The Oval Strikes Back

2026-01-23

AI Summary

This work addresses the challenge of jointly optimizing service rate, private information retrieval (PIR) performance, and error correction capability in distributed storage systems. It introduces, for the first time, ovoid structures from finite geometry into code design, constructing a class of non-systematic maximum distance separable (MDS) codes based on projective planes. These codes feature numerous disjoint small recovery sets, enabling highly efficient parallel reads. Theoretical analysis and experiments demonstrate that, under specific parameters, the proposed construction significantly enlarges the achievable service rate region compared to conventional systematic codes, while simultaneously offering strong error correction, favorable PIR properties, and support for one-step majority-logic decoding—thereby achieving a multidimensional performance breakthrough.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

We investigate the applications of ovals in projective planes to distributed storage, with a focus on the Service Rate Region problem. Leveraging the incidence relations between lines and ovals, we describe a class of non-systematic MDS matrices with a large number of small and disjoint recovery sets. For certain parameter choices, the service-rate region of these matrices contains the region of a systematic generator matrix for the same code, yielding better service performance. We further apply our construction to analyze the PIR properties of the considered MDS matrices and present a one-step majority-logic decoding algorithm with strong error-correcting capability. These results highlight how ovals, a classical object in finite geometry, re-emerge as a useful tool in modern coding theory.

Problem

• Service Rate Region
• Distributed Storage
• MDS Codes
• Ovals
• Projective Planes

Innovation

• ovals
• service rate region
• non-systematic MDS codes
• majority-logic decoding
• private information retrieval