In the modern era of large-scale computing systems, a crucial use of error correcting codes is to judiciously introduce redundancy to ensure recoverability from failure. To get the most out of every byte, practitioners and theorists have introduced the framework of maximal recoverability (MR) to study optimal error-correcting codes in various architectures. In this survey, we dive into the study of two families of MR codes: MR locally recoverable codes (LRCs) (also known as partial MDS codes) and grid codes (GCs). For each of these two families of codes, we discuss the primary recoverability guarantees as well as what is known concerning optimal constructions. Along the way, we discuss many surprising connections between MR codes and broader questions in computer science and mathematics. For MR LRCs, the use of skew polynomial codes has unified many previous constructions. For MR GCs, the theory of higher order MDS codes shows that MR GCs can be used to construct optimal list-decodable codes. Furthermore, the optimally recoverable patterns of MR GCs have close ties to long-standing problems on the structural rigidity of graphs.

翻译：在大规模计算系统的现代时代，纠错码的一个关键应用是明智地引入冗余以确保从故障中恢复。为了充分利用每个字节，实践者和理论家引入了最大可恢复性框架，以研究各种架构中的最优纠错码。在本综述中，我们深入探讨了两类MR码的研究：MR局部可恢复码（也称为部分MDS码）和网格码。对于这两类码，我们讨论了主要的可恢复性保证以及关于最优构造的已知结果。在此过程中，我们探讨了MR码与计算机科学和数学中更广泛问题之间的许多惊人联系。对于MR LRCs，斜多项式码的使用统一了许多先前的构造。对于MR GCs，高阶MDS码理论表明MR GCs可用于构造最优列表可译码。此外，MR GCs的最优可恢复模式与图结构刚性的长期问题有着密切关联。