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.

Relay Load Balancing in Queued Cooperative Wireless Networks with Rateless Codes

Relay selection combined with buffering of packets of relays can substantially increase the throughput of a cooperative network that uses rateless codes. However, buffering also increases the end-to-end delays due to the additional queuing delays at the relay nodes. In this paper we propose a novel method that exploits a unique property of rateless codes that enables a receiver to decode a packet from non-contiguous and unordered portions of the received signal. In it, each relay, depending on its queue length, ignores its received coded bits with a given probability. We show that this substantially reduces the end-to-end delays while retaining almost all of the throughput gain achieved by buffering. In effect, the method increases the odds that the packet is first decoded by a relay with a smaller queue. Thus, the queuing load is balanced across the relays and traded off with transmission times. We derive explicit necessary and sufficient conditions for the stability of this system when the various channels undergo fading. Despite encountering analytically intractable G/GI/1 queues in our system, we also gain insights about the method by analyzing a similar system with a simpler model for the relay-to-destination transmission times.

Network Error Correction for Unit-Delay, Memory-Free Networks Using Convolutional Codes

A single source network is said to be memory-free if all of the internal nodes (those except the source and the sinks) do not employ memory but merely send linear combinations of the symbols received at their incoming edges on their outgoing edges. In this work, we introduce network-error correction for single source, acyclic, unit-delay, memory-free networks with coherent network coding for multicast. A convolutional code is designed at the source based on the network code in order to correct network- errors that correspond to any of a given set of error patterns, as long as consecutive errors are separated by a certain interval which depends on the convolutional code selected. Bounds on this interval and the field size required for constructing the convolutional code with the required free distance are also obtained. We illustrate the performance of convolutional network error correcting codes (CNECCs) designed for the unit-delay networks using simulations of CNECCs on an example network under a probabilistic error model.

X-Codes: A Low Complexity Full-Rate High-Diversity Achieving Precoder for TDD MIMO Systems

We consider a time division duplex multiple-input multiple-output (nt × nr MIMO). Using channel state information (CSI) at the transmitter, singular value decomposition (SVD) of the channel matrix is performed. This transforms the MIMO channel into parallel subchannels, but has a low overall diversity order. Hence, we propose X-Codes which achieve a higher diversity order by pairing the subchannels, prior to SVD preceding. In particular, each pair of information symbols is encoded by a fixed 2 × 2 real rotation matrix. X-Codes can be decoded using nr very low complexity two-dimensional real sphere decoders. Error probability analysis for X-Codes enables us to choose the optimal pairing and the optimal rotation angle for each pair. Finally, we show that our new scheme outperforms other low complexity precoding schemes.

Implications of the half-duplex constraint on relay-aided cooperation using rateless codes

The half-duplex constraint, which mandates that a cooperative relay cannot transmit and receive simultaneously, considerably simplifies the demands made on the hardware and signal processing capabilities of a relay. However, the very inability of a relay to transmit and receive simultaneously leads to a potential under-utilization of time and bandwidth resources available to the system. We analyze the impact of the half-duplex constraint on the throughput of a cooperative relay system that uses rateless codes to harness spatial diversity and efficiently transmit information from a source to a destination. We derive closed-form expressions for the throughput of the system, and show that as the number of relays increases, the throughput approaches that of a system that uses more sophisticated full-duplex nodes. Thus, half-duplex nodes are well suited for cooperation using rateless codes despite the simplicity of both the cooperation protocol and the relays.