however, the main focus was to provide an efficient enumeration and classification of the large number of specific small instances (all instances considered require 7 or fewer random variables), where each instance itself poses little computation issue. Beyond computing outer bounds, the problem of computationally generating inner bounds was also explored [23,24].…”

Entropic Matroids and Their Representation

“…Matroid representations and forbidden minors were studied in Reference [ 14 ] for GF(3), Reference [ 15 , 16 ] for GF(4) and some results for general fields were obtained in References [ 17 , 18 , 19 ]. Linear representable matroids are also intimately related to linear solutions to network coding problems, in particular in Reference [ 20 ], in which a network-constrained matroid enumeration algorithm is developed, as well as Reference [ 21 ] that considers integer-valued polymatroids and representable polymatroids in References [ 22 , 23 ]. Matroid’s minors and the connection to Zhang-Yeung inequality was discussed in Reference [ 24 ], which shows in particular that almost entropic matroids have infinitely many excluded minor.…”

Multilevel Diversity Coding Systems: Rate Regions, Codes, Computation, & Forbidden Minors

“…We will introduce these bounds in the next subsection. For details on the polyhedral computation methods used to obtain these bounds, interested readers are referred to [2], [1], [22].…”