Codes for storage systems aim to minimize the repair locality, which is the number of disks (or nodes) that participate in the repair of a single failed disk. Simultaneously, the code must sustain a high rate, operate on a small finite field to be practically significant and be tolerant to a large number of erasures. To this end, we construct new families of binary linear codes that have an optimal dimension (rate) for a given minimum distance and locality. Specifically, we construct cyclic codes that are locally repairable for locality 2 and distances 2, 6 and 10. In doing so, we discover new upper bounds on the code dimension, and prove the optimality of enabling local repair by provisioning disjoint groups of disks. Finally, we extend our construction to build codes that have multiple repair sets for each disk.

Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. Specifically, an (n, k) MDS code stores k symbols in n disks such that the overall system is tolerant to a failure of up to n − k disks. However, access to at least k disks is still required to repair a single erasure. To reduce repair bandwidth, array codes are used where the stored symbols or packets are vectors of length ℓ. MDS array codes have the potential to repair a single erasure using a fraction 1/(n−k) of data stored in the remaining disks. We introduce new methods of analysis which capitalize on the translation of the storage system problem into a geometric problem on a set of operators and subspaces. In particular, we ask the following question: for a given (n, k), what is the minimum vector-length or sub-packetization factor ℓ required to achieve this optimal fraction? For exact recovery of systematic disks in an MDS code of low redundancy, i.e. k/n > 1/2, the best known explicit codes [1] have a sub-packetization factor ℓ which is exponential in k. It has been conjectured [2] that for a fixed number of parity nodes, it is in fact necessary for ℓ to be exponential in k. In this paper, we provide a new log-squared converse bound on k for a given ℓ, and prove that k ≤ 2 log 2 ℓ (log δ ℓ + 1), for an arbitrary number of parity nodes r = n − k, where δ = r/(r − 1).

Abstract-We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalizes the classical construction of ReedSolomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. IT, no. 8, 2014). In this paper we focus on the optimal cyclic codes that arise from the general construction. We give a characterization of these codes in terms of their zeros, and observe that there are many equivalent ways of constructing optimal cyclic LRC codes over a given field. We also study subfield subcodes of cyclic LRC codes (BCH-like LRC codes) and establish several results about their locality and minimum distance.

The problem of securing data against eavesdropping in distributed storage systems is studied. The focus is on systems that use linear codes and implement exact repair to recover from node failures. The maximum file size that can be stored securely is determined for systems in which all the available nodes help in repair (i.e., repair degree d = n − 1, where n is the total number of nodes) and for any number of compromised nodes. Similar results in the literature are restricted to the case of at most two compromised nodes. Moreover, new explicit upper bounds are given on the maximum secure file size for systems with d < n − 1. The key ingredients for the contribution of this paper are new results on subspace intersection for the data downloaded during repair. The new bounds imply the interesting fact that the maximum data that can be stored securely decreases exponentially with the number of compromised nodes.

Abstract-Regenerating codes for distributed storage have attracted much research interest in the past decade. Such codes trade the bandwidth needed to repair a failed node with the overall amount of data stored in the network. Minimum storage regenerating (MSR) codes are an important class of optimal regenerating codes that minimize (first) the amount of data stored per node and (then) the repair bandwidth. Specifically, an [n, k, d]-(α) MSR code C over F q stores a file F consisting of αk symbols over F q among n nodes, each storing α symbols, in such a way that:• the file F can be recovered by downloading the content of any k of the n nodes; and • the content of any failed node can be reconstructed by accessing any d of the remaining n − 1 nodes and downloading α/(d−k+1) symbols from each of these nodes. In practice, the file F is typically available in uncoded form on some k of the n nodes, known as systematic nodes, and the defining node-repair condition above can be relaxed to requiring the optimal repair bandwidth for systematic nodes only. Such codes are called systematic-repair MSR codes.Unfortunately, finite-α constructions of [n, k, d] MSR codes are known only for certain special cases: either low rate, namely k/n 0.5, or high repair connectivity, namely d = n − 1. Our main result in this paper is a finite-α construction of systematic-repair [n, k, d] MSR codes for all possible values of parameters n, k, d. We also introduce a generalized construction for [n, k] MSR codes to achieve the optimal repair bandwidth for all values of d simultaneously.