Burst Erasure Research Articles

An Explicit Rate-Optimal Streaming Code for Channels With Burst and Arbitrary Erasures

This paper considers the transmission of an infinite sequence of messages (a streaming source) over a packet erasure channel, where every source message must be recovered perfectly at the destination subject to a fixed decoding delay. While the capacity of a channel that introduces only bursts of erasures has been known for more than fifteen years, only recently, the capacity of a channel with either one burst of erasures or multiple arbitrary erasures in any fixed-sized sliding window has been established. However, explicit codes shown to achieve this capacity for any admissible set of parameters require a large field size that scales exponentially with the delay. Recently, it has been shown that there exist codes that achieve the capacity with field size which scales quadratically with the delay. However, explicit constructions were shown only to specific combinations of parameters. This work describes an explicit rate-optimal construction for all admissible channel and delay parameters over a field size that scales quadratically with the delay.

Efficient Parallel Decoding Architecture for Cluster Erasure Correcting 2-D LDPC Codes for 2-D Data Storage

Native 2-D low-density parity-check (LDPC) codes provide 2-D burst erasure correction capability and have promising applications in two-dimensional magnetic recording (TDMR) technology. Though carefully constructed rastered 1-D LDPC codes can provide 2-D burst erasure correction, they are not as efficient as 2-D native codes constructed for handling the 2-D span of burst erasures. Our contributions are twofold. First, we propose a new 2-D LDPC code with girth greater than 4 by generating a parity-check tensor through stacking permutation tensors of size $p\times p\times p$ along with the $i,j,k$ -axes. The permutations are achieved through circular shifts on an identity tensor along different coordinate axes in such a way that it provides a burst erasure correction capability of at least $p\times p$ . Second, we propose a fast, efficient, and scalable hardware architecture for a parallel 2-D LDPC decoder based on the proposed code construction for data storage applications. Through efficient indexing of the received messages in an RAM, we propose novel routing mechanisms for messages between the check nodes and variable nodes through a set of two barrel shifters, producing shifts along two axes. Through simulations, we show that the performance of the proposed 2-D LDPC codes matches a 1-D QC-LDPC code, with a sharp waterfall drop of three to four orders of magnitude over ~0.3 dB, for random errors over code sizes of ~32 kbits or equivalently ~ $180\times 180 2$ -D arrays. Furthermore, we prove that the proposed native 2-D LDPC codes outperform their 1-D counterparts in terms of 2-D cluster erasure correction ability. For $p=16$ and code arrays of size $48\times 48$ , we implemented the proposed design architecture on a Kintex-7 KC-705 field-programmable gate array (FPGA) kit, achieving a significantly high worst case throughput of 12.52 Gb/s at a clock frequency of 163 MHz.

Quasi-Cyclic LDPC Codes With Parity-Check Matrices of Column Weight Two or More for Correcting Phased Bursts of Erasures

In his pioneering work on LDPC codes, Gallager dismissed codes with parity-check matrices of weight two after proving that their minimum Hamming distances grow at most logarithmically with their code lengths. In spite of their poor minimum Hamming distances, it is shown that quasi-cyclic LDPC codes with parity-check matrices of column weight two have good capability to correct phased bursts of erasures which may not be surpassed by using quasi-cyclic LDPC codes with parity-check matrices of column weight three or more. By modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. Quasi-cyclic LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel are also considered. Examples of such codes based on Reed-Solomon and Gabidulin codes are presented.

Complete j-MDP Convolutional Codes

Maximum distance profile (MDP) convolutional codes have been proven to be very suitable for transmission over an erasure channel. In addition, the subclass of complete MDP convolutional codes has the ability to restart decoding after a burst of erasures. However, there is a lack of constructions of these codes over fields of small size. In this article, we introduce the notion of complete 3-MDP convolutional codes, which are a generalization of complete MDP convolutional codes, and describe their decoding properties. In particular, we present a decoding algorithm for decoding erasures within a given time delay T and show that complete T-MDP convolutional codes are optimal for this algorithm. Moreover, using a computer search with the MAPLE software, we determine the minimal binary and non-binary field size for the existence of (2, 1, 2) complete 3-MDP convolutional codes and provide corresponding constructions. We give a description of all (2, 1, 2) complete MDP convolutional codes over the smallest possible fields, namely F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</sub> and F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">16</sub> and we also give constructions for (2, 1, 3) complete 4-MDP convolutional codes over F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">128</sub> obtained by a randomized computer search.

