Matroid Theory and Storage Codes: Bounds and Constructions

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, δ), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

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“…Bounds for codes with availability were established in [8], [12], [21], [22]. For a summary on various bounds for LRCs, see [23].…”

Section: Introductionmentioning

confidence: 99%

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Grezet

Freij-Hollanti

Westerbäck

et al. 2019

IEEE Trans. Inform. Theory

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“…Here, we choose to present matroids via their rank functions. Much of the contents in this part can be found in more detail in [5].…”

Section: Preliminaries On Matroidsmentioning

confidence: 99%

The complete hierarchical locality of the punctured Simplex code

Grezet

Hollanti

2020

Des. Codes Cryptogr.

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“…Here, we choose to define matroids via their rank functions. Much of the contents in this section can be found in more detail in [10].…”

Section: Preliminariesmentioning

confidence: 99%

Uniform Minors in Maximally Recoverable Codes

et al. 2019

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In this letter, locally recoverable codes with maximal recoverability are studied with a focus on identifying the MDS codes resulting from puncturing and shortening. By using matroid theory and the relation between MDS codes and uniform minors, the list of all the possible uniform minors is derived.This list is used to improve the known non-asymptotic lower bound on the required field size of a maximally recoverable code. I. INTRODUCTIONWith the exponential growth of data needed to be stored remotely, distributed storage systems (DSSs) using erasure-correcting codes have become attractive due to their high reliability and low storage overhead. A class of codes called locally recoverable codes (LRCs) has been introduced in [1], [2] as an alternative to traditional maximum distance separable (MDS) codes to improve node repair efficiency by allowing one failed node to be repaired by only accessing a few other nodes.A linear (n, k, r)-LRC is a linear code of length n and dimension k over F q such that every codeword symbol i ∈ [n] = {1, . . . , n} is contained in a repair set R i ⊆ [n] with |R i | ≤ r + 1 and the minimum Hamming distance of the restriction of the code to R i is at least 2. In other words, any symbol can be

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Matroid Theory and Storage Codes: Bounds and Constructions

Cited by 10 publications

References 55 publications

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

The complete hierarchical locality of the punctured Simplex code

Uniform Minors in Maximally Recoverable Codes

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