Abstract:
This paper presents and analyzes a novel concatenated coding scheme for enabling error resilience in two distributed storage settings: one being storage using existing regenerating codes and the second being storage using locally repairable codes. The concatenated coding scheme brings together a maximum rank distance (MRD) code as an outer code and either a globally regenerating or a locally repairable code as an inner code. Also, error resilience for combination of locally repairable codes with regenerating codes is considered. This concatenated coding system is designed to handle two different types of adversarial errors: the first type includes an adversary that can replace the content of an affected node only once; while the second type studies an adversary that is capable of polluting data an unbounded number of times. The paper establishes an upper bound on the resilience capacity for a locally repairable code and proves that this concatenated coding coding scheme attains the upper bound on resilience capacity for the first type of adversary. Further, the paper presents mechanisms that combine the presented concatenated coding scheme with subspace signatures to achieve error resilience for the second type of errors.

Abstract:
Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that {\it i)} are repair efficient and {\it ii)} achieve arbitrarily high data rates. In this paper, we explore the repair metric of {\it locality}, which corresponds to the number of disk accesses required during a {\color{black}single} node repair. Under this metric we characterize an information theoretic trade-off that binds together locality, code distance, and the storage capacity of each node. We show the existence of optimal {\it locally repairable codes} (LRCs) that achieve this trade-off. The achievability proof uses a locality aware flow-graph gadget which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data-rates. Our locality optimal construction is based on simple combinations of Reed-Solomon blocks.

Abstract:
In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. Also, 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. The proof of this also gives a constructive method to find locally repairable codes. Constructions are given of three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size, not depending on the size of the code.

Abstract:
In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. Also, 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. The proof of this gives also a constructive method to find locally repairable codes.

Abstract:
Locally repairable codes (LRCs) are error correcting codes used in distributed data storage. A traditional approach is to look for codes which simultaneously maximize error tolerance and minimize storage space consumption. However, this tends to yield codes for which error correction requires an unrealistic amount of communication between storage nodes. LRCs solve this problem by allowing errors to be corrected locally. This thesis reviews previous results on the subject presented in [1]. These include that every almost affine LRC induces a matroid such that the essential properties of the code are determined by the matroid. Also, the generalized Singleton bound for LRCs can be extended to matroids as well. Then, matroid theory can be used to find classes of matroids that either achieve the bound, meaning they are optimal in a certain sense, or at least come close to the bound. This thesis presents an improvement to the results of [1] in both of these cases. [1] T. Westerb\"ack, R. Freij, T. Ernvall and C. Hollanti, "On the Combinatorics of Locally Repairable Codes via Matroid Theory", arXiv:1501.00153 [cs.IT], 2014.

Abstract:
The locally repairable codes (LRCs) were introduced to correct erasures efficiently in distributed storage systems. LRCs are extensively studied recently. In this paper, we first deal with the open case remained in \cite{q} and derive an improved upper bound for the minimum distances of LRCs. We also give an explicit construction for LRCs attaining this bound. Secondly, we consider the constructions of LRCs with any locality and availability which have high code rate and minimum distance as large as possible. We give a graphical model for LRCs. By using the deep results from graph theory, we construct a family of LRCs with any locality $r$ and availability $2$ with code rate $\frac{r-1}{r+1}$ and optimal minimum distance $O(\log n)$ where $n$ is the length of the code.

Abstract:
This paper presents a construction for several families of optimal binary locally repairable codes (LRCs) with small locality (2 and 3). This construction is based on various anticodes. It provides binary LRCs which attain the Cadambe-Mazumdar bound. Moreover, most of these codes are optimal with respect to the Griesmer bound.

Abstract:
Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$, i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this work, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$, dimension $k$, locality $r$, availability $t$, and field size $q$. Then, we study binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed-Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared to other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and $4$-cycle free regular low-density parity-check (LDPC) codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code.

Abstract:
Constructions of optimal locally repairable codes (LRCs) in the case of $(r+1) \nmid n$ and over small finite fields were stated as open problems for LRCs in [I. Tamo \emph{et al.}, "Optimal locally repairable codes and connections to matroid theory", \emph{2013 IEEE ISIT}]. In this paper, these problems are studied by constructing almost optimal linear LRCs, which are proven to be optimal for certain parameters, including cases for which $(r+1) \nmid n$. More precisely, linear codes for given length, dimension, and all-symbol locality are constructed with almost optimal minimum distance. `Almost optimal' refers to the fact that their minimum distance differs by at most one from the optimal value given by a known bound for LRCs. In addition to these linear LRCs, optimal LRCs which do not require a large field are constructed for certain classes of parameters.

Abstract:
We study properties of rank metric and codes in rank metric over finite fields. We show that in rank metric perfect codes do not exist. We derive an existence bound that is the equivalent of the Gilbert--Varshamov bound in Hamming metric. We study the asymptotic behavior of the minimum rank distance of codes satisfying GV. We derive the probability distribution of minimum rank distance for random and random $\F{q}$-linear codes. We give an asymptotic equivalent of their average minimum rank distance and show that random $\F{q}$-linear codes are on GV bound for rank metric. We show that the covering density of optimum codes whose codewords can be seen as square matrices is lower bounded by a function depending only on the error-correcting capability of the codes. We show that there are quasi-perfect codes in rank metric over fields of characteristic 2.