On the Combinatorics of Locally Repairable Codes via Matroid Theory

Westerbäck, Thomas; Freij-Hollanti, Ragnar; Ernvall, Toni; Hollanti, Camilla

doi:10.1109/tit.2016.2598149

Cited by 40 publications

(79 citation statements)

References 47 publications

(80 reference statements)

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“…Several constructions of LRCs achieving the bound (2) already exist, for example in [6], [14], [15], [19], [20]. Many of these constructions require an alphabet size to be exponential in the code length.…”

Section: Achievability Resultsmentioning

confidence: 99%

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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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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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Section: Achievability Resultsmentioning

confidence: 99%

“…The two bounds have been proven to be tight for large alphabet size with constructions provided in [5]- [7], [12]- [20]. Bounds for codes with availability were established in [8], [12], [21], [22].…”

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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“…The following proposition for linear LRCs follows immediately from Theorem V.5 (i) and (ii) in [23]. …”

Section: Constructionmentioning

confidence: 89%

A connection between locally repairable codes and exact regenerating codes

Ernvall¹,

Westerbäck²,

Freij-Hollanti³

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Abstract-Typically, locally repairable codes (LRCs) and regenerating codes have been studied independently of each other, and it has not been clear how the parameters of one relate to those of the other. In this paper, a novel connection between locally repairable codes and exact regenerating codes is established. Via this connection, locally repairable codes are interpreted as exact regenerating codes. Further, some of these codes are shown to perform better than time-sharing codes between minimum bandwidth regenerating and minimum storage regenerating codes.

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“…It was shown in [18] that the (r, δ = 2)-locality of a linear LRC is a matroid invariant. The connection between matroid theory and linear LRCs was examined in more detail in [20]. In addition, the parameters (n, k, d, r, δ) for linear LRCs were generalised to matroids, and new results for both matroids and linear LRCs were given therein.…”

Section: Introductionmentioning

confidence: 99%

“…It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of F2-linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.The parameters (n, k, d, r, δ) can immediately be defined and studied for matroids in general, as in [18,20].…”

mentioning

confidence: 99%

On Binary Matroid Minors and Applications to Data Storage over Small Fields

Grezet

Freij-Hollanti

Westerbäck

et al. 2017

Coding Theory and Applications

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Locally repairable codes for distributed storage systems have gained a lot of interest recently, and various constructions can be found in the literature. However, most of the constructions result in either large field sizes and hence too high computational complexity for practical implementation, or in low rates translating into waste of the available storage space. In this paper we address this issue by developing theory towards code existence and design over a given field. This is done via exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields. In particular, nonexistence can be shown by finding certain forbidden uniform minors within the lattice of cyclic flats. It is shown that the lattice of cyclic flats of binary matroids have additional structure that significantly restricts the possible locality properties of F2-linear storage codes. Moreover, a collection of criteria for detecting uniform minors from the lattice of cyclic flats of a given matroid is given, which is interesting in its own right.The parameters (n, k, d, r, δ) can immediately be defined and studied for matroids in general, as in [18,20]. Matroid fundamentalsMatroids were first introduced by Whitney in 1935, to capture and generalise the notion of linear dependence in purely combinatorial terms.

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On the Combinatorics of Locally Repairable Codes via Matroid Theory

Cited by 40 publications

References 47 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

A connection between locally repairable codes and exact regenerating codes

On Binary Matroid Minors and Applications to Data Storage over Small Fields

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