Bounds on Fractional Repetition Codes using Hypergraphs

TL;DR

This paper explores the relationship between fractional repetition codes in distributed storage and hypergraphs, deriving bounds and conditions for their existence, and identifying properties of codes linked to linear hypergraphs.

Contribution

It establishes a hypergraph equivalence for FR codes, derives new bounds, and characterizes codes associated with linear hypergraphs as universally good.

Findings

Hypergraph properties map directly to FR code properties.

New bounds on the existence of FR codes are derived.

FR codes from linear hypergraphs are universally good.

Abstract

In the \textit{Distributed Storage Systems} (DSSs), an encoded fraction of information is stored in the distributed fashion on different chunk servers. Recently a new paradigm of \textit{Fractional Repetition} (FR) codes have been introduced, in which, encoded data information is stored on distributed servers, where encoding is done using a \textit{Maximum Distance Separable} (MDS) code and a smart replication of packets. In this work, we have shown that an FR code is equivalent to a hypergraph. Using the correspondence, the properties and the bounds of a hypergraph are directly mapped to the associated FR code. In general, the necessary and sufficient conditions for the existence of an FR code is obtained by using the correspondence. Some of the bounds are new and FR codes meeting these bounds are unknown. It is also shown that any FR code associated with a linear hypergraph is universally good.