Correct Burst Research Articles

The Burst Error Correcting Capabilities of a Simple Array Code

We propose a simple decoder for a widely used array code, known as the EVENODD code, which is originally designed to correct phased burst errors, to make it useful for correcting nonphased errors. The proposed scheme is capable of correcting almost all bursts up to a certain length. We show that the failure rate is sufficiently small and approaches zero as the block length increases. The redundancy of the code is twice the maximal burst length, which is a lower bound for the redundancy of a true burst-error-correcting code. Both the encoder and the decoder have very low complexity, both in terms of number of operations and in terms of computer code size

Burst Erasure Correction Codes With Low Decoding Delay

We present a new class of systematic, time-invariant, convolutional encoders suitable for low delay burst erasure correction. Specifically, we show that the new encoders have the shortest possible decoding delay required to correct all bursts of a given length with a fixed redundancy. By comparing the new encoders to maximum distance separable (MDS) codes, we show that the latter generally require either more redundancy or more delay to correct bursts of a given length. In addition, we show that the new encoders can achieve better performance than MDS codes on a simple two-state Markov erasure channel. Thus, we demonstrate the advantages of using cross packet coding for delay-sensitive applications such as Voice over Internet Protocol, video-conferencing, etc., on bursty packet networks. Finally, we discuss suitable performance measures for encoders designed to correct both burst and random erasures and report the results of a computer search for such hybrid encoders.

Interleaving for combating bursts of errors

To ensure data fidelity, a number of random error correction codes (ECCs) have been developed. ECC is, however, not efficient in combating bursts of errors, i.e., a group of consecutive (in one-dimensional (1-D) case) or connected (in two- and three- dimensional (2-D and 3-D) case) erroneous code symbols owing to the bursty nature of errors. Interleaving is a process to rearrange code symbols so as to spread bursts of errors over multiple code-words that can be corrected by ECCs. By converting bursts of errors into random-like errors, interleaving thus becomes an effective means to combat error bursts. In this article, we first illustrate the philosophy of interleaving by introducing a 1-D block interleaving technique. Then multi-dimensional (M-D) bursts of errors and optimality of interleaving are defined. The fundamentals and algorithms of the state of the art of M-D interleaving - the t-interleaved array approach by Blaum, Bruck and Vardy and the successive packing approach by Shi and Zhang-are presented and analyzed. In essence, a t-interleaved array is constructed by closely tiling a building block, which is solely determined by the burst size t. Therefore, the algorithm needs to be implemented each time for a different burst size in order to maintain either the error burst correction capability or optimality. Since the size of error bursts is usually not known in advance, the application of the technique is somewhat limited. The successive packing algorithm, based on the concept of 2 /spl times/ 2 basis array, only needs to be implemented once for a given square 2-D array, and yet it remains optimal for a set of bursts of errors having different sizes. The performance comparison between different approaches is made. Future research on the successive packing approach is discussed. Finally, applications of 2-D/3-D successive packing interleaving in enhancing the robustness of image/video data hiding are presented as examples of practical utilization of interleaving.

Reed-solomon codes for correcting phased error bursts

A code structure is introduced that represents a Reed-Solomon (RS) code in two-dimensional format. Based on this structure, a novel approach to multiple error burst correction using RS codes is proposed. For a model of phased error bursts, where each burst can affect one of the columns in a two-dimensional transmitted word, it is shown that the bursts can be corrected using a known multisequence shift-register synthesis algorithm. It is further shown that the resulting codes posses nearly optimal burst correction capability, under certain probability of decoding failure. Finally, low-complexity systematic encoding and syndrome computation algorithms for these codes are discussed. The proposed scheme may also find use in decoding of different coding schemes based on RS codes, such as product or concatenated codes.

