Matroid Theory and Storage Codes: Bounds and Constructions

@article{Freij2017MatroidTA,
  title={Matroid Theory and Storage Codes: Bounds and Constructions},
  author={Ragnar Freij and Camilla Hollanti and Thomas Westerb{\"a}ck},
  journal={ArXiv},
  year={2017},
  volume={abs/1704.04007}
}
Recent research on distributed storage systems (DSSs) has revealed interesting connections between matroid theory and locally repairable codes (LRCs). The goal of this chapter is to introduce the reader to matroids and polymatroids, and illustrate their relation to distributed storage systems. While many of the results are rather technical in nature, effort is made to increase accessibility via simple examples. The chapter embeds all the essential features of LRCs, namely locality, availability… 
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9 Citations
Highly Influential Citations
1
Background Citations
7
Methods Citations
1

9 Citations

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

This paper develops theory towards code existence and design over a given field by exploiting recently established connections between linear locally repairable codes and matroids, and using matroid-theoretic characterisations of linearity over small fields.

Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures

This work studies linear locally repairable codes over the binary field, tolerating multiple local erasures, and derives bounds on the minimum distance on such codes, and gives examples of LRCs achieving these bounds.

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

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, and 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.

The natural matroid of an integer polymatroid

A BSTRACT . The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate,…

Decompositions of q-Matroids Using Cyclic Flats

We study the direct sum of q-matroids by way of their cyclic flats. Using that the rank function of a q-matroid is fully determined by the cyclic flats and their ranks, we show that the cyclic flats…

The complete hierarchical locality of the punctured Simplex code

A new alphabet-dependent bound for codes with hierarchical locality is presented, and the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code.

Cyclic Flats of a Polymatroid

Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to…

C O ] 1 3 O ct 2 01 9 Cyclic flats of a polymatroid

Polymatroids can be considered as “fractional matroids” where the rank function is not required to be integer valued. Many, but not every notion in matroid terminology translates naturally to…

Uniform Minors in Maximally Recoverable Codes

The list of all the possible uniform minors is derived by using matroid theory and the relation between MDS codes and uniform minors to improve the known non-asymptotic lower bound on the required field size of a maximally recoverable code.

References

SHOWING 1-10 OF 51 REFERENCES

Locally repairable codes with availability and hierarchy: matroid theory via examples

The goal of this paper is to illustrate the essential features of LRCs, namely locality, availability, and hierarchy alongside with related generalized Singleton bounds.

Optimal locally repairable codes and connections to matroid theory

This work presents an explicit and simple to implement construction of optimal LRCs, for code parameters previously established by existence results, and derives a new result on the matroid represented by the code's generator matrix.

Applications of polymatroid theory to distributed storage systems

A link between polymatroid theory and locally repairable codes (LRCs) is established, and a generalized Singelton bound on the parameters for these three classes of polymatroids and LRCs is given.

Optimal locally repairable codes via rank-metric codes

This paper presents a new explicit construction for locally repairable codes (LRCs) for distributed storage systems which possess all-symbol locality and the largest possible minimum distance, or…

Improved Singleton-type Bounds for Locally Repairable Codes

Two improvements to matroid theory are presented, proved to be optimal for the main class of matroids used to derive the existence bounds in [T.

Locally Repairable Codes

This paper explores the repair metric of locality, which corresponds to the number of disk accesses required during a single node repair, and shows the existence of optimal locally repairable codes (LRCs) that achieve this tradeoff.

On the Combinatorics of Locally Repairable Codes via Matroid Theory

It is shown that the given bound is not tight for certain classes of parameters, implying a nonexistence result for the corresponding locally repairable almost affine codes that are coined perfect in this paper.

Constructions and Properties of Linear Locally Repairable Codes

It is shown that, with high probability, a random matrix with a few extra columns guaranteeing the locality property is a generator matrix for a locally repairable code with a good minimum distance.

Pyramid Codes: Flexible Schemes to Trade Space for Access Efficiency in Reliable Data Storage Systems

To flexibly explore the trade-offs between storage space and access efficiency in reliable data storage systems, we describe two classes of erasure resilient coding schemes: basic and generalized…

An upper bound on the size of locally recoverable codes

This paper bound the minimum distance of a code in terms of of its length, size and locality from a significantly simple analysis and depends on the size of the alphabet being used.
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