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.

Burst error correction using partial fourier matrices and block sparse representation

There is a strong relation between sparse signal recovery and error control coding. It is known that burst errors are block sparse in nature. So, here we attempt to solve burst error correction problem using block sparse signal recovery methods. We construct partial Fourier based encoding and decoding matrices using results on difference sets. These constructions offer guaranteed and efficient error correction when used in conjunction with reconstruction algorithms which exploit block sparsity.

Jamming to foil an eavesdropper

We consider key-less secure communication against a passive adversary, by allowing the legitimate receiver to selectively jam transmitted bits. The channel between the transmitter and legitimate receiver is assumed to be half-duplex (i.e., one cannot jam and receive simultaneously), while the only degradation seen by the eavesdropper is due to jamming done by the legitimate receiver. However, jamming must be done without knowledge of the transmitted sequence, and the transmitted sequence must be recovered exactly by the receiver from the unjammed bits alone. We study the resulting coding problem in this setup, both under complete equivocation (CE) and partial equivocation (PE) of the eavesdropper. For (CE), we give explicit code-constructions that achieve the maximum transmission rate, while for (PE) we compute upper and lower bounds on the maximum possible transmission rate.

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.

Codes With Local Regeneration and Erasure Correction

Regenerating codes and codes with locality are two coding schemes that have recently been proposed, which in addition to ensuring data collection and reliability, also enable efficient node repair. In a situation where one is attempting to repair a failed node, regenerating codes seek to minimize the amount of data downloaded for node repair, while codes with locality attempt to minimize the number of helper nodes accessed. This paper presents results in two directions. In one, this paper extends the notion of codes with locality so as to permit local recovery of an erased code symbol even in the presence of multiple erasures, by employing local codes having minimum distance >2. An upper bound on the minimum distance of such codes is presented and codes that are optimal with respect to this bound are constructed. The second direction seeks to build codes that combine the advantages of both codes with locality as well as regenerating codes. These codes, termed here as codes with local regeneration, are codes with locality over a vector alphabet, in which the local codes themselves are regenerating codes. We derive an upper bound on the minimum distance of vector-alphabet codes with locality for the case when their constituent local codes have a certain uniform rank accumulation property. This property is possessed by both minimum storage regeneration (MSR) and minimum bandwidth regeneration (MBR) codes. We provide several constructions of codes with local regeneration which achieve this bound, where the local codes are either MSR or MBR codes. Also included in this paper, is an upper bound on the minimum distance of a general vector code with locality as well as the performance comparison of various code constructions of fixed block length and minimum distance.

A FUNCTIONAL MODEL FOR PURE Gamma-CONTRACTIONS

A pair of commuting operators (S,P) defined on a Hilbert space H for which the closed symmetrized bidisc Gamma = {(z(1) + z(2), z(1)z(2)) : vertical bar z(1)vertical bar <= 1, vertical bar z(2)vertical bar <= 1} subset of C-2 is a spectral set is called a Gamma-contraction in the literature. A Gamma-contraction (S, P) is said to be pure if P is a pure contraction, i.e., P*(n) -> 0 strongly as n -> infinity Here we construct a functional model and produce a set of unitary invariants for a pure Gamma-contraction. The key ingredient in these constructions is an operator, which is the unique solution of the operator equation S - S*P = DpXDp, where X is an element of B(D-p), and is called the fundamental operator of the Gamma-contraction (S, P). We also discuss some important properties of the fundamental operator.

Codes with Locality for Two Erasures

In this paper, we study codes with locality that can recover from two erasures via a sequence of two local, parity-check computations. By a local parity-check computation, we mean recovery via a single parity-check equation associated with small Hamming weight. Earlier approaches considered recovery in parallel; the sequential approach allows us to potentially construct codes with improved minimum distance. These codes, which we refer to as locally 2-reconstructible codes, are a natural generalization along one direction, of codes with all-symbol locality introduced by Gopalan et al, in which recovery from a single erasure is considered. By studying the generalized Hamming weights of the dual code, we derive upper bounds on the minimum distance of locally 2-reconstructible codes and provide constructions for a family of codes based on Turan graphs, that are optimal with respect to this bound. The minimum distance bound derived here is universal in the sense that no code which permits all-symbol local recovery from 2 erasures can have larger minimum distance regardless of approach adopted. Our approach also leads to a new bound on the minimum distance of codes with all-symbol locality for the single-erasure case.

