No 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.Comment: 32 pages. Second code construction in Section V is corrected in this version. Also, some typos are corrected. The results remain the same. Submitted to IEEE Transactions on Information Theory. This is extended, generalized, and clarified version of arXiv:1408.018

Abstract-We study the capacity of heterogeneous distributed storage systems under repair dynamics. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. We give lower and upper bounds on the system capacity. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, we study the case in which nodes may be compromised by an eavesdropper, and give bounds on the system secrecy capacity. One implication of our results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature.

No abstract

Constructions of optimal locally repairable codes (LRCs) in the case of (r + 1) n and over small finite fields were stated as open problems for LRCs in [I. Tamo et al., "Optimal locally repairable codes and connections to matroid theory", 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) 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.

We study the capacity of heterogeneous distributed storage systems under repair dynamics. Examples of these systems include peer-to-peer storage clouds, wireless, and Internet caching systems. Nodes in a heterogeneous system can have different storage capacities and different repair bandwidths. We give lower and upper bounds on the system capacity. These bounds depend on either the average resources per node, or on a detailed knowledge of the node characteristics. Moreover, we study the case in which nodes may be compromised by an eavesdropper, and give bounds on the system secrecy capacity. One implication of our results is that symmetric repair maximizes the capacity of a homogeneous system, which justifies the model widely used in the literature.

In this paper we provide a link between matroid theory and locally repairable codes (LRCs) that are almost affine. The parameters (n, k, d, r) of LRCs are generalized to matroids. A bound on the parameters (n, k, d, r), similar to the bound in [P. Gopalan et al., "On the locality of codeword symbols," IEEE Trans. Inf. Theory] for linear LRCs, is given for matroids. We prove that the given bound is not tight for a certain class of parameters, which implies a non-existence result for a certain class of optimal locally repairable almost affine codes. Constructions of optimal LRCs over small finite fields were stated as an open problem in [I. Tamo et al., "Optimal locally repairable codes and connections to matroid theory", 2013 IEEE ISIT]. In this paper optimal LRCs which do not require a large field are constructed for certain classes of parameters.

No abstract