Single Burst Error Research Articles

New Design of Error Control Codes Resilient to Single Burst Error or Two Random Bit Errors Using Constacyclic Codes

In this paper, we introduce a new design method of burst error control codes (BECCs), which can correct single burst error or two random bit errors by using the maximum likelihood syndrome decoder (MLSD), where the proposed BECCs are designed by a modification of the well-known Fire codes using constacyclic codes. Also, for the existing low-latency burst error-correcting decoder, it is shown that the proposed BECCs have the single burst error correction capability together with two random bit error detection capability with no additional parity bits and comparable complexity with the existing BECCs. For this, the complexity and latency is numerically analyzed by register-transfer level (RTL) synthesis.

Power efficient error correction coding for on‐chip interconnection links

A configurable self-calibrated power efficient five-bit error correction code is proposed to correct both single bit random and burst errors up to five bits; providing 100% error correction probability with crosstalk avoidance. It can also correct higher-order error up to 9 bits with an error correction probability tolerance of 73% for on-chip interconnection links. Single error correction and double error detection with extended Hamming code (22,16) is utilised along with standard triplication error correction methods in the proposed code. Self-calibration algorithm and data stream rerouting block are integrated into the error correction code to achieve power efficiency. Reliability, link power consumption, and link swing voltage are estimated using an analytical model used in a network-on-chip. Area, power, and delay of the codec are obtained using Synopsys tools utilising UMC 90 nm technology. The proposed method provides 32–73% power saving and 22.3–60.6% delay reduction with negligible area overhead compared with the state-of-the-art works. Estimated results prove that it provides a 40.5–50% reduction in link swing voltage and link power consumption compared with the state-of-the-art works. The proposed code is more appropriate for on-chip interconnect links where it provides high reliability and low swing voltage with high error correction capability compared with existing codes.

A New Class of Single Burst Error Correcting Codes with Parallel Decoding

With technology scaling, burst errors or clustered errors are becoming increasingly common in different types of memories. Multiple bit upsets due to particle strikes, write disturbance errors, and magnetic field coupling are a few of the mechanisms which cause clustered errors. In this article, a new class of single burst error correcting codes are presented which correct a single burst of any size b within a codeword. A code construction methodology is presented which enables us to construct the proposed scheme from existing codes, e.g., Hamming codes. A new single step decoding methodology for the proposed class of codes is also presented which enables faster decoding. Different code constructions using Hamming codes, and BCH codes have been presented in this paper and a comparison is made with existing schemes in terms of decoding complexity and data redundancy. The proposed scheme in all cases reduces the decoder complexity for little to no increase in data redundancy, specifically for higher burst error sizes.

On the List Decodability of Burst Errors

Burst errors are a type of distortion in many data communications and data storage channels. In this paper, we consider the list decodability of codes for single burst error case and phased-burst error case independently. Firstly, we analyze the list decodability of random codes, and we show that the burst list decoding radius and the rate of random codes achieve the Singleton bound and the Gilbert-Varshamov bound for single case and phased case, respectively. Second, we illustrate that cyclic codes and algebraic geometry codes are good burst list-decodable codes for single case and phased case, respectively.

Novel Burst Error Correction Algorithms for Reed-Solomon Codes

In this paper, we present three novel burst error correcting algorithms for an (n, k) Reed-Solomon code. The algorithmic complexities are of the same order as error-and-erasure decoding, O(rn), where r=n-k. In particular, their hardware implementation shares elements of Blahut error-and-erasure decoding. In contrast, all existing single-burst error correcting algorithms, which are equivalent to the proposed first algorithm, have complexity O(r2n). The first algorithm corrects the shortest single-burst with length f up to r-1. The algorithm follows the key characterization that the ending locations of all candidate bursts can be purely determined by the roots of a polynomial which is a linear function of syndromes, and moreover, the shortest burst is associated with the longest sequence of consecutive roots. The algorithmic miscorrection probability is bounded by q-(r-1-f), where q denotes the field size. The second algorithm extends the first one to correct the shortest burst with length f ≤ r-3 and additionally a random symbol error. The algorithmic miscorrection probability is bounded by q-(r-3-f). The third algorithm probabilistically corrects the shortest burst with length f ≤ r-1-2δ and additionally δ (a small constant) random symbol errors. The algorithmic miscorrection and failure probabilities are both bounded by q-(r-1-2δ-f). Our simulation results for (60, 40) and (30, 16) shortened Reed-Solomon codes verify that the miscorrection probability for three algorithms and the failure probability for the third algorithm all decay exponentially (at the rate of q-1) with respect to the length of burst.

Burst Error Recovery for Huffman Coding

Although data compression is popularly used, compressed data have a problem that they are very sensitive to errors. This paper proposes a single burst error recovery method for Huffman coding by using the bidirectionally decodable Huffman coding. Computer simulation shows that the proposed method can recover 2.5lburst bits burst error with high probability, where lburst is the maximum length of burst errors which the proposed method is expected to be able to recover.

