Sreechakra Goparaju

Binary cyclic codes that are locally repairable

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.

An Improved Sub-Packetization Bound for Minimum Storage Regenerating Codes

Distributed storage systems employ codes to provide resilience to failure of multiple storage disks. In particular, an (n, k) maximum distance separable (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. The 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 subpacketization 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 have a subpacketization factor ℓ, which is exponential in k. It has been conjectured 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 log2 I(logδ ℓ + 1), for an arbitrary number of parity nodes r = n - k, where δ = r/(r - 1).

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 amount of data that can be stored securely decreases exponentially with the number of compromised nodes. Whether this exponential decrease is fundamental or is a consequence of the exactness and linearity constraints remains an open question.

Minimum storage regenerating codes for all parameters

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 Fq is defined as follows. Using such a code C, a file F consisting of αk symbols over Fq can be distributed 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. A common practical requirement for regenerating codes is to have the original file F available in uncoded form on some k of the n nodes, known as systematic nodes. In this case, several authors relax the defining node-repair condition above, requiring the optimal repair bandwidth of dα/(d-k+1) symbols for systematic nodes only. We shall call such codes systematic-repair MSR codes. Unfortunately, explicit 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. Although setting d = n - 1 minimizes the repair bandwidth, it may be impractical to connect to all the remaining nodes in order to repair a single failed node. Our main result in this paper is an explicit construction of systematic-repair [n, k, d] MSR codes for all possible values of parameters n, k, d. In particular, we construct systematic-repair MSR codes of high rate k/n > 0.5 and low repair connectivity k ≤ d ≤ n - 1. Such codes were not previously known to exist. In order to construct these codes, we solve simultaneously several repair scenarios, each of which is expressible as an interference alignment problem. Extension of our results beyond systematic repair remains an open problem.

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 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.

Synchronizing edits in distributed storage networks

We consider the problem of synchronizing data in distributed storage networks under edits that include deletions and insertions. We present modifications of codes on distributed storage systems that allow updates in the parity-check values to be performed with one round of communication at low bit rates and a small storage overhead. Our main contributions are novel protocols for synchronizing both frequently updated and semi-static data, and protocols for data deduplication applications, based on intermediary coding using permutation and Vandermonde matrices.

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.

A new sub-packetization bound for minimum storage regenerating codes

Codes for distributed storage systems are often designed to sustain 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 can potentially repair a single erasure using a fraction l/(n - k) of data stored in the surviving nodes. 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 I 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 new converse bounds on k for a given ℓ We prove that k ≤ ℓ2 for an arbitrary but fixed number of parity nodes r = n ™ k. For the practical case of 2 parity nodes, we prove a stronger result that k ≤ 4ℓ.

Synchronization and Deduplication in Coded Distributed Storage Networks

We consider the problem of synchronizing coded data in distributed storage networks undergoing insertion and deletion edits. We present modifications of distributed storage codes that allow updates in the parity-check values to be performed with one round of communication at low bit rates and with small storage overhead. Our main contributions are novel protocols for synchronizing frequently updated and semi-static data based on functional intermediary coding involving permutation and Vandermonde matrices.

Can linear minimum storage regenerating codes be universally secure

We study the problem of making a distributed storage system information-theoretically secure against a passive eavesdropper, and aim to characterize coding schemes that are universally secure for up to a given number of eavesdropped nodes. Specifically, we consider minimum storage regenerating (MSR) codes and ask the following question: For an MSR code where a failed node is repaired using all the remaining nodes, is it possible to simultaneously be optimally secure using a single linear coding scheme? We define a pareto-optimality associated with this simultaneity and show that there exists at least one linear coding scheme that is pareto-optimal.