In this paper, we continue the study of Maximally Recoverable (MR) Grid Codes initiated by Gopalan et al. [SODA 2017]. More precisely, we study codes over an $m \times n$ grid topology with one parity check per row and column of the grid along with $h \ge 1$ global parity checks. Previous works have largely focused on the setting in which $m = n$, where explicit constructions require field size which is exponential in $n$. Motivated by practical applications, we consider the regime in which $m,h$ are constants and $n$ is growing. In this setting, we provide a number of new explicit constructions whose field size is polynomial in $n$. We further complement these results with new field size lower bounds.

翻译：本文延续了Gopalan等人[SODA 2017]开创的最大可恢复（MR）网格码研究。具体而言，我们研究在$m \times n$网格拓扑上的编码方案，该方案包含网格每行每列的一个奇偶校验位以及$h \ge 1$个全局奇偶校验位。先前研究主要集中于$m = n$的情形，其显式构造所需域规模相对于$n$呈指数增长。受实际应用驱动，我们考虑$m,h$为常数而$n$增长的参数范围。在此设定下，我们提出了若干域规模为$n$的多项式级的新显式构造，并进一步通过新的域规模下界结果对这些构造进行了补充。