Correctable Erasure Patterns in Product Topologies

TL;DR

This paper explores the correction of erasure patterns in product topologies of storage codes, introducing a method to add global parity checks and relating these patterns to tensor-product codes.

Contribution

It presents a generic approach for incorporating global parity checks into product topologies and characterizes the correctable erasure patterns in this setting.

Findings

Derived new results on correctable erasure patterns with global constraints

Established a relationship between correctable patterns in product topologies and tensor-product codes

Provided a framework for enhancing local recovery with global parity checks

Abstract

Locality enables storage systems to recover failed nodes from small subsets of surviving nodes. The setting where nodes are partitioned into subsets, each allowing for local recovery, is well understood. In this work we consider a generalization introduced by Gopalan et al., where, viewing the codewords as arrays, constraints are imposed on the columns and rows in addition to some global constraints. Specifically, we present a generic method of adding such global parity-checks and derive new results on the set of correctable erasure patterns. Finally, we relate the set of correctable erasure patterns in the considered topology to those correctable in tensor-product codes.