Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction

Regenerating codes are a class of distributed storage codes that allow for efficient repair of failed nodes, as compared to traditional erasure codes. An [n, k, d] regenerating code permits the data to be recovered by connecting to any k of the n nodes in the network, while requiring that a failed node be repaired by connecting to any d nodes. The amount of data downloaded for repair is typically much smaller than the size of the source data. Previous constructions of exact-regenerating codes have been confined to the case n = d + 1. In this paper, we present optimal, explicit constructions of (a) Minimum Bandwidth Regenerating (MBR) codes for all values of [n, k, d] and (b) Minimum Storage Regenerating (MSR) codes for all [n, k, d >= 2k - 2], using a new product-matrix framework. The product-matrix framework is also shown to significantly simplify system operation. To the best of our knowledge, these are the first constructions of exact-regenerating codes that allow the number n of nodes in the network, to be chosen independent of the other parameters. The paper also contains a simpler description, in the product-matrix framework, of a previously constructed MSR code with [n = d + 1, k, d >= 2k - 1].

Efficient space-time codes from cyclic division algebras

An overview of space-time code construction based on cyclic division algebras (CDA) is presented. Applications of such space-time codes to the construction of codes optimal under the diversity-multiplexing gain (D-MG) tradeoff, to the construction of the so-called perfect space-time codes, to the construction of optimal space-time codes for the ARQ channel as well as to the construction of codes optimal for the cooperative relay network channel are discussed. We also present a construction of optimal codes based on CDA for a class of orthogonal amplify and forward (OAF) protocols for the cooperative relay network

A general construction of space-time trellis codes for PSK signal sets

The diversity order and coding gain are crucial for the performance of a multiple antenna communication system. It is known that space-time trellis codes (STTC) can be used to achieve these objectives. In particular, we can use STTCs to obtain large coding gains. Many attempts have been made to construct STTCs which achieve full-diversity and good coding gains, though a general method of construction does not exist. Delay diversity code (rate-1) is known to achieve full-diversity, for any number of transmit antennas and any signal set, but does not give a good coding gain. A product distance code based delay diversity scheme (Tarokh, V. et al., IEEE Trans. Inform. Theory, vol.44, p.744-65, 1998) enables one to improve the coding gain and construct STTCs for any given number of states using coding in conjunction with delay diversity; it was stated as an open problem. We achieve such a construction. We assume a shift register based model to construct an STTC for any state complexity. We derive a sufficient condition for this STTC to achieve full-diversity, based on the delay diversity scheme. This condition provides a framework to do coding in conjunction with delay diversity for any signal constellation. Using this condition, we provide a formal rate-1 STTC construction scheme for PSK signal sets, for any number of transmit antennas and any given number of states, which achieves full-diversity and gives a good coding gain.

Matrix Processors Using p-adic Arithmetic for Exact Linear Computations

A unique code (called Hensel's code) is derived for a rational number by truncating its infinite p-adic expansion. The four basic arithmetic algorithms for these codes are described and their application to rational matrix computations is demonstrated by solving a system of linear equations exactly, using the Gaussian elimination procedure.

Distributed Storage Codes With Repair-by-Transfer and Nonachievability of Interior Points on the Storage-Bandwidth Tradeoff

Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any subset of k nodes within the n-node network. However, regenerating codes possess in addition, the ability to repair a failed node by connecting to an arbitrary subset of d nodes. It has been shown that for the case of functional repair, there is a tradeoff between the amount of data stored per node and the bandwidth required to repair a failed node. A special case of functional repair is exact repair where the replacement node is required to store data identical to that in the failed node. Exact repair is of interest as it greatly simplifies system implementation. The first result of this paper is an explicit, exact-repair code for the point on the storage-bandwidth tradeoff corresponding to the minimum possible repair bandwidth, for the case when d = n-1. This code has a particularly simple graphical description, and most interestingly has the ability to carry out exact repair without any need to perform arithmetic operations. We term this ability of the code to perform repair through mere transfer of data as repair by transfer. The second result of this paper shows that the interior points on the storage-bandwidth tradeoff cannot be achieved under exact repair, thus pointing to the existence of a separate tradeoff under exact repair. Specifically, we identify a set of scenarios which we term as ``helper node pooling,'' and show that it is the necessity to satisfy such scenarios that overconstrains the system.