Parallel decoding for burst error control codes

AbstractIt is well known that sequential decoding methods for burst error control codes use Linear Feedback Shift Registers (LFSR). However, a parallel decoding method employing only combinational logic is preferable in certain applications, such as semiconductor memory systems and holographic memory systems, where a huge amount of data needs to be read or written at ultrahigh speed. This paper proposes a parallel decoding method for linear burst error control codes including the Fire code. In order to obtain the error pattern, this decoding method uses an inverse matrix of a nonsingular matrix which includes a sub‐matrix of the parity check matrix. Burst error correction is possible because correct burst error patterns are obtained only from frames that completely include the burst error pattern. Furthermore, this decoding method can also be applied to single byte error correcting codes with less hardware. The implementation of decoding circuits for burst and byte error control codes are illustrated in this paper. And their hardware complexity is also evaluated to show that the proposed method requires less hardware than the existing parallel decoding method. Finally, we will discuss about how the proposed decoding method can be extended to correct multiple burst or byte errors. © 2003 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 87(1): 38–48, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10094

Greedy loop transversal codes for correcting error bursts

The greedy loop transversal algorithm is used to construct linear codes in a binary channel for the correction of error bursts. The dimensions of the codes constructed are compared with the dimensions of the corresponding white noise greedy loop transversal codes, and with the dimensions of the few previously best-known codes for the burst-error patterns. The dimensions of the greedy loop transversal codes for burst-error correction match or exceed the dimensions of these previously known codes.

Two-dimensional interleaving schemes with repetitions: constructions and bounds

Two-dimensional interleaving schemes with repetitions are considered. These schemes are required for the correction of two-dimensional bursts (or clusters) of errors in applications such as optical recording and holographic storage. We assume that a cluster of errors may have an arbitrary shape, and is characterized solely by its area t. Thus, an interleaving scheme A(t,r) of strength t with r repetitions is an (infinite) array of integers defined by the property that every integer appears no more than r times in any connected component of area t. The problem is to minimize, for a given t and r, the interleaving degree deg A(t,r), which is the total number of distinct integers contained in the array. Optimal interleaving schemes for r=1 (no repetitions) have been devised in earlier work. Here, we consider interleaving schemes for r/spl ges/2. Such schemes reduce the overall redundancy, yet are considerably more difficult to construct and analyze. To this end, we generalize the concept of L/sub 1/-distance and introduce the notions of tristance, quadristance, and more generally r-dispersion. We focus on the special class of interleaving schemes, called lattice interleavers, that is akin to the class of linear codes in coding theory. We construct efficient lattice interleavers for r=2,3,4 and some higher values of r. For r=2,3 we show that these lattice interleavers are either optimal for all t or asymptotically optimal for t/spl rarr//spl infin/. We present the results of an extensive computer search that yields the optimal lattice interleavers for r=2,3,4,5,6 and t up to about 1000. Finally, we consider an alternative connectivity model for clusters, where two elements in an array are connected if they are adjacent horizontally, vertically, or diagonally. We establish relations between interleavers for this model and interleavers for the standard horizontal/vertical connectivity model, and show that these models become equivalent for t/spl rarr//spl infin/. We conclude with some conjectures and open problems.

Spatially correlated qubit errors and burst-correcting quantum codes

We explore the design of quantum error-correcting codes for cases where the decoherence events of qubits are correlated. In particular, we consider the case where only spatially contiguous qubits decohere, which is analogous to the case of burst errors in classical coding theory. We present several different efficient schemes for constructing families of such codes. For example, one can find one-dimensional quantum codes of length n=13 and 15 that correct burst errors of width b<3; as a comparison, a random-error correcting quantum code that corrects t=3 errors must have length n/spl ges/17. In general, we show that it is possible to build quantum burst-correcting codes that have near optimal dimension. For example, we show that for any constant b, there exist b-burst-correcting quantum codes with length n, and dimension k=n-log n-O(1); as a comparison, the Hamming bound for the case with t (constant) random errors yields k/spl les/n-tlog n+O(1).

Multiplexing digital television sources on ATM networks

This paper investigates the transmission of digital television signals on ATM networks. The key characteristic of an ATM network is its ability of statistically multiplexing variable bit rate sources in queueing systems called multiplexers. These queues are optimally located at the output port of the switching fabrics and emptied at constant bit rate by the destined transmission links. In that framework, the paper intends to analyze the multiplexing of digital television sources which are connected at a given ATM node and share the same transmission resources. Consequently, the study starts with analyzing the packet traffic generated when pooling television sources. The paper proceeds with modeling the queue behaviors and, eventually, with deriving the multiplexing performances. These latter parameters refer to three main topics namely, the cell losses and their configurations, the transmission jitter and the statistical multiplexing. Both theoretical approaches and simulations are employed to yield the final performances. As an important issue, substantial multiplexing gains is also put into evidence through theoretical and experimental arguments. Cell loss configurations are presented to suggest what policy to implement for cell loss burst correction. It is further shown how random transmission delay and jitter turn out to be minor problems when dealing with the transmission of television services.

