Explicit Construction of Optimal Exact Regenerating Codes for Distributed Storage

Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, Wu et al. characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A sub-space based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular.

Quantum Error Correction via Codes over GF(2)

It is well known that n-length stabilizer quantum error correcting codes (QECCs) can be obtained via n-length classical error correction codes (CECCs) over GF(4), that are additive and self-orthogonal with respect to the trace Hermitian inner product. But, most of the CECCs have been studied with respect to the Euclidean inner product. In this paper, it is shown that n-length stabilizer QECCs can be constructed via 371 length linear CECCs over GF(2) that are self-orthogonal with respect to the Euclidean inner product. This facilitates usage of the widely studied self-orthogonal CECCs to construct stabilizer QECCs. Moreover, classical, binary, self-orthogonal cyclic codes have been used to obtain stabilizer QECCs with guaranteed quantum error correcting capability. This is facilitated by the fact that (i) self-orthogonal, binary cyclic codes are easily identified using transform approach and (ii) for such codes lower bounds on the minimum Hamming distance are known. Several explicit codes are constructed including two pure MDS QECCs.

Convolutional Codes for Network-Error Correction

In this work, we introduce convolutional codes for network-error correction in the context of coherent network coding. We give a construction of convolutional codes that correct a given set of error patterns, as long as consecutive errors are separated by a certain interval. We also give some bounds on the field size and the number of errors that can get corrected in a certain interval. Compared to previous network error correction schemes, using convolutional codes is seen to have advantages in field size and decoding technique. Some examples are discussed which illustrate the several possible situations that arise in this context.

On BCH codes over arbitrary integer rings

Bose-C-Hocquenghem (BCH) atdes with symbols from an arbitrary fhite integer ring are derived in terms of their generator polynomials. The derivation is based on the factohation of x to the power (n) - 1 over the unit ring of an appropriate extension of the fiite integer ring. lke eomtruetion is thus shown to be similar to that for BCH codes over fink flelda.

Regenerating Codes for Distributed Storage Networks

In a storage system where individual storage nodes are prone to failure, the redundant storage of data in a distributed manner across multiple nodes is a must to ensure reliability. Reed-Solomon codes possess the reconstruction property under which the stored data can be recovered by connecting to any k of the n nodes in the network across which data is dispersed. This property can be shown to lead to vastly improved network reliability over simple replication schemes. Also of interest in such storage systems is the minimization of the repair bandwidth, i.e., the amount of data needed to be downloaded from the network in order to repair a single failed node. Reed-Solomon codes perform poorly here as they require the entire data to be downloaded. Regenerating codes are a new class of codes which minimize the repair bandwidth while retaining the reconstruction property. This paper provides an overview of regenerating codes including a discussion on the explicit construction of optimum codes.

On Computation of Minimum Distance of Linear Block Codes Above 1/2 Rate Coding

The minimum distance of linear block codes is one of the important parameter that indicates the error performance of the code. When the code rate is less than 1/2, efficient algorithms are available for finding minimum distance using the concept of information sets. When the code rate is greater than 1/2, only one information set is available and efficiency suffers. In this paper, we investigate and propose a novel algorithm to find the minimum distance of linear block codes with the code rate greater than 1/2. We propose to reverse the roles of information set and parity set to get virtually another information set to improve the efficiency. This method is 67.7 times faster than the minimum distance algorithm implemented in MAGMA Computational Algebra System for a (80, 45) linear block code.

