Applications of Polymatroid Theory to Distributed Storage Systems

TL;DR

This paper establishes a theoretical link between polymatroid theory and locally repairable codes (LRCs), generalizing bounds and providing new constructions for perfect linear LRCs using gammoids.

Contribution

It introduces a polymatroid-based framework for LRCs, generalizes Singleton bounds, and constructs new perfect linear LRCs from gammoids.

Findings

Generalized Singleton bounds for polymatroid-based LRCs

Introduction of perfect LRCs that achieve these bounds

New linear LRC constructions from gammoids

Abstract

In this paper, a link between polymatroid theory and locally repairable codes (LRCs) is established. The codes considered here are completely general in that they are subsets of $A^n$, where $A$ is an arbitrary finite set. Three classes of LRCs are considered, both with and without availability, and for both information-symbol and all-symbol locality. The parameters and classes of LRCs are generalized to polymatroids, and a general- ized Singelton bound on the parameters for these three classes of polymatroids and LRCs is given. This result generalizes the earlier Singleton-type bounds given for LRCs. Codes achieving these bounds are coined perfect, as opposed to the more common term optimal used earlier, since they might not always exist. Finally, new constructions of perfect linear LRCs are derived from gammoids, which are a special class of matroids. Matroids, for their part, form a subclass of polymatroids and have proven useful in analyzing and constructing linear LRCs.