Trellis-coded modulation on digital microwave radio systems-simulations for multipath fading channels

Extensive simulations have been made to examine the performance of digital microwave radio systems based on TCM (trellis-coded modulation). Two coded systems have been simulated: the coded 1024-QAM (quadrature amplitude modulation) system uses separate TCM schemes for the I and Q channels, while the coded 512-CR system treats I and Q channels together. Theoretically predicted lower bounds on error performance were approached by both systems for additive white Gaussian noise channels. With residual ISI (intersymbol interference) the coded system achieved larger coding gains than with additive white Gaussian noise alone, due to the ability of the Viterbi decoder to correct short burst errors. The coded 512-CR system performed better than the coded 1024-QAM system in all circumstances. This is attributed to the use of combined I and Q channel coding and the more complex convolutional encoder used in the coded 512-CR system. It is concluded that TMC used in conjunction with adaptive equalization can achieve significant improvements in error performance on digital microwave radio systems. >

Two-dimensional burst identification codes and their use in burst correction

A new class of codes, called burst identification codes, is defined and studied. These codes can be used to determine the patterns of burst errors. Two-dimensional burst correcting codes can be easily constructed from burst identification codes. The resulting class of codes is simple to implement and has lower redundancy than other comparable codes. The results are pertinent to the study of radiation effects on VLSI RAM chips, which can cause two-dimensional bursts of errors.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Microprocessor based codec for a new class of burst error correcting codes

In this paper, an entirely new class of burst error correcting codes are described, and a design of a codec (coder/decoder) based on a microprocessor is given. The codes described are applicable not only to the conventional 1-D telecommunications channels but also to the less conventional 2-D and 3-D channels in which the burst errors manifest themselves as patches and clusters respectively. The codes being cyclic, may be generated by the usual shift register arrangement and similarly for the decoder which consists of a programmable counter, some logic circuits and shift registers. It will be shown that great improvements can be achieved in the codec by the use of a microprocessor based system. Since the codec is dependent on the burst shape and the channel dimensions, an efficient way to cope with variations in these parameters is to design an adaptive codec which will cope with the various expected burst shapes in a given channel. For this purpose an experimental codec has been designed based on the 6809 microprocessor. In this design each component of the hardware design has been emulated by software routines. Each codec being stored in an individual ROM. This provides a highly versatile system in that for any given communication channel and burst shape, only the appropriate ROM need be selected to ensure effective decoding of erroneous data. The efficiency of the codes allows for correction of large bursts of errors for relatively small memory requirements. the experimental codec is to be used to investigate fully delays due to processing time in the decoder. The delays incurred set limitations on the bit rate capabilities of the system, which is a key factor in comparison of performance against the equivalent hardware design. The transition from a hardware orientated design to a software orientated design offers advantages not only in versatility, but also in reduced complexity and increased compatibility within a communications network.

Efficient burst-correcting cyclic codes

A class of single burst error-correcting cyclic codes is introduced. The binary version of these codes has block length n = 2m − 1, number of parity check digits n − k = 2m−1, and correct bursts of length b = 2m−2, where m is an integer. These codes achieve the upper bound b ≤(n − k)/2, and normally have more information digits than interleaved codes of the same burst-correcting power. The multilevel case is also treated in the letter.

An error-correcting system for mobile radio data transmission

An error-correcting system for mobile radio data transmission with improved reliability and simple implementation is presented here. The new rate one-half code absolutely corrects two errors within 12 consecutive bits, while the (15, 7, 2) Bose-Chaudhuri-Hocquenquem (BCH) code corrects two errors within 15 bits and Hagelbarger's code corrects two errors within 14 bits. Error propagation in the feedback majority logic decoder is discussed, and it is proved empirically that the new code does not propagate infinite errors. In order to correct burst errors, a 12-column interleaving is proposed for fading channels.

