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.

翻译：本文研究了射影平面中椭圆在分布式存储中的应用，重点关注服务率区域问题。利用直线与椭圆之间的关联关系，我们描述了一类具有大量小型且互不相交恢复集的非系统MDS矩阵。对于特定参数选择，这些矩阵的服务率区域包含相同编码的系统生成矩阵的区域，从而提供更优的服务性能。我们进一步应用该构造分析所研究MDS矩阵的PIR特性，并提出一种具有强大纠错能力的一步大数逻辑解码算法。这些成果表明，有限几何中的经典对象——椭圆，如何作为现代编码理论的有效工具重新焕发活力。