A spectral element for wave propagation in honeycomb sandwich construction considering core flexibility

Spectral elements are found to be extremely resourceful to study the wave propagation characteristics of structures at high frequencies. Most of the aerospace structures use honeycomb sandwich constructions. The existing spectral elements use single layer theories for a sandwich construction wherein the two face sheets vibrate together and this model is sufficient for low frequency excitations. At high frequencies, the two face sheets vibrate independently. The Extended Higher order SAndwich Plate theory (EHSaPT) is suitable for representing the independent motion of the face sheets. A 1D spectral element based on EHSaPT is developed in this work. The wave number and the wave speed characteristics are obtained using the developed spectral element. It is shown that the developed spectral element is capable of representing independent wave motions of the face sheets. The propagation speeds of a high frequency modulated pulse in the face sheets and the core of a honeycomb sandwich are demonstrated. Responses of a typical honeycomb sandwich beam to high frequency shock loads are obtained using the developed spectral element and the response match very well with the finite element results. It is shown that the developed spectral element is able to represent the flexibility of the core resulting into independent wave motions in the face sheets, for which a finite element method needs huge degrees of freedom. (C) 2015 Elsevier Ltd. All rights reserved.

The influence of termites on soil sheeting properties varies depending on the materials on which they feed

Fungus-growing termites are involved in many ecological processes and play a central role in influencing soil dynamics in the tropics. The physical and chemical properties of their nest structures have been largely described; however less information is available concerning the relatively temporary structures made above-ground to access food items and protect the foraging space (the soil `sheetings'). This study investigated whether the soil physical and chemical properties of these constructions are constant or if they vary depending on the type of food they cover. Soil samples and soil sheetings were collected in a forest in India, from leaves on the ground (LEAF), fallen branches (WOOD), and vertical soil sheetings covering the bark of trees (TREE). In this environment, termite diversity was dominated by Odontotermes species, and especially Odontotermes feae and Odontotermes obesus. However, there was no clear niche differentiation and, for example, O. feae termites were found on all the materials. Compared with the putative parent soil (control), TREE sheetings showed the greatest (and most significant) differences (higher clay content and smaller clay particle sizes, lower C and N content and smaller delta C-13 and delta N-15), while LEAF sheetings were the least modified, though still significantly different than the control soil. We suggest that the termite diversity is a less important driver of potential soil modification than sheeting diversity. Further, there is evidence that construction properties are adapted to their prospective life-span, with relatively long-lasting structures being most different from the parent soil. (C) 2015 Elsevier Masson SAS. All rights reserved.

Evaluation of the Grounding Resistance of Conductors Buried in Steplike Terrain

Availability of land for conventional air-insulated substations is becoming increasingly difficult not only in urban but also in semiurban areas. When the land made available is highly uneven, the associated technoeconomic factors favors the erection of substations on a steplike-formed ground surface and such constructions are in service for more than ten years in some parts of southern India. Noting that the literature on the performance of ground grids in such a construction is rather scarce, the present work was taken up. Evaluation of the performance of earthing elements in steplike ground forms the main goal of the present work. For the numerical evaluation, a suitable boundary-based methodology is employed. This method retains the classical Galerkin approach for the conductors, while the interfaces are replaced by equivalent fictitious surface sources defined over unstructured mesh. Details of the implementation of this numerical method, along with special measures to minimize the computation, are presented. The performance of basic earthing elements, such as the driven rod, counterpoise, and simple grids buried in steplike ground, are analyzed and compared with that for the case with uniform soil surface. It is shown that more than the earthing resistances, the step potentials can get significantly affected.