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.

Improved perfect space-time block codes

Perfect space-time block codes (STBCs) are based on four design criteria-full-rateness, nonvanishing determinant, cubic shaping, and uniform average transmitted energy per antenna per time slot. Cubic shaping and transmission at uniform average energy per antenna per time slot are important from the perspective of energy efficiency of STBCs. The shaping criterion demands that the generator matrix of the lattice from which each layer of the perfect STBC is carved be unitary. In this paper, it is shown that unitariness is not a necessary requirement for energy efficiency in the context of space-time coding with finite input constellations, and an alternative criterion is provided that enables one to obtain full-rate (rate of complex symbols per channel use for an transmit antenna system) STBCs with larger normalized minimum determinants than the perfect STBCs. Further, two such STBCs, one each for 4 and 6 transmit antennas, are presented and they are shown to have larger normalized minimum determinants than the comparable perfect STBCs which hitherto had the best-known normalized minimum determinants.

Asymptotically-good, multigroup decodable space-time block codes

For a family of Space-Time Block Codes (STBCs) C-1, C-2,..., with increasing number of transmit antennas N-i, with rates R-i complex symbols per channel use, i = 1, 2,..., we introduce the notion of asymptotic normalized rate which we define as lim(i ->infinity) R-i/N-i, and we say that a family of STBCs is asymptotically-good if its asymptotic normalized rate is non-zero, i. e., when the rate scales as a non-zero fraction of the number of transmit antennas. An STBC C is said to be g-group decodable, g >= 2, if the information symbols encoded by it can be partitioned into g groups, such that each group of symbols can be ML decoded independently of the others. In this paper we construct full-diversity g-group decodable codes with rates greater than one complex symbol per channel use for all g >= 2. Specifically, we construct delay-optimal, g-group decodable codes for number of transmit antennas N-t that are a multiple of g2left perpendicular(g-1/2)right perpendicular with rate N-t/g2(g-1) + g(2)-g/2N(t). Using these new codes as building blocks, we then construct non-delay-optimal g-group decodable codes with rate roughly g times that of the delay-optimal codes, for number of antennas N-t that are a multiple of 2left perpendicular(g-1/2)right perpendicular, with delay gN(t) and rate Nt/2(g-1) + g-1/2N(t). For each g >= 2, the new delay-optimal and non-delay- optimal families of STBCs are both asymptotically-good, with the latter family having the largest asymptotic normalized rates among all known families of multigroup decodable codes with delay T <= gN(t). Also, for g >= 3, these are the first instances of g-group decodable codes with rates greater than 1 reported in the literature.

Minimizing the Complexity of Fast Sphere Decoding of STBCs

Decoding of linear space-time block codes (STBCs) with sphere-decoding (SD) is well known. A fast-version of the SD known as fast sphere decoding (FSD) was introduced by Biglieri, Hong and Viterbo. Viewing a linear STBC as a vector space spanned by its defining weight matrices over the real number field, we define a quadratic form (QF), called the Hurwitz-Radon QF (HRQF), on this vector space and give a QF interpretation of the FSD complexity of a linear STBC. It is shown that the FSD complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when SD is used depends on the channel realization) or the number of receive antennas. It is also shown that the FSD complexity is completely captured into a single matrix obtained from the HRQF. Moreover, for a given set of weight matrices, an algorithm to obtain an optimal ordering of them leading to the least FSD complexity is presented. The well known classes of low FSD complexity codes (multi-group decodable codes, fast decodable codes and fast group decodable codes) are presented in the framework of HRQF.

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.

Fast-Decodable MIDO Codes With Large Coding Gain

In this paper, a new method is proposed to obtain full-diversity, rate-2 (rate of two complex symbols per channel use) space-time block codes (STBCs) that are full-rate for multiple input double output (MIDO) systems. Using this method, rate-2 STBCs for 4 x 2, 6 x 2, 8 x 2, and 12 x 2 systems are constructed and these STBCs are fast ML-decodable, have large coding gains, and STBC-schemes consisting of these STBCs have a non-vanishing determinant (NVD) so that they are DMT-optimal for their respective MIDO systems. It is also shown that the Srinath-Rajan code for the 4 x 2 system, which has the lowest ML-decoding complexity among known rate-2 STBCs for the 4x2 MIDO system with a large coding gain for 4-/16-QAM, has the same algebraic structure as the STBC constructed in this paper for the 4 x 2 system. This also settles in positive a previous conjecture that the STBC-scheme that is based on the Srinath-Rajan code has the NVD property and hence is DMT-optimal for the 4 x 2 system.

Impedance spectroscopy of novel hybrid composite films of polyvinylbutyral (PVB)/functionalized mesoporous silica

Polyvinyl butyral/functionalized mesoporous silica hybrid composite films have been fabricated by solution casting technique with various weight percentages of functionalized silica. A polyol (tripentaerythritol-electron rich component), which acts as an electron donor to the polymer backbone, was added to enhance the conductivity. The prepared composites were characterized by Fourier transformed infrared spectroscopy and the morphology was evaluated by scanning electron microscopy. Dielectric properties of these freestanding composites were studied using the two-probe method. The dielectric constant and impedance value decreased with the increase in applied frequency as well as with the increase in functionalized silica content in the polyvinyl butyral matrix. An increase in conductivity of the PVB/functionalized silica composites was also observed. (C) 2013 Elsevier Ltd. All rights reserved.

