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In recent years, the rapidly increasing amounts of data created and processedthrough the internet resulted in distributed storage systems employing erasurecoding based schemes. Aiming to balance the tradeoff between data recovery forcorrelated failures and efficient encoding and decoding, distributed storagesystems employing maximally recoverable codes came up. Unifying a number oftopologies considered both in theory and practice, Gopalan et al.\cite{Gopalan2017} initiated the study of maximally recoverable codes forgrid-like topologies.  In this paper, we focus on the maximally recoverable codes that instantiategrid-like topologies $T_{m\times n}(1,b,0)$. To characterize the property ofcodes for these topologies, we introduce the notion of \emph{pseudo-paritycheck matrix}. Then, using the Combinatorial Nullstellensatz, we establish thefirst polynomial upper bound on the field size needed for achieving the maximalrecoverability in topologies $T_{m\times n}(1,b,0)$. And using hypergraphindependent set approach, we further improve this general upper bound fortopologies $T_{4\times n}(1,2,0)$ and $T_{3\times n}(1,3,0)$. By relating theproblem to generalized \emph{Sidon sets} in $\mathbb{F}_q$, we also obtainnon-trivial lower bounds on the field size for maximally recoverable codes thatinstantiate topologies $T_{4\times n}(1,2,0)$ and $T_{3\times n}(1,3,0)$.

New Bounds on the Field Size for Maximally Recoverable Codes Instantiating Grid-like Topologies