New Bounds on the Field Size for Maximally Recoverable Codes Instantiating Grid-like Topologies
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In recent years, the rapidly increasing amounts of data created and processed through the internet resulted in distributed storage systems employing erasure coding based schemes. Aiming to balance the tradeoff between data recovery for correlated failures and efficient encoding and decoding, distributed storage systems employing maximally recoverable codes came up. Unifying a number of topologies considered both in theory and practice, Gopalan Gopalan2017 initiated the study of maximally recoverable codes for grid-like topologies. In this paper, we focus on the maximally recoverable codes that instantiate grid-like topologies T_m× n(1,b,0). To characterize the property of codes for these topologies, we introduce the notion of pseudo-parity check matrix. Then, using the hypergraph independent set approach, we establish the first polynomial upper bound on the field size needed for achieving the maximal recoverability in topologies T_m× n(1,b,0), when n is large enough. And we further improve this general upper bound for topologies T_4× n(1,2,0) and T_3× n(1,3,0). By relating the problem to generalized Sidon sets in F_q, we also obtain non-trivial lower bounds on the field size for maximally recoverable codes that instantiate topologies T_4× n(1,2,0) and T_3× n(1,3,0).
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