Reactive diffusion in the Ti-Si system and the significance of the parabolic growth constant

Solid diffusion couple experiments are conducted to analyse the growth mechanism of the phases and the diffusion mechanism of the components in the Ti-Si system. The calculation of the parabolic growth constants and the integrated diffusion coefficients substantiates that the analysis is intrinsically prone to erroneous conclusions if it is based on just the parabolic growth constants determined for a multiphase interdiffusion zone. The location of the marker plane is detected based on the uniform grain morphology in the TiSi2 phase, which indicates that this phase grows mainly because of Si diffusion. The growth mechanism of the phases and morphological evolution in the interdiffusion zone are explained with the help of imaginary diffusion couples. The activation enthalpies for the integrated diffusion coefficient of TiSi2 and the Si tracer diffusion are calculated as 190 +/- 9 and 197 +/- 8 kJ/mol, respectively. The crystal structure, details on the nearest neighbours of the components, and their relative mobilities indicate that the vacancies are mainly present on the Si sublattice.

Nearly Constant Switching Frequency Hysteresis Current Controller with Fast Online Computation of Boundary for a 2-Level Induction Motor Drive

A nearly constant switching frequency current hysteresis controller for a 2-level inverter fed induction motor drive is proposed in this paper: The salient features of this controller are fast dynamics for the current, inherent protection against overloads and less switching frequency variation. The large variation of switching frequency as in the conventional hysteresis controller is avoided by defining a current-error boundary which is obtained from the current-error trajectory of the standard space vector PWM. The current-error boundary is computed at every sampling interval based on the induction machine parameters and from the estimated fundamental stator voltage. The stator currents are always monitored and when the current-error exceeds the boundary, voltage space vector is switched to reduce the current-error. The proposed boundary computation algorithm is applicable in linear and over-modulation region and it is simple to implement in any standard digital signal processor: Detailed experimental verification is done using a 7.5 kW induction motor and the results are given to show the performance of the drive at various operating conditions and validate the proposed advantages.

Hysteresis current controller for a general n-level inverter fed drive with online current error boundary computation and nearly constant switching frequency

A space vector-based hysteresis current controller for any general n-level three phase inverter fed induction motor drive is proposed in this study. It offers fast dynamics, inherent overload protection and low harmonic distortion for the phase voltages and currents. The controller performs online current error boundary calculations and a nearly constant switching frequency is obtained throughout the linear modulation range. The proposed scheme uses only the adjacent voltage vectors of the present sector, similar to space vector pulse-width modulation and exhibits fast dynamic behaviour under different transient conditions. The steps involved in the boundary calculation include the estimation of phase voltages from the current ripple, computation of switching time and voltage error vectors. Experimental results are given to show the performance of the drive at various speeds, effect of sudden change of the load, acceleration, speed reversal and validate the proposed advantages.

A Nearly Constant Switching Frequency Current Error Space Vector Based Hysteresis Controller for an IM Drive with 12-sided Polygonal Voltage Space Vectors

In this paper, a current error space vector (CESV) based hysteresis controller for a 12-sided polygonal voltage space vector inverter fed induction motor (IM) drive is proposed, for the first time. An open-end winding configuration is used for the induction motor. The proposed controller uses parabolic boundary with generalized vector selection logic for all sectors. The drive scheme is first studied with a space vector based PWM (SVPWM) control and from this the current error space phasor boundary is obtained. This current error space phasor boundary is approximated with four parabolas and then the system is run with space phasor based hysteresis PWM controller by limiting the CESV within the parabolic boundary. The proposed controller has increased modulation range, absence of 5th and 7th order harmonics for the entire modulation range, nearly constant switching frequency, fast dynamic response with smooth transition to the over modulation region and a simple controller implementation.

A Gibbs-ensemble based technique for Monte Carlo simulation of electric double layer capacitors (EDLC) at constant voltage

Current methods for molecular simulations of Electric Double Layer Capacitors (EDLC) have both the electrodes and the electrolyte region in a single simulation box. This necessitates simulation of the electrode-electrolyte region interface. Typical capacitors have macroscopic dimensions where the fraction of the molecules at the electrode-electrolyte region interface is very low. Hence, large systems sizes are needed to minimize the electrode-electrolyte region interfacial effects. To overcome these problems, a new technique based on the Gibbs Ensemble is proposed for simulation of an EDLC. In the proposed technique, each electrode is simulated in a separate simulation box. Application of periodic boundary conditions eliminates the interfacial effects. This in addition to the use of constant voltage ensemble allows for a more convenient comparison of simulation results with experimental measurements on typical EDLCs. (C) 2014 AIP Publishing LLC.

Flame surface statistics of constant-pressure turbulent expanding premixed flames

In this paper we investigate the local flame surface statistics of constant-pressure turbulent expanding flames. First the statistics of local length ratio is experimentally determined from high-speed planar Mie scattering images of spherically expanding flames, with the length ratio on the measurement plane, at predefined equiangular sectors, defined as the ratio of the actual flame length to the length of a circular-arc of radius equal to the average radius of the flame. Assuming isotropic distribution of such flame segments we then convolute suitable forms of the length-ratio probability distribution functions (pdfs) to arrive at the corresponding area-ratio pdfs. It is found that both the length ratio and area ratio pdfs are near log-normally distributed and shows self-similar behavior with increasing radius. Near log-normality and rather intermittent behavior of the flame-length ratio suggests similarity with dissipation rate quantities which stimulates multifractal analysis. (C) 2014 AIP Publishing LLC.

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.