Maximal Recoverability: A Nexus of Coding Theory

2026-02-25

AI Summary

This study addresses the problem of optimal constructions of Maximally Recoverable (MR) codes in large-scale fault-tolerant storage systems, with a focus on MR Local Reconstruction Codes (LRCs) and Grid Codes (GCs). By uncovering an intrinsic connection between MR LRCs and skew polynomial codes, and by establishing deep links between MR GCs and mathematical theories such as higher-order MDS codes and graph rigidity, the work integrates tools from skew polynomial coding, higher-order MDS theory, combinatorial design, and graph theory to propose a unified framework for constructing MR codes. This approach not only enables efficient constructions of both classes of MR codes but also advances theoretical understanding in related areas, including list-decodable codes and structural rigidity.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Applications of cryptography

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.

Problem

• Maximal Recoverability
• Error-Correcting Codes
• Locally Recoverable Codes
• Grid Codes
• Optimal Constructions

Innovation

• Maximal Recoverability
• Locally Recoverable Codes
• Grid Codes
• Skew Polynomial Codes
• Higher Order MDS