Maximally recoverable codes: The bounded case

Gandikota, Venkata; Grigorescu, Elena; Thomas, Clayton G.; Zhu, Minshen

doi:10.1109/allerton.2017.8262862

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Denote E as the set of all the types of regular irreducible erasure patterns for topology T m×n (1, b, 0). Assume the parity check matrix of the code C row is H row , then the pseudo-parity check matrix H is of the form in (12). Thus, our goal is to construct a b × n matrix H row such that:…”

Section: B Regular Irreducible Erasure Patternsmentioning

confidence: 99%

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

Kong

2019

Preprint

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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 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).

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Section: B Regular Irreducible Erasure Patternsmentioning

confidence: 99%

“…As for other related works, Gandikota et al [12] considered the maximal recoverability for erasure patterns of bounded size. Shivakrishna et al [35] considered the recoverability of a special kind of erasure patterns called extended erasure patterns for topologies T (m+m ′ )×n (2, b, 0).…”

mentioning

confidence: 99%

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

Kong

2019

Preprint

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“…MRC for grid-like topologies have been studied in [2] and a super-polynomial lower bound on the field size of these MRCs has been derived. MRC for grid-like topologies which can recover from all bounded erasures (bounded by a constant) have been investigated in [3]. In [4], explicit MRC for T m,n (1, 0, h) are constructed over a field size of the order of n h−1 , the order is calculated assuming that h, r are constants.…”

Section: A Mrc For Grid-like Topologiesmentioning

confidence: 99%

On Maximally Recoverable Codes for Product Topologies

Shivakrishna

Rameshwar

Lalitha

et al. 2018

2018 Twenty Fourth National Conference on Communications (NCC)

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Given a topology of local parity-check constraints, a maximally recoverable code (MRC) can correct all erasure patterns that are information-theoretically correctable. In a grid-like topology, there are a local constraints in every column forming a column code, b local constraints in every row forming a row code, and h global constraints in an (m × n) grid of codeword. Recently, Gopalan et al. initiated the study of MRCs under grid-like topology, and derived a necessary and sufficient condition, termed as the regularity condition, for an erasure pattern to be recoverable when a = 1, h = 0.In this paper, we consider MRCs for product topology (h = 0). First, we construct a certain bipartite graph based on the erasure pattern satisfying the regularity condition for product topology (any a, b, h = 0) and show that there exists a complete matching in this graph. We then present an alternate direct proof of the sufficient condition when a = 1, h = 0. We later extend our technique to study the topology for a = 2, h = 0, and characterize a subset of recoverable erasure patterns in that case. For both a = 1, 2, our method of proof is uniform, i.e., by constructing tensor product Gcol ⊗ Grow of generator matrices of column and row codes such that certain square sub-matrices retain full rank. The full-rank condition is proved by resorting to the matching identified earlier and also another set of matchings in erasure sub-patterns.

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

Kong

2021

J Algebr Comb

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Maximally recoverable codes: The bounded case

Cited by 4 publications

References 21 publications

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

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

On Maximally Recoverable Codes for Product Topologies

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

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