The Oval Strikes Back

TL;DR

This paper explores the use of ovals in projective planes to design non-systematic MDS matrices with enhanced service rate regions and PIR properties, demonstrating their practical benefits in distributed storage systems.

Contribution

It introduces a novel construction of MDS matrices based on ovals, improving service performance and decoding capabilities in distributed storage applications.

Findings

Constructed non-systematic MDS matrices with many small, disjoint recovery sets.

Demonstrated that the service-rate region can surpass that of systematic generator matrices.

Developed a one-step majority-logic decoding algorithm with strong error correction.

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.