Error statistics and protocol construction using general column and diagonal matrix burst correcting codes

This paper deals with : The use of the General Column and Diagonal Matrix Burst Correcting Codes (GCDM3CC) for burst and random correction errors of noisy digital communication channels and synchronization of messages used in a computer communication network. Statistics for the GCDMBCC, The construction strategies for the protocol packets used for a computer communication network with the aid of the redundancy cyole.

Two-dimensional Fire codes

To improve the reliability of two-dimensional information, codes that can correct two-dimensional bursts (or spots) may be useful. In this paper a class of two-dimensional burst-correcting codes, called two-dimensional Fire codes, is proposed. The definition of these codes is a natural extension of that of the conventional Fire codes. The two-dimensional Fire code is a two-dimensional cyclic code designed for single two-dimensional burst correction. Several important properties such as the burst-correcting capability and the positions of the check symbols are presented. Also, encoding and decoding methods are presented. It is shown that the encoding and decoding are easily implemented by using two-dimensional feedback shift registers.

Random error and burst correction by iterated codes

We give a decoding algorithm for iterated codes that can correct up to the number of errors guaranteed by the product minimum distance, rather than about half that number when the iterated codes are decoded independently. This result is achieved by adapting Forney's generalized minimum distance decoding for use with iterated codes. We derive results on the simultaneous burst- and random-error-correction capability of iterated codes that improve considerably on known results.

On type-B1 burst-error-correcting convolutional codes

Two new classes of type-B1 burst-error-correcting convolutional codes are introduced. One of them requires a shorter length of guard space and a smaller number of shift register stages than optimum type-B2 codes used for type-B1 burst correction. Another class of codes improves the required number of shift register stages considerably when the correctable burst length is very large. In addition, these codes require a very short length of additional guard space to restore the decoder to correct operation after a decoding failure. Both classes of codes are derived in a straightforward manner and their implementations are also very simple. Thus, we can avoid type-B2 code procedures to correct type-B1 bursts. The codes derived here result in the more efficient and simply implemented type-B1 burst-correcting convolutional codes.

Cyclic product codes

A new class of cyclic codes, cyclic product codes, is characterized. These codes enjoy the implementation advantages of cyclic codes and, in addition, possess the important structural properties of product (iterated) codes. The main results are as follows: \begin{enumerate} \item Conditions are given which ensure that the product of two, and, hence, arbitrarily many, cyclic codes is itself a cyclic code. \item Cyclic product codes are shown to be capable of unambiguous correction of both bursts and random errors. \item The generator polynomial of the cyclic product code is derived and shown to be a simple function of the generator polynomials of the subcodes. \end{enumerate} These results have the following various applications: a) the codes effect a compromise between random and burst-error-correcting codes and, therefore, appear to be well-suited to error correction on channels in which both types of errors occur, b) many codes in a particular subclass of the class of cyclic product codes are efficient burst-error correctors and are more easily implemented than the equivalent codes formed by interleaving short codes, c) the previously mentioned results 1) and 3) can be applied to some cyclic codes to show that they are product codes as well; then known results on the minimum distance of product codes can be applied. This improves on the Bose-Chaudhuri-Hocquenghem lower bound in many cases.

Cyclic codes from irreducible polynomials for correction of multiple errors

A class of shift-register codes is studied for correction of multiple errors. The defining matrix will satisfy an irreducible polynomial equation with coefficients 0 or 1. The errors are classified in terms of error cycles and a simple procedure is suggested to determine if error cycles are distinguishable. To facilitate the analyses, another classification of errors is introduced leaving the number of erroneous digits (weight) invariant. A number of specific codes have been investigated either for correction of error bursts or correction of all multiple errors of certain weights. Burst lengths 2, 3, 4, 5 and 6 are tried, mainly with as long codes as are theoretically possible. Codes are found for burst lengths 2, 3 and 4. The degree of the polynomial varies from 6 to 18 giving a wide range of code lengths. With single and double adjacent errors, a sufficient condition is given for the existence of a code of maximal theoretical length. Correction of all single and double errors in a code of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^p + 1</tex> is studied in some detail. As a special case of triple error correction, the Golay code of length 23 also is analyzed. Finally these and other known examples of shift-register codes are compared with the result of an exhaustive search for suitable polynomials of degrees 8, 9, 10 and 11.