Rate-Optimal Streaming Codes for Channels With Burst and Random Erasures

In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$ . One needs to recover any erased packet with a strict decoding delay deadline of $\tau $ , where erasures are as per the channel model. Presently existing rate-optimal constructions in the literature require, in general, a field-size which grows exponential in $\tau $ , as long as $\frac {a}{\tau }$ remains a constant. In this work, we present a new rate-optimal code construction covering all channel and delay parameters, which requires an $O(\tau ^{2})$ field-size. As a special case, when $(b-a)=1$ , we have a field-size linear in $\tau $ . We also present two other constructions having linear field-size, under certain constraints on channel and decoding delay parameters. As a corollary, we obtain low field-size, rate-optimal convolutional codes for any given column distance and column span. Simulations indicate that the newly proposed streaming code constructions offer lower packet-loss probabilities compared to existing schemes, for selected instances of Gilbert-Elliott and Fritchman channels.

Optimal Multiplexed Erasure Codes for Streaming Messages With Different Decoding Delays

This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , and every source message in the second sequence subject to a shorter decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> ,. We assume that the channel loss model introduces a burst erasure of a fixed length B on the discrete timeline. Under this channel loss assumption, the capacity region for the case where T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B was previously solved. In this paper, we fully characterize the capacity region for the remaining case T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ) T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B. The key step in the achievability proof is achieving the non-trivial corner point of the capacity region through using a multiplexed streaming code constructed by superimposing two single-stream codes. The main idea in the converse proof is obtaining a genie-aided bound when the channel is subject to a periodic erasure pattern where each period consists of a length-B burst erasure followed by a length-T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> noiseless duration.

Recent Results on the Implementation of a Burst Error and Burst Erasure Channel Emulator Using an FPGA Architecture

The behaviour of a transmission channel may be simulated using the performance abilities of current generation multiprocessing hardware, namely, a multicore Central Processing Unit (CPU), a general purpose Graphics Processing Unit (GPU), or a Field Programmable Gate Array (FPGA). These were investigated by Cullinan et al. in a recent paper (published in 2012) where these three devices capabilities were compared to determine which device would be best suited towards which specific task. In particular, it was shown that, for the application which is objective of our work (i.e., for a transmission channel simulation), the FPGA is 26.67 times faster than the GPU and 10.76 times faster than the CPU. Motivated by these results, in this paper we propose and present a direct hardware emulation. In particular, a Cyclone II FPGA architecture is implemented to simulate a burst error channel behaviour, in which errors are clustered together, and a burst erasure channel behaviour, in which the erasures are clustered together. The results presented in the paper are valid for any FPGA architecture that may be considered for this scope.

Block Toeplitz matrices for burst-correcting convolutional codes

In this paper we study a problem in the area of coding theory. In particular, we focus on a class of error-correcting codes called convolutional codes. We characterize convolutional codes that can correct bursts of erasures with the lowest possible delay. This characterization is given in terms of a block Toeplitz matrix with entries in a finite field that is built upon a given generator matrix of the convolutional code. This result allows us to provide a concrete construction of a generator matrix of a convolutional code with entries being only zeros or ones that can recover bursts of erasures with low delay. This construction admits a very simple decoding algorithm and, therefore, simplifies the existing schemes proposed recently in the literature.

Optimal Streaming Codes for Channels With Burst and Arbitrary Erasures

This paper considers transmitting a sequence of messages (streaming messages) over a packet erasure channel. In each time slot, the source constructs a packet based on the current and the previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. Every source message must be recovered perfectly at the destination subject to a fixed decoding delay. We assume that the channel loss model introduces either one burst erasure or multiple arbitrary erasures in any fixed-sized sliding window. Under this channel loss assumption, we fully characterize the maximum achievable rate by constructing streaming codes that achieve the optimal rate. In addition, our construction of optimal streaming codes implies the full characterization of the maximum achievable rate for convolutional codes with any given column distance, column span, and decoding delay. Numerical results demonstrate that the optimal streaming codes outperform existing streaming codes of comparable complexity over some instances of the Gilbert–Elliott channel and the Fritchman channel.

Protograph-Based Folded Spatially Coupled LDPC Codes for Burst Erasure Channels

In this letter, protograph-based folded spatially coupled (FSC) LDPC codes are proposed. The new folded-type structure is obtained by folding the spatial coupling chain of a conventional spatially coupled (SC) LDPC protograph and interlacing the nodes at staggered spatial positions. The proposed codes outperform the SC LDPC codes over single and multiple random-burst erasure channels. We extend the construction of the folded-type structure by connecting multiple one-sided SC LDPC chains for higher resilience to burst erasure channels. The FSC LDPC codes are also compatible with windowed decoder and outperform conventional SC LDPC codes.