Interference Alignment in Regenerating Codes for Distributed Storage: Necessity and Code Constructions

Regenerating codes are a class of recently developed codes for distributed storage that, like Reed-Solomon codes, permit data recovery from any arbitrary of nodes. However regenerating codes possess in addition, the ability to repair a failed node by connecting to any arbitrary nodes and downloading an amount of data that is typically far less than the size of the data file. This amount of download is termed the repair bandwidth. Minimum storage regenerating (MSR) codes are a subclass of regenerating codes that require the least amount of network storage; every such code is a maximum distance separable (MDS) code. Further, when a replacement node stores data identical to that in the failed node, the repair is termed as exact. The four principal results of the paper are (a) the explicit construction of a class of MDS codes for d = n - 1 >= 2k - 1 termed the MISER code, that achieves the cut-set bound on the repair bandwidth for the exact repair of systematic nodes, (b) proof of the necessity of interference alignment in exact-repair MSR codes, (c) a proof showing the impossibility of constructing linear, exact-repair MSR codes for d < 2k - 3 in the absence of symbol extension, and (d) the construction, also explicit, of high-rate MSR codes for d = k+1. Interference alignment (IA) is a theme that runs throughout the paper: the MISER code is built on the principles of IA and IA is also a crucial component to the nonexistence proof for d < 2k - 3. To the best of our knowledge, the constructions presented in this paper are the first explicit constructions of regenerating codes that achieve the cut-set bound.

Optimal Linear Codes with a Local-Error-Correction Property

Motivated by applications to distributed storage, Gopalan et al recently introduced the interesting notion of information-symbol locality in a linear code. By this it is meant that each message symbol appears in a parity-check equation associated with small Hamming weight, thereby enabling recovery of the message symbol by examining a small number of other code symbols. This notion is expanded to the case when all code symbols, not just the message symbols, are covered by such ``local'' parity. In this paper, we extend the results of Gopalan et. al. so as to permit recovery of an erased code symbol even in the presence of errors in local parity symbols. We present tight bounds on the minimum distance of such codes and exhibit codes that are optimal with respect to the local error-correction property. As a corollary, we obtain an upper bound on the minimum distance of a concatenated code.

Information-theoretically secure regenerating codes for distributed storage

Regenerating codes are a class of codes for distributed storage networks that provide reliability and availability of data, and also perform efficient node repair. Another important aspect of a distributed storage network is its security. In this paper, we consider a threat model where an eavesdropper may gain access to the data stored in a subset of the storage nodes, and possibly also, to the data downloaded during repair of some nodes. We provide explicit constructions of regenerating codes that achieve information-theoretic secrecy capacity in this setting.

A nested linear codes approach to distributed function computation over subspaces

In this paper, we consider a distributed function computation setting, where there are m distributed but correlated sources X1,...,Xm and a receiver interested in computing an s-dimensional subspace generated by [X1,...,Xm]Γ for some (m × s) matrix Γ of rank s. We construct a scheme based on nested linear codes and characterize the achievable rates obtained using the scheme. The proposed nested-linear-code approach performs at least as well as the Slepian-Wolf scheme in terms of sum-rate performance for all subspaces and source distributions. In addition, for a large class of distributions and subspaces, the scheme improves upon the Slepian-Wolf approach. The nested-linear-code scheme may be viewed as uniting under a common framework, both the Korner-Marton approach of using a common linear encoder as well as the Slepian-Wolf approach of employing different encoders at each source. Along the way, we prove an interesting and fundamental structural result on the nature of subspaces of an m-dimensional vector space V with respect to a normalized measure of entropy. Here, each element in V corresponds to a distinct linear combination of a set {Xi}im=1 of m random variables whose joint probability distribution function is given.

