# 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}
}
• Published 13 April 2017
• Computer Science
• ArXiv
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…
• Computer Science, Mathematics
ICMCTA
• 2017
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.
• Computer Science
ArXiv
• 2017
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.
• Computer Science
IEEE Transactions on Information Theory
• 2019
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.
• Mathematics
• 2022
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,
• Mathematics
ArXiv
• 2023
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
• Computer Science
2019 IEEE International Symposium on Information Theory (ISIT)
• 2019
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.
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
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
• Computer Science
IEEE Communications Letters
• 2019
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

• Computer Science
• 2016
The goal of this paper is to illustrate the essential features of LRCs, namely locality, availability, and hierarchy alongside with related generalized Singleton bounds.
• Computer Science
2013 IEEE International Symposium on Information Theory
• 2013
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.
• Mathematics
2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton)
• 2015
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.
• Computer Science
2013 IEEE International Symposium on Information Theory
• 2013
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
• Computer Science
ArXiv
• 2016
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.
• Computer Science
IEEE Transactions on Information Theory
• 2014
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.
• Mathematics
IEEE Transactions on Information Theory
• 2016
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.
• Computer Science
IEEE Transactions on Information Theory
• 2016
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.
• Computer Science
Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007)
• 2007
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
• Computer Science
2013 International Symposium on Network Coding (NetCod)
• 2013
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.