Explicit and Optimal Exact-Regenerating Codes for the Minimum-Bandwidth Point in Distributed Storage

In the distributed storage setting that we consider, data is stored across n nodes in the network such that the data can be recovered by connecting to any subset of k nodes. Additionally, one can repair a failed node by connecting to any d nodes while downloading beta units of data from each. Dimakis et al. show that the repair bandwidth d beta can be considerably reduced if each node stores slightly more than the minimum required and characterize the tradeoff between the amount of storage per node and the repair bandwidth. In the exact regeneration variation, unlike the functional regeneration, the replacement for a failed node is required to store data identical to that in the failed node. This greatly reduces the complexity of system maintenance. The main result of this paper is an explicit construction of codes for all values of the system parameters at one of the two most important and extreme points of the tradeoff - the Minimum Bandwidth Regenerating point, which performs optimal exact regeneration of any failed node. A second result is a non-existence proof showing that with one possible exception, no other point on the tradeoff can be achieved for exact regeneration.

A Flexible Class of Regenerating Codes for Distributed Storage

In the distributed storage setting introduced by Dimakis et al., B units of data are stored across n nodes in the network in such a way that the data can be recovered by connecting to any k nodes. Additionally one can repair a failed node by connecting to any d nodes while downloading at most beta units of data from each node. In this paper, we introduce a flexible framework in which the data can be recovered by connecting to any number of nodes as long as the total amount of data downloaded is at least B. Similarly, regeneration of a failed node is possible if the new node connects to the network using links whose individual capacity is bounded above by beta(max) and whose sum capacity equals or exceeds a predetermined parameter gamma. In this flexible setting, we obtain the cut-set lower bound on the repair bandwidth along with a constructive proof for the existence of codes meeting this bound for all values of the parameters. An explicit code construction is provided which is optimal in certain parameter regimes.

On Network-Error Correcting Convolutional Codes under the BSC Edge Error Model

Convolutional network-error correcting codes (CNECCs) are known to provide error correcting capability in acyclic instantaneous networks within the network coding paradigm under small field size conditions. In this work, we investigate the performance of CNECCs under the error model of the network where the edges are assumed to be statistically independent binary symmetric channels, each with the same probability of error pe(0 <= p(e) < 0.5). We obtain bounds on the performance of such CNECCs based on a modified generating function (the transfer function) of the CNECCs. For a given network, we derive a mathematical condition on how small p(e) should be so that only single edge network-errors need to be accounted for, thus reducing the complexity of evaluating the probability of error of any CNECC. Simulations indicate that convolutional codes are required to possess different properties to achieve good performance in low p(e) and high p(e) regimes. For the low p(e) regime, convolutional codes with good distance properties show good performance. For the high p(e) regime, convolutional codes that have a good slope ( the minimum normalized cycle weight) are seen to be good. We derive a lower bound on the slope of any rate b/c convolutional code with a certain degree.

Queued Cooperative Wireless Networks With Rateless Codes

Cooperative communication using rateless codes, in which the source transmits an infinite number of parity bits to the destination until the receipt of an acknowledgment, has recently attracted considerable interest. It provides a natural and efficient mechanism for accumulating mutual information from multiple transmitting relays. We develop an analysis of queued cooperative relay systems that combines the communication-theoretic transmission aspects of cooperative communication using rateless codes over Rayleigh fading channels with the queuing-theoretic aspects associated with buffering messages at the relays. Relay cooperation combined with queuing reduces the message transmission times and also helps distribute the traffic load in the network, which improves throughput significantly.

Code Generation Paradigms On Optical Orthogonal And Other Codes In a Fiber Optic CDMA Lan

Code Division Multiple Access (CDMA) techniques, by far, had been applied to LAN problems by many investigators, An analytical study of well known algorithms for generation of Orthogonal codes used in FO-CDMA systems like those for prime, quasi-Prime, Optical Orthogonal and Matrix codes has been presented, Algorithms for OOCs like Greedy/Modified Greedy/Accelerated Greedy algorithms are implemented. Many speed-up enhancements. for these algorithms are suggested. A novel Synthetic Algorithm based on Difference Sets (SADS) is also proposed. Investigations are made to vectorise/parallelise SADS to implement the source code on parallel machines. A new matrix for code families of OOCs with different seed code-words but having the same (n,w,lambda) set is formulated.

