Bounds on generalized FR codes using hypergraphs

Abstract

In the Distributed Storage Systems (DSSs), the encoded fraction of information is stored in a distributed fashion on different chunk servers. Recently a new paradigm of Fractional Repetition (FR) codes has been introduced, in which, encoded data information is stored on distributed servers using an encoding involving a Maximum Distance Separable (MDS) code and a Repetition code. In this work, we have considered FR codes with asymmetric parameters known as Generalized Fractional Repetition (GFR) codes. We have shown that any GFR code is equivalent to a hypergraph. Using the correspondence, the properties and the bounds of a hypergraph are directly mapped to the associated GFR code. In general, using the correspondence, the necessary and sufficient conditions for the existence of a GFR code is obtained. It is also shown that any GFR code associated with a linear hypergraph is universally good, and any locally repairable GFR code is associated with a class of non-linear hypergraph.