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

Abstract

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 et al. [15] 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 Tm×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 Combinatorial Nullstellensatz, we establish the first polynomial upper bound on the field size needed for achieving the maximal recoverability in topologies Tm×n(1, b, 0). And using hypergraph independent set approach, we further improve this general upper bound for topologies T4×n(1, 2, 0) and T3×n(1, 3, 0). By relating the problem to generalized Sidon sets in Fq, we also obtain non-trivial lower bounds on the field size for maximally recoverable codes that instantiate topologies T4×n(1, 2, 0) and T3×n(1, 3, 0).

Correctable Erasure Patterns in Product Topologies

“…Even in this simplification of the general setting, it is a highly non-trivial problem to determine the erasure patterns that are correctable, i.e., the patterns that an MR code must be able to correct. The seminal works of [6], [8] initiated studies of the classification of these patterns for some restricted cases (see Table I) [11] and bounds on the required field size [12], [13]. In particular, [8] established a necessary condition for an erasure pattern to be correctable, which is then shown to also be sufficient for the case of one column constraint and no global constraints.…”