Codes with local regeneration

Regenerating codes and codes with locality are schemes recently proposed for a distributed storage network. While regenerating codes minimize the data downloaded for node repair, codes with locality minimize the number of nodes accessed during repair. In this paper, we provide some constructions of codes with locality, in which the local codes are regenerating codes, thereby combining the advantages of both classes of codes. The proposed constructions achieve an upper bound on minimum distance and are hence optimal. The constructions include both the cases when the local regenerating codes correspond to the MSR point as well as the MBR point on the storage repair-bandwidth tradeoff curve.

Regenerating codes for errors and erasures in distributed storage

Regenerating codes are a class of codes proposed for providing reliability of data and efficient repair of failed nodes in distributed storage systems. In this paper, we address the fundamental problem of handling errors and erasures at the nodes or links, during the data-reconstruction and node-repair operations. We provide explicit regenerating codes that are resilient to errors and erasures, and show that these codes are optimal with respect to storage and bandwidth requirements. As a special case, we also establish the capacity of a class of distributed storage systems in the presence of malicious adversaries. While our code constructions are based on previously constructed Product-Matrix codes, we also provide necessary and sufficient conditions for introducing resilience in any regenerating code.

Codes for solving three dimensional linear elastostatic problems using constant boundary elements while ignoring body forces

Three codes, that can solve three dimensional linear elastostatic problems using constant boundary elements while ignoring body forces, are provided here. The file 'bemconst.m' contains a MATLAB code for solving three dimensional linear elastostatic problems using constant boundary elements while ignoring body forces. The file 'bemconst.f90' is a Fortran translation of the MATLAB code contained in the file 'bemconst.m'. The file 'bemconstp.f90' is a parallelized version of the Fortran code contained in the file 'bemconst.f90'. The file 'inbem96.txt' is the input file for the Fortran codes contained in the files 'bemconst.f90' and 'bemconstp.f90'. Author hereby declares that the present codes are the original works of the author. Further, author hereby declares that any of the present codes, in full or in part, is not a translation or a copy of any of the existing codes written by someone else. Author's institution (Indian Institute of Science) has informed the author in writing that the institution is not interested in claiming any copyright on the present codes. Author is hereby distributing the present codes under the MIT License; full text of the license is included in each of the files that contain the codes.

On t-designs and bounds relating query complexity to error resilience in locally correctable codes

An n-length block code C is said to be r-query locally correctable, if for any codeword x ∈ C, one can probabilistically recover any one of the n coordinates of the codeword x by querying at most r coordinates of a possibly corrupted version of x. It is known that linear codes whose duals contain 2-designs are locally correctable. In this article, we consider linear codes whose duals contain t-designs for larger t. It is shown here that for such codes, for a given number of queries r, under linear decoding, one can, in general, handle a larger number of corrupted bits. We exhibit to our knowledge, for the first time, a finite length code, whose dual contains 4-designs, which can tolerate a fraction of up to 0.567/r corrupted symbols as against a maximum of 0.5/r in prior constructions. We also present an upper bound that shows that 0.567 is the best possible for this code length and query complexity over this symbol alphabet thereby establishing optimality of this code in this respect. A second result in the article is a finite-length bound which relates the number of queries r and the fraction of errors that can be tolerated, for a locally correctable code that employs a randomized algorithm in which each instance of the algorithm involves t-error correction.

Regenerating codes: A reformulated storage-bandwidth trade-off and a new construction

In this paper, the storage-repair-bandwidth (SRB) trade-off curve of regenerating codes is reformulated to yield a tradeoff between two global parameters of practical relevance, namely information rate and repair rate. The new information-repair-rate (IRR) tradeoff provides a different and insightful perspective on regenerating codes. For example, it provides a new motivation for seeking to investigate constructions corresponding to the interior of the SRB tradeoff. Interestingly, each point on the SRB tradeoff corresponds to a curve in the IRR tradeoff setup. We characterize completely, functional repair under the IRR framework, while for exact repair, an achievable region is presented. In the second part of this paper, a rate-half regenerating code for the minimum storage regenerating point is constructed that draws upon the theory of invariant subspaces. While the parameters of this rate-half code are the same as those of the MISER code, the construction itself is quite different.