# 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…

## 9 Citations

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

- Computer Science, MathematicsICMCTA
- 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.

### Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures

- Computer ScienceArXiv
- 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.

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

- Computer ScienceIEEE 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.

### The natural matroid of an integer polymatroid

- 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,…

### Decompositions of q-Matroids Using Cyclic Flats

- MathematicsArXiv
- 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…

### The complete hierarchical locality of the punctured Simplex code

- Computer Science2019 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.

### Cyclic Flats of a Polymatroid

- MathematicsAnnals of Combinatorics
- 2020

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

- Mathematics
- 2019

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

- Computer ScienceIEEE 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.

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