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 \(T_{m\times n}(1,b,0)\). 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 new upper bound on the field size needed for achieving the maximal recoverability in topologies \(T_{m\times n}(1,b,0)\). By relating the problem to generalized Sidon sets in \({\mathbb {F}}_q\), we obtain a polynomial lower bound on the field size for maximally recoverable codes that instantiate topologies \(T_{m\times n}(1,2,0)\). Moreover, using hypergraph independent set approach, we further improve our general upper bound for topologies \(T_{4\times n}(1,2,0)\) and \(T_{3\times n}(1,3,0)\).

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Notes

  1. 1.

    Recall the condition of \({\mathbf {B}}\) having full rank and the polynomial \(f(x_1,x_2,\ldots ,x_6)\) we defined in the proof of Theorem 5.3, the condition we obtain here for the general case is actually the same.

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Correspondence to Gennian Ge.

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Research supported by the National Natural Science Foundation of China under Grant No. 11971325, National Key Research and Development Program of China under Grant Nos. 2020YFA0712100 and 2018YFA0704703, and Beijing Scholars Program.

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Kong, X., Ma, J. & Ge, G. New bounds on the field size for maximally recoverable codes instantiating grid-like topologies. J Algebr Comb (2021). https://doi.org/10.1007/s10801-021-01013-1

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Keywords

  • Maximally recoverable codes
  • Grid-like topologies
  • Pseudo-parity check matrix
  • Hypergraph independent set
  • Distributed storage systems

Mathematics Subject Classification

  • 94B60
  • 11T71