Analysis of Coding Strategies Within File Delivery Protocol Framework for HbbTV Based Push-VoD Services Over DVB Networks

Hybrid broadcast broadband TV is a technique providing Push-VoD services over an interactive hybrid TV. These services are broadcast using a file delivery protocol (FDP), which includes different coding strategies to ensure reliable delivery. This protocol is characterized by three levels of data representation giving rise to segmentation of packet losses, which may result in poor recovery capabilities. This paper provides a first thorough investigation of the coding FDP framework for reliable delivery of Push-VoD service over DVB networks. We propose Markov modeling for characterizing inter-layer loss propagation within FDP on a wide variety of burst erasure channels. Based on this analytical analysis and a simulation study, we determine the possible recovering areas and the accurate loss measurements within FDP. The latter is then used to effectively investigate and configure the different coding strategies provided within FDP. In addition, we present a suitable recovering strategy for FDP, which guarantees transmission robustness against the broadcast network impairments.

Streaming Codes for Multiplicative-Matrix Channels With Burst Rank Loss

The burst rank loss network is an extension of the burst erasure channel, where the channel matrix between the sender and receiver becomes rank-deficient for a certain period of consecutive time-slots. We study streaming communication over the burst rank loss channel, propose a new class of codes, ROBIN codes, and establish their optimality. Our construction uses the maximum rank distance and maximum sum rank codes from previous works as constituent codes and combines them in a layered fashion. Our results generalize previous works on both the single-link and multiple-parallel-link streaming setups over burst erasure channels. We perform simulations over statistical network models to show that ROBIN codes attain low packet loss rates in comparison with the existing codes.

Multiplexed Coding for Multiple Streams With Different Decoding Delays

We consider a communication setup where two source streams with different decoding deadlines, must be simultaneously transmitted over a single channel subjected to burst erasures. The encoder multiplexes the two source streams into a single stream of channel-packets. The decoder must recover the two source streams sequentially by their corresponding deadlines. One of the streams, the urgent stream, has a smaller delay than the other stream. We study the capacity region for such a setting for a certain range of system parameters. We divide the system into three different cases based on the relative values of the delays. For each case we provide achievability and converse bounds, which match under certain conditions. Our proposed coding scheme involves a careful construction of the parity check packets by jointly coding across the two streams despite different deadlines. Interestingly it is possible to transmit the urgent stream at a certain positive rate even when the sum rate equals the capacity associated with the less urgent stream. A separation based approach where we apply separate single-stream codes to each stream is suboptimal. Although our capacity results assume a simplistic channel model with a single erasure burst, we further demonstrate that our proposed code constructions also provide significant performance gains in simulations over statistical channel models with random bursts.

Finite-Length Analysis of Spatially-Coupled Regular LDPC Ensembles on Burst-Erasure Channels

Regular spatially-coupled low-density parity-check ensembles have gained significant interest, since they were shown to universally achieve the capacity of binary memoryless channels under low-complexity belief-propagation decoding. In this paper, we focus primarily on the performance of these ensembles over binary channels affected by bursts of erasures. We first develop an analysis of the finite length performance for a single burst per code word and no errors otherwise. We first assume that the burst erases a complete spatial position, modeling for instance node failures in distributed storage. We provide new tight lower bounds for the block erasure probability (PB) at finite block length and bounds on the coupling parameter for being asymptotically able to recover the burst. We further show that expurgating the ensemble can improve the block erasure probability by several orders of magnitude. Later we extend our methodology to more general channel models. In a first extension, we consider bursts that can start at a random location in the code word and span across multiple spatial positions. Besides the finite length analysis, we determine by means of density evolution the maximum correctable burst length. In a second extension, we consider the case where in addition to a single burst, random bit erasures may occur. Finally, we consider a block erasure channel model which erases each spatial position independently with some probability p, potentially introducing multiple bursts simultaneously. All results are verified using Monte-Carlo simulations.

