New Codes and Inner Bounds for Exact Repair in Distributed Storage Systems

We study the exact-repair tradeoff between storage and repair bandwidth in distributed storage systems (DSS). We give new inner bounds for the tradeoff region and provide code constructions that achieve these bounds.Comment: Submitted to the IEEE International Symposium on Information Theory (ISIT) 2014. This draft contains 8 pages and 4 figure

Cyclic LRC Codes and their Subfield Subcodes

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 Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Transactions on Information Theory, no. 8, 2014; arXiv:1311.3284). 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.Comment: Submitted for publicatio

Cyclic LRC Codes, binary LRC codes, and upper bounds on the distance of cyclic codes

We consider linear cyclic codes with the locality property, or locally recoverable codes (LRC codes). A family of LRC codes that generalize the classical construction of Reed-Solomon codes was constructed in a recent paper by I. Tamo and A. Barg (IEEE Trans. Inform. Theory, no. 8, 2014). In this paper we focus on optimal cyclic codes that arise from this 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 locality parameter of a cyclic code is related to the dual distance of this code, and we phrase our results in terms of upper bounds on the dual distance.Comment: 12pp., submitted for publication. An extended abstract of this submission was posted earlier as arXiv:1502.01414 and was published in Proceedings of the 2015 IEEE International Symposium on Information Theory, Hong Kong, China, June 14-19, 2015, pp. 1262--126

Data Secrecy in Distributed Storage Systems under Exact Repair

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.Comment: Submitted to Netcod 201

Erasure Codes for Optimal Node Repairs in Distributed Storage Systems

Google, Amazon, and other services store data in multiple geographically separated disks called nodes, among other reasons, to safeguard the data from node failures. Standard techniques for such a distributed way of storage include multiple backups (typically triple replication) or using erasure codes such as Reed-Solomon codes. The latter codes are the most space-efficient for a targeted worst-case number of simultaneous node failures. They are extremely inefficient how- ever for repairing the frequently occurring single node failure. Replication provides the most cost-effective repair in this scenario but ultimately is an unwise option in today's data proliferation. New erasure codes are therefore required to simultaneously optimize storage efficiency, worst-case resilience and repair costs for single node failures. This dissertation looks at two such erasure codes: regenerating codes, which optimize the communication costs, and locally repairable codes (LRCs), which optimize the I/O costs (number of nodes contacted). Regenerating codes store a file of size M on n nodes and trade-off the amount of data stored &alpha; per node for the amount of bandwidth &gamma; used to repair a node. This dissertation presents new code constructions and thereby, state-of-the-art inner bounds for this trade-off region. A lower bound is also provided for &alpha; for codes achieving the optimal storage point in the trade-off, signifying the necessity of storing an exponential number of symbols. Ideas developed in this analysis have been applied to establish the optimal file size that can be securely stored in the presence of an eavesdropper, when the corresponding regenerating code is at the optimal storage point. Locally repairable codes, on the other hand, can be viewed as classical erasure codes of dimension k, length n, distance d and a new parameter r, called locality. In storage parlance, an LRC of locality r stores a file of size k on n nodes such that when a node fails, there exist r other nodes that suffice to reconstruct the failed node. Previous considerations on optimality have largely ignored the finite field involved. This dissertation provides codes on the binary field that optimize k for certain families of parameters n, d, and r

Hierarchical zonation technique to extract common boundaries of a layered earth model

Earth formations can be modeled as a layered medium with each layer having its own electrical, mechanical, and geometrical properties. In a typical exploration environment, a well-bore (about 15 to 30 cm in diameter) is drilled to a depth that may extend to a fe