On the Combinatorics of Locally Repairable Codes via Matroid Theory

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

doi:10.48550/arxiv.1501.00153

Cited by 2 publications

(19 citation statements)

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“…This paper strengthens several results given in [10]. Firstly, using the matroid-based construction we extend the class of linear perfect (n, k, d, r, δ)-LRCs with ⌈k/r⌉ = 2.…”

Section: B Contributionssupporting

confidence: 80%

“…Matroid theory was used in [9] in order to prove that the minimum distance of a class of linear LRCs achieves the generalized Singleton bound. It was proved in [10] that every almost affine LRC induces a matroid such that the parameters (n, k, d, r, δ) of the LRC appear as matroid invariants. Consequently, the parameters (n, k, d, r, δ) were generalized to matroids and the bound (1) was proven to also hold for all matroids, which is nontrivial since not all matroids are induced by almost affine codes.…”

Section: A Related Workmentioning

confidence: 99%

“…Codes or matroids achieving the generalized Singleton bound are here called perfect. Using the lattice of cyclic flats of matroids, the non-existence results of [12] were strengthened in [10].…”

Section: A Related Workmentioning

confidence: 99%

“…see [3], [9], [12] [13], [14]. Using a matroid-based construction in [10], classes of linear LRCs with a large span on the parameters (n, k, d, r, δ) and local repair sets were given. By this construction, linear perfect (n, k, d, r, δ)-LRCs were constructed for all the parameters from the existence results given in [12].…”

Section: A Related Workmentioning

confidence: 99%

“…) in the theorem below. Theorem 3.2 ( [10]): Let M = (E, ρ) be a matroid with 0 < ρ(E) and 1 Z = E. Then…”

Section: A Relationship Between Matroids and Almost Affine Lrcsmentioning

confidence: 99%

See 4 more Smart Citations

Bounds on the maximal minimum distance of linear locally repairable codes

Pöllänen

Westerbäck

Freij-Hollanti

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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, which also provided a general lower bound on the maximal achievable minimum distance dmax(n, k, r, δ) that a linear LRC with parameters (n, k, r, δ) can have. This article expands the class of parameters (n, k, d, r, δ) for which there exist perfect linear LRCs and improves the lower bound for dmax(n, k, r, δ). Further, this bound is proved to be optimal for the class of matroids that is used to derive the existence bounds of linear LRCs.

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“…This paper strengthens several results given in [10]. Firstly, using the matroid-based construction we extend the class of linear perfect (n, k, d, r, δ)-LRCs with ⌈k/r⌉ = 2.…”

Section: B Contributionssupporting

confidence: 80%

Section: A Related Workmentioning

confidence: 99%

Section: A Related Workmentioning

confidence: 99%

Section: A Related Workmentioning

confidence: 99%

“…) in the theorem below. Theorem 3.2 ( [10]): Let M = (E, ρ) be a matroid with 0 < ρ(E) and 1 Z = E. Then…”

Section: A Relationship Between Matroids and Almost Affine Lrcsmentioning

confidence: 99%

See 3 more Smart Citations

Bounds on the maximal minimum distance of linear locally repairable codes

Pöllänen

Westerbäck

Freij-Hollanti

et al. 2016

2016 IEEE International Symposium on Information Theory (ISIT)

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Applications of polymatroid theory to distributed storage systems

Westerbäck

Freij-Hollanti

Hollanti

2015

2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Abstract-In this paper, a link between polymatroid theory and locally repairable codes (LRCs) is established. The codes considered here are completely general in that they are subsets of A n , where A is an arbitrary finite set. Three classes of LRCs are considered, both with and without availability, and for both information-symbol and all-symbol locality. The parameters and classes of LRCs are generalized to polymatroids, and a generalized Singelton bound on the parameters for these three classes of polymatroids and LRCs is given. This result generalizes the earlier Singleton-type bounds given for LRCs. Codes achieving these bounds are coined perfect, as opposed to the more common term optimal used earlier, since they might not always exist. Finally, new constructions of perfect linear LRCs are derived from gammoids, which are a special class of matroids. Matroids, for their part, form a subclass of polymatroids and have proven useful in analyzing and constructing linear LRCs.

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On the Combinatorics of Locally Repairable Codes via Matroid Theory

Cited by 2 publications

References 0 publications

Bounds on the maximal minimum distance of linear locally repairable codes

Bounds on the maximal minimum distance of linear locally repairable codes

Applications of polymatroid theory to distributed storage systems

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