A Truncated Prediction Framework for Streaming Over Erasure Channels

We propose a new coding technique for sequential transmission of a stream of Gauss–Markov sources over erasure channels under a zero decoding delay constraint. Our proposed scheme is a combination (hybrid) of predictive coding with truncated memory, and quantization-and-binning. We study the optimality of our proposed scheme using an information theoretic model. In our setup, the encoder observes a stream of source vectors that are spatially independent and identically distributed (i.i.d.) and temporally sampled from a first-order stationary Gauss–Markov process. The channel introduces an erasure burst of a certain maximum length $B$ , starting at an arbitrary time, not known to the transmitter. The reconstruction of each source vector at the destination must be with zero delay and satisfy a quadratic distortion constraint with an average distortion of $D$ . The decoder is not required to reconstruct those source vectors that belong to the period spanning the erasure burst and a recovery window of length $W$ following it. We study the minimum compression rate $R(B,W,D)$ in this setup. As our main result, we establish upper and lower bounds on the compression rate. The upper bound (achievability) is based on our hybrid scheme. It achieves significant gains over baseline schemes such as (leaky) predictive coding, memoryless binning, a separation-based scheme, and a group of pictures-based scheme. The lower bound is established by observing connection to a network source coding problem. The bounds simplify in the high resolution regime, where we provide explicit expressions whenever possible, and identify conditions when the proposed scheme is close to optimal. We finally discuss the interplay between the parameters of our burst erasure channel and the statistical channel models and explain how the bounds in the former model can be used to derive insights into the simulation results involving the latter. In particular, our proposed scheme outperforms the baseline schemes over the i.i.d. erasure channel and the Gilbert–Elliott channel, and achieves performance close to a lower bound in some regimes.

Burst-Erasure Correcting Codes with Optimal Average Delay

The objective of low-delay codes is to protect communication streams from erasure bursts by minimizing the time between the packet erasure and its reconstruction. Previous work has concentrated on the constant-delay scenario, where all erased packets need to exhibit the same decoding delay. We consider the case of heterogeneous delay, where the objective is to minimize the average delay across the erased packets in a burst. We derive delay lower bounds for the average case, and show that they match the constant-delay bounds only at a single rate point $R=0.5$ . We then construct codes with optimal average delays for the entire range of code rates. The construction for rates $R\leq 0.5$ achieves optimality for every erasure instance, while the construction for rates $R>0.5$ is optimal for a $(1-R)/R$ fraction of all burst instances and close to optimal for the remaining fraction. The paper also studies the benefits of delay heterogeneity within the application of sensor communications. It is shown that a carefully designed code can significantly improve the temporal precision at the receiving node following erasure-burst events.

Band Splitting Permutations for Spatially Coupled LDPC Codes Achieving Asymptotically Optimal Burst Erasure Immunity

Band Splitting Permutations for Spatially Coupled LDPC Codes Achieving Asymptotically Optimal Burst Erasure Immunity

Defect Detection and Burst Erasure Correction for TDMR

Media defects are common in magnetic recording, resulting in the bursts of erasures. The use of error correcting codes can be aided by a defect detector for indicating the location of erasures. In this paper, we propose a model to study 2-D media defects that can occur in 2-D magnetic recording (TDMR). We also propose a novel defect identification and detection algorithm to identify the location of defects by the magnitude thresholding of the readback samples and identifying defective data patterns at the output of a 2-D signal detector. The defect detector identifies a 2-D edge-connected region of defects for burst erasure correction. Using simulations, we show that the defect detector can be designed to identify the location of 2-D defects with >90% accuracy for deep defects. The effect of the defect detector in TDMR is studied over a Voronoi-based media model where low density parity check (LDPC) codes are used for error correction. The defect detector is observed to achieve >2 dB gain in signal-to-noise ratio (SNR) for a $38\times 38$ burst erasure. We also study simple interleaving schemes for LDPC codes that give a further gain of >0.3 dB in SNR and can correct $76\times 76$ and larger burst erasures.

Corrección de borrado en ráfaga utilizando códigos LDPC construidos sobre matrices generadas por grupos combinados, polígonos anidados y matrices circulantes superpuestas

&lt;p&gt;En este artículo son propuestos procedimientos para la construcción de matrices base embazado en el álgebra moderna y en la geometría. Estas matrices sirven de plataforma para generar las matrices de verificación de paridad en la corrección de borrado en ráfaga a través de códigos LDPC, por medio de superposición en las matrices base y movimientos de las matrices circulantes. La construcción de las matrices es realizada por concatenación, siendo de fácil implementación y de menor aleatoriedad. Para demonstrar el potencial de la técnica, fue elaborado un conjunto de simulaciones que utiliza codificación de baja complejidad, bien como algoritmo soma y producto. Fueron generados varios códigos LDPC (matrices) y los resultados obtenidos comparados con otros abordajes. Son también presentados los resultados de la simulación de la recuperación de borrados resultantes de la transmisión de una imagen a través de un canal ruidoso.&lt;/p&gt;

On the Burst Erasure Efficiency of Array LDPC Codes

In this paper a family of LDPC codes based on the shortened and superposed array LDPC codes is proposed for burst erasure channels. The design of constructions is based on identifying and analyzing...