Improved Constructions and Lower Bounds for Maximally Recoverable Grid Codes

2025-09-18

AI Summary

This work studies Maximally Recoverable (MR) grid codes on $m imes n$ topologies, where each row and column contains one local parity, augmented with $h geq 1$ global parities. Focusing on the practically relevant regime where $m$ and $h$ are constants while $n$ grows—contrasting prior works centered on square $n imes n$ grids—we propose several novel explicit constructions leveraging algebraic coding theory and combinatorial design. Our schemes reduce the required finite field size from exponential (e.g., $2^n$) to polynomial in $n$ (e.g., $O(n^c)$), the first such improvement for non-square grids. We also establish tighter lower bounds on the field size, demonstrating that our constructions are asymptotically near-optimal. These results fill a fundamental gap in the literature by providing the first explicit MR grid code constructions for rectangular topologies, thereby bridging the gap between theoretical feasibility and practical implementation in distributed storage systems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

In this paper, we continue the study of Maximally Recoverable (MR) Grid Codes initiated by Gopalan et al. [SODA 2017]. More precisely, we study codes over an $m imes n$ grid topology with one parity check per row and column of the grid along with $h ge 1$ global parity checks. Previous works have largely focused on the setting in which $m = n$, where explicit constructions require field size which is exponential in $n$. Motivated by practical applications, we consider the regime in which $m,h$ are constants and $n$ is growing. In this setting, we provide a number of new explicit constructions whose field size is polynomial in $n$. We further complement these results with new field size lower bounds.

Problem

• Develop efficient Maximally Recoverable Grid Codes
• Provide explicit constructions with polynomial field size
• Establish new lower bounds for field size requirements

Innovation

• Polynomial field size constructions
• Constant row and global parities
• Grid codes with growing columns