Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures

Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers, which makes the systems computationally expensive and difficult to maintain. Here, we study linear locally repairable codes over the binary field, tolerating multiple local erasures. We derive bounds on the minimum distance on such codes, and give examples of LRCs achieving these bounds. Our main technical tools come from matroid theory, and as a byproduct of our proofs, we show that the lattice of cyclic flats of a simple binary matroid is atomic.Comment: 9 pages, 1 figure. Parts of this paper were presented at IZS 2018. This extended arxiv version includes corrected versions of Theorem 1.4 and Proposition 6 that appeared in the IZS 2018 proceeding

On Matroid Theory and Distributed Data Storage

The fast development of web services and cloud computing has generated an enormous amount of digital data. Huge data centers were therefore built to store and remotely access this data. A distributed storage system consists of a network of storage servers where the data is distributed among these servers. The main challenge in the design of such systems is to guarantee that the data is reliably stored. In fact, given the required large number of storage servers, server failures happen on a daily basis. To prevent from data loss, redundant data is stored alongside the initial data, by either replicating or encoding the data. The amount of redundant data is referred to as the storage overhead. While, for the same failure tolerance, encoding the data requires a smaller storage overhead than replicating the data, it implies a more complex repair process of the failed servers. Therefore, on top of the storage overhead and the failure tolerance, two notions are of particular interest: the repair bandwidth and the locality, which is the number of servers contacted for repairing a few failed servers. This thesis focuses on the notion of locality. More precisely, the main goal is to derive a tradeoff between the storage overhead, the failure tolerance, and the locality when the underlying code alphabet is fixed. Deriving a tradeoff is important in practice as it characterizes the best possible codes. Furthermore, since the alphabet relates to the repair complexity and affects the different aforementioned notions, it is interesting to derive alphabet-dependent tradeoffs. To approach this problem, we use the internal structure of the storage codes and the relation between codes and matroids. Matroids are interesting mathematical objects on their own right and provide useful tools related to the alphabet. In the articles composing this thesis, we derive alphabet-dependent tradeoffs with various locality assumptions. Furthermore, we study the impact of the alphabet on the substructures of a representable matroid. We also analyse the hierarchical locality of a specific code construction. Finally, we examine nonlinear codes with locality on mixed alphabets and derive their tradeoff using polymatroids

The complete hierarchical locality of the punctured Simplex code

This paper presents a new alphabet-dependent bound for codes with hierarchical locality. Then, the complete list of possible localities is derived for a class of codes obtained by deleting specific columns from a Simplex code. This list is used to show that these codes are optimal codes with hierarchical locality.Peer reviewe

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

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 $(r,\delta)$ , 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.Peer reviewe

Cyclic flats of binary matroids

Funding Information: The work of M. Grezet, C. Hollanti, and T. Westerbäck was supported in part by The Academy of Finland [grant numbers 276031 , 282938 , and 303819 ] and by the Technical University of Munich – Institute for Advanced Study, funded by the German Excellence Initiative and the EU 7th Framework Programme [grant number 291763 ], via a Hans Fischer Fellowship. R. Freij-Hollanti was supported by the German Research Foundation ( Deutsche Forschungsgemeinschaft , DFG) [grant number WA3907/1-1 ]. Publisher Copyright: © 2021 The Author(s) Copyright: Copyright 2021 Elsevier B.V., All rights reserved.In this paper, first steps are taken towards characterizing rank-decorated lattices of cyclic flats Z(M) that belong to matroids M that can be represented over a prescribed finite field Fq. Two natural maps from Z(M) to the lattice of cyclic flats of a minor of M are given. Binary matroids are characterized via their lattice of cyclic flats. It is shown that the lattice of cyclic flats of a simple binary matroid without isthmuses is atomic.Peer reviewe