Constrained Linear Representability of Polymatroids and Algorithms for Computing Achievability Proofs in Network Coding

TL;DR

This paper introduces algorithms based on group theory to solve the constrained linear representability problem for polymatroids, enabling the verification of achievability in network coding and secret sharing scenarios.

Contribution

It develops a novel group-theoretic approach for solving CLRP, providing an information theoretic achievability prover with computational experiments.

Findings

Successfully tested on network coding instances

Validated for secret sharing schemes

Demonstrated utility of the algorithms

Abstract

The constrained linear representability problem (CLRP) for polymatroids determines whether there exists a polymatroid that is linear over a specified field while satisfying a collection of constraints on the rank function. Using a computer to test whether a certain rate vector is achievable with vector linear network codes for a multi-source network coding instance and whether there exists a multi-linear secret sharing scheme achieving a specified information ratio for a given secret sharing instance are shown to be special cases of CLRP. Methods for solving CLRP built from group theoretic techniques for combinatorial generation are developed and described. These techniques form the core of an information theoretic achievability prover, an implementation accompanies the article, and several computational experiments with interesting instances of network coding and secret sharing demonstrating the utility of the method are provided.