New bounds on the field size for maximally recoverable codes instantiating grid-like topologies

In recent years, the rapidly increasing amounts of data created and processed through internet resulted in distributed storage systems employing erasure coding based schemes. Aiming to balance tradeoff between recovery for correlated failures efficient encoding decoding, maximally recoverable codes came up. Unifying a number topologies considered both theory practice, Gopalan et al. [15] initiated study grid-like topologies. this paper, we focus on that instantiate $$T_{m\times n}(1,b,0)$$ . To characterize property these topologies, introduce notion pseudo-parity check matrix. Then, using Combinatorial Nullstellensatz, establish new upper bound field size needed achieving maximal recoverability By relating problem generalized Sidon sets $${\mathbb {F}}_q$$ , obtain polynomial lower n}(1,2,0)$$ Moreover, hypergraph independent set approach, further improve our general $$T_{4\times $$T_{3\times n}(1,3,0)$$