The Oval Strikes Back
Andrea Di Giusto, Alberto Ravagnani, Emina Soljanin
TL;DR
The paper shows how oval geometry in $PG(2,q)$ can be used to design non-systematic generator matrices for $[n,3,n-2]_q$ MDS codes with maximum length $n=m(2,q)$, optimizing the service-rate region (SRR) in distributed storage. It constructs a generator matrix $\mathbf{G}$ such that the minimal recovery sets have size $2$ and the per-object recovery sets $\mathcal{R}_i^2(\mathbf{G})$ partition $[n]$, yielding $\Lambda(\mathbf{G})=\Delta_3(n/2)$ and a constant cost $C(\lambda)=2$. For some $q$, this non-systematic SRR strictly contains the SRR of a systematic generator $\mathbf{S}$ for the same code, and the framework provides an $n/2$-PIR code and a 1S-MLD that can correct up to $t\leq \left\lfloor \frac{n-2}{4} \right\rfloor$ errors. The work links classical finite-geometry objects to practical coding problems and suggests extensions to other geometric configurations.
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.