MIMO Precoding With X- and Y- Codes

We consider a slow fading multiple-input multiple-output (MIMO) system with channel state information at both the transmitter and receiver. A well-known precoding scheme is based upon the singular value decomposition (SVD) of the channel matrix, which transforms the MIMO channel into parallel subchannels. Despite having low maximum likelihood decoding (MLD) complexity, this SVD precoding scheme provides a diversity gain which is limited by the diversity gain of the weakest subchannel. We therefore propose X- and Y-Codes, which improve the diversity gain of the SVD precoding scheme but maintain the low MLD complexity, by jointly coding information across a pair of subchannels. In particular, subchannels with high diversity gain are paired with those having low diversity gain. A pair of subchannels is jointly encoded using a 2 2 real matrix, which is fixed a priori and does not change with each channel realization. For X-Codes, these rotation matrices are parameterized by a single angle, while for Y-Codes, these matrices are left triangular matrices. Moreover, we propose X-, Y-Precoders with the same structure as X-, Y-Codes, but with encoding matrices adapted to each channel realization. We observed that X-Codes/Precoders are good for well-conditioned channels, while Y-Codes/Precoders are good for ill-conditioned channels.

Wavelength/Time orthogonal codes with FEC offer dual advantage

To meet the growing demands of the high data rate applications, suitable asynchronous schemes such as Fiber-Optic Code Division Multiple Access (FO-CDMA) are required in the last mile. FO-CDMA scheme offers potential benefits and at the same time it faces many challenges. Wavelength/Time (W/T) 2-D codes for use in FO-CDMA, can be classified mainly into two types: 1) hybrid codes and 2) matrix codes, to reduce the 'time' like property, have been proposed. W/T single-pulse-per-row (SPR) are energy efficient codes as this family of codes have autocorrelation sidelobes of '0', which is unique to this family and the important feature of the W/T multiple-pulses-per-row (MPR) codes is that the aspect ratio can be varied by trade off between wavelength and temporal lengths. These W/T codes have improved cardinality and spectral efficiency over other W/T codes and at the same time have lowest crosscorrelation values. In this paper, we analyze the performances of the FO-CDMA networks using W/T SPR codes and W/T MPR codes, with and without forward error correction (FEC) coding and show that with FEC there is dual advantage of error correction and reduced spread sequence length.

Fast-Group-Decodable STBCs via Codes over GF(4)

In this paper we construct low decoding complexity STBCs by using the Pauli matrices as linear dispersion matrices. In this case the Hurwitz-Radon orthogonality condition is shown to be easily checked by transferring the problem to $\mathbb{F}_4$ domain. The problem of constructing low decoding complexity STBCs is shown to be equivalent to finding certain codes over $\mathbb{F}_4$. It is shown that almost all known low complexity STBCs can be obtained by this approach. New codes are given that have the least known decoding complexity in particular ranges of rate.

Precoding with X-codes to increase capacity with discrete input alphabets

We consider Gaussian multiple-input multiple-output (MIMO) channels with discrete input alphabets. We propose a non-diagonal precoder based on X-Codes in to increase the mutual information. The MIMO channel is transformed into a set of parallel subchannels using Singular Value Decomposition (SVD) and X-codes are then used to pair the subchannels. X-Codes are fully characterized by the pairings and the 2 × 2 real rotation matrices for each pair (parameterized with a single angle). This precoding structure enables to express the total mutual information as a sum of the mutual information of all the pairs. The problem of finding the optimal precoder with the above structure, which maximizes the total mutual information, is equivalent to i) optimizing the rotation angle and the power allocation within each pair and ii) finding the optimal pairing and power allocation among the pairs. It is shown that the mutual information achieved with the proposed pairing scheme is very close to that achieved with the optimal precoder by Cruz et al., and significantly better than mercury/waterfilling strategy by Lozano et al.. Our approach greatly simplifies both the precoder optimization and the detection complexity, making it suitable for practical applications.