Maximal Recoverability: A Nexus of Coding Theory

TL;DR

This survey explores maximal recoverability in error-correcting codes, focusing on locally recoverable codes and grid codes, highlighting their optimal constructions, recoverability guarantees, and connections to broader mathematical problems.

Contribution

It provides a comprehensive overview of MR locally recoverable codes and grid codes, discussing their optimal constructions and revealing connections to other areas in computer science and mathematics.

Findings

Skew polynomial codes unify MR LRC constructions

Higher order MDS codes enable optimal list decoding of MR GCs

MR GCs relate to graph rigidity problems

Abstract

In the modern era of large-scale computing systems, a crucial use of error correcting codes is to judiciously introduce redundancy to ensure recoverability from failure. To get the most out of every byte, practitioners and theorists have introduced the framework of maximal recoverability (MR) to study optimal error-correcting codes in various architectures. In this survey, we dive into the study of two families of MR codes: MR locally recoverable codes (LRCs) (also known as partial MDS codes) and grid codes (GCs). For each of these two families of codes, we discuss the primary recoverability guarantees as well as what is known concerning optimal constructions. Along the way, we discuss many surprising connections between MR codes and broader questions in computer science and mathematics. For MR LRCs, the use of skew polynomial codes has unified many previous constructions. For MR GCs, the theory of higher order MDS codes shows that MR GCs can be used to construct optimal list-decodable codes. Furthermore, the optimally recoverable patterns of MR GCs have close ties to long-standing problems on the structural rigidity of graphs.