A statistical decoding of simultaneous burst and random errors beyond the BCH bound using reed‐solomon codes

AbstractThis paper proposes a stochastic decoding method that can correct the composite errors beyond the BCH bound. It does this by tolerating that some of the composite errors cannot be corrected. Considering the case where the t‐error correcting Reed‐Solomon code with designed distance d = 2t+1 on GF(q) and code length n(⩽q‐1), it is shown that the proposed decoding method can correct the simultaneous composite errors composed of a single burst error of length v(<2t‐1) or less and t′(⩽t‐(1/2)v) or less random errors, with the probability not less than 1‐(n/(q‐1)2t‐v‐2t′).(1/t′). In other words, by setting v and t′ in the range tvt′<2t, the proposed method can correct the simultaneous composite errors beyond the BCH bound with the above probability. Then, a stochastic decoding system and a decoder circuit configuration are proposed based on the remainder decoding, and it is shown that a high‐speed stochastic decoder can be realized by a relatively small‐scale circuit when t′ is small (0⩽t′⩽2∼3).

The single burst error detection performance of binary cyclic codes

Using a new statistical model for burst errors, the authors calculate the conditional probability of undetected error for CRC codes with generator polynomials of the form g(x)=(1+x)p(x), p(x) a primitive polynomial. They show that the choice of p(x) affects the performance of these codes. >

Phased burst error-correcting array codes

Various aspects of single-phased burst-error-correcting array codes are explored. These codes are composed of two-dimensional arrays with row and column parities with a diagonally cyclic readout order; they are capable of correcting a single burst error along one diagonal. Optimal codeword sizes are found to have dimensions n/sub 1/*n/sub 2/ such that n/sub 2/ is the smallest prime number larger than n/sub 1/. These codes are capable of reaching the Singleton bound. A new type of error, approximate errors, is defined; in q-ary applications, these errors cause data to be slightly corrupted and therefore still close to the true data level. Phased burst array codes can be tailored to correct these codes with even higher rates than before. >

New decoding methods of interleaved burst error‐correcting codes

AbstractThis paper proposes new methods of burst error correction for block interleaved error‐correcting codes. First, a probabilistic method of a single burst error correction, using syndrome correlation of subcodes which constitute the interleaved code, is presented. With this method, the high capability of correcting burst errors potentially owned by the interleaved codes is realized with less decoding delay. Second, by generalizing the above method, a probabilistic method of multiple (m‐fold) burst error correction is presented. With this method, after estimating the burst error positions using syndrome correlation of subcodes which are interleaved m‐fold burst error detecting codes, erasure errors are corrected in each subcode and m‐fold burst errors are corrected successfully. the code constructed by interleaving m‐fold burst error‐detecting codes is shown to be superior to the Reed‐Solomon code for 2‐fold burst error correction with respect to both error correction capability and decoding delay. the performances of these methods (single burst error correction method and multiple burst error correction method) are analyzed by computer simulation and their effectiveness is clarified.

A class of majority decodable multiple solid burst-error correcting cyclic codes†

Abstract In this paper, a class of majority decodable solid burst-error correcting cyclic codes will be presented. The advantage of the cyclic codes presented is that the complexity of the decoding algorithm is independent of the multiplicity of the solid burst-error patterns. For the correction of single and double solid burst-error patterns, special parity check codes with modified decoding algorithm will be presented so that the number of check digits will be further reduced.

Cellular arrays for the parallel implementation of binary error-correcting codes

A cellular array is a logical network of identical or almost identical cells, each of which contains a small amount of logic and storage, and, except for a few buses to the edge of the array, is connected only to its immediate neighbors. The cellular approach offers special advantages for realization by the forthcoming large-scale-integrated (LSI) technology. Such arrays are shown to be applicable for the encoding and decoding of binary error-correcting codes, and also for identifying the possibilities of tradeoffs between decoding time and equipment complexity. Arrays are presented for the decoding of single errors, burst errors, and erasures; the decoding of erasures is accomplished by the equation-solution approach, and it is shown for several code families that the Gauss elimination procedure is not required.

A new class of recurrent codes

Recurrent codes for burst-error correction are defined to be a type B1 code. A type B1 code is capable of correcting all the single burst errors of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> or less. A type B2 code is a subclass of a type B1 code with the restriction that the single burst error of length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> or less occurs within <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</tex> consecutive blocks (it is assumed that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> is divisible by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l = rb</tex> ). In this paper, the author will present a class of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> -nary recurrent codes--both type B1 and B2 codes--for burst-error correction. The construction procedures for this class of codes are simple and systematic. An interesting relation exists between a type B1 and a type B2 code. If a code is type B2 (with block length <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</tex> , burst-error-correction capability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l = rb</tex> ), then by reducing the block length by one symbol, the type B2 code becomes a type B1 code, while the burst-error-correction capability remains the same. Both types of codes can be used for the correction of binary burst errors. The correction of binary burst errors will be briefly discussed in this paper.

Synchronization of cyclic codes in the presence of burst errors

This paper deals with the ability of coset codes derivable from ( n, k ) cyclic codes to detect and correct loss of synchronization, or slip, in a burst-error channel. Results are presented both for general cyclic codes and burst error correcting cyclic codes. In particular, the Fire codes, which correct single burst errors, are considered. Two classes of burst-error channels are considered, namely, Type I and Type II. In Type I channels, slip and additive burst errors cannot occur simultaneously in any received n -tuple. In the more general Type II channels, this restriction is removed. It is shown that certain coset codes can correct slip and burst errors even when they occur simultaneously. Furthermore, both the magnitude and the direction of the slip can be determined by examining the syndrome of the received n -tuple.