Hsin-Chiu Chang

Algebraic decoding of the (41, 21, 9) Quadratic Residue code

In this paper, an algebraic decoding algorithm is proposed to correct all patterns of four or fewer errors in the binary (41, 21, 9) Quadratic Residue (QR) code. The technique needed here to decode the (41, 21, 9) QR code is different from the algorithms developed in [I.S. Reed, T.K. Truong, X. Chen, X. Yin, The algebraic decoding of the (41, 21, 9) Quadratic Residue code, IEEE Transactions on Information Theory 38 (1992 ) 974-986]. This proposed algorithm does not require to solve certain quadratic, cubic, and quartic equations and does not need to use any memory to store the five large tables of the fundamental parameters in GF(2^2^0) to decode this QR code. By the modification of the technique developed in [R. He, I.S. Reed, T.K. Truong, X. Chen, Decoding the (47, 24, 11) Quadratic Residue code, IEEE Transactions on Information Theory 47 (2001) 1181-1186], one can express the unknown syndromes as functions of the known syndromes. With the appearance of known syndromes, one can solve Newtons identities to obtain the coefficients of the error-locator polynomials. Besides, the conditions for different number of errors of the received words will be derived. Computer simulations show that the proposed decoding algorithm requires about 22% less execution time than the syndrome decoding algorithm. Therefore, this proposed decoding scheme developed here is more efficient to implement and can shorten the decoding time.

Decoding of the (31, 16, 7) quadratic residue code

Abstract An algebraic decoding algorithm is proposed to correct all error patterns of up to three errors in the binary (31, 16, 7) Quadratic Residue (QR) code with reducible generator polynomial. The decoding technique, a modification of the decoding algorithm given by Reed et al., is based on the application of the decoding algorithm proposed by Truong et al. The computation of all syndromes is done in a small field, namely, GF(25). Thus, the computational complexity can be reduced. A full simulation shows that this novel decoding method is superior to the algebraic decoding algorithm given by Reed et al.

High speed decoding of the binary (47,24,11) quadratic residue code

An efficient table lookup decoding algorithm (TLDA) is presented to decode up to five possible errors in a binary systematic (47,24,11) quadratic residue (QR) code. The main idea of the TLDA is based on the weight of syndrome, the syndrome decoder together with a reduced-size lookup table (RSLT), and the shift-search method given by Reed et al. Thus, the size of the lookup table and computational complexity in a finite field can be significantly reduced. The memory size of the proposed condensed lookup table (CLT) consists of only 36.6Kbytes and is only about 0.24% of the full lookup table (FLT) and 3.2% of the lookup up table given by Chen et al., respectively. These facts lead to significant reduction of computational time and the decoding complexity. A simulation result shows that the decoding speed of the proposed TLDA is much faster than all existing decoding algorithms. Moreover, it can be extended to decode all QR codes, including the class of the cyclic codes when the code length is moderate. The CLT makes this new decoding algorithm suitable for hardware or firmware implementations.

On the decoding of the (24,12,8) Golay code

An improved syndrome shift-register decoding algorithm, called the syndrome-weight decoding algorithm, is proposed for decoding three possible errors and detecting four errors in the (24,12,8) Golay code. This method can also be extended to decode two other short codes, such as the (15,5,7) cyclic code and the (31,16,7) quadratic residue (QR) code. The proposed decoding algorithm makes use of the properties of cyclic codes, the weight of syndrome, and the syndrome decoder with a reduced-size lookup table (RSLT) in order to reduce the number of syndromes and their corresponding coset leaders. This approach results in a significant reduction in the memory requirement for the lookup table, thereby yielding a faster decoding algorithm. Simulation results show that the decoding speed of the proposed algorithm is approximately 3.6 times faster than that of the algebraic decoding algorithm.

Asymptotic performance of amplify‐and‐forward cooperative diversity networks with the Nth best relay over Rician fading channels

This paper analyzes and derives the asymptotic symbol error rate expression of amplify-and-forward cooperative communications with the Nth best relay over independent and non-identical Rician fading channels. Rician fading takes the line-of-sight effects into account and often exists in microcellular mobile and wireless mesh networks. In practice, the Nth best relay is selected when the best relay is unavailable because of fairness and load balancing issues. Performance analysis reveals that the diversity gain of the Nth best-relay scheme equals R–N+2, while the best relay and all-participate schemes achieve the diversity gain of R+1, where R is the number of relays. Simulation and numerical evaluation show the tightness of the asymptotic symbol error rate expression and the accuracy of our theoretical analysis. Copyright

Fast decoding of the (23, 12, 7) Golay code with four‐error‐correcting capability

In this paper, a fast and efficient decoding algorithm for correcting the (23, 12, 7) Golay code up to four errors is presented. The aim of this paper is to develop a fast syndrome-group search method for finding the candidate codewords by utilizing the property that the syndromes of the weight-4 error patterns are identical to that of the weight-3 error pattern. When the set of the candidate codewords is constructed, the most likely one is determined by assessing the corresponding correlation metrics. The well-known Chase-II decoder, which needs to perform the hard-decision decoder multiple times, acts as a comparison basis. Simulation results over the additive white Gaussian noise channel show that the decoding complexity of the proposed method is averagely reduced by at least 86% in terms of the decoding time. Furthermore, the successful decoding percentage of the new decoder in the case of four errors is always superior to Chase-II decoder. At the signal-to-noise ratio of 0 dB, the proposed algorithm still can correct up to 97.40% weight-4 error patterns. The overall bit error rate performance of the proposed decoder is close to that of Chase-II decoder. It implies that the new decoder is beneficial to the practical implementation. Copyright

Impulsive noise suppression in the case of frequency estimation by exploring signal sparsity

The frequency estimation problem is addressed in this work in the presence of impulsive noise. Two typical scenarios are considered; that is, the received data are assumed to be uniformly sampled, i.e., without data missing for the first case and data are randomly missed for the second case. The main objective of this work is to explore the signal sparsity in the frequency domain to perform frequency estimation under the impulsive noise. Therefore, to that end, a DFT-like matrix is created in which the frequency sparsity is provided. The missing measurements are modeled by a sparse representation as well, where missing samples are set to be zeros. Based on this model, the missing pattern represented by a vector is indeed sparse since it only contains zeros and ones. The impulsive noise is remodeled as a superposition of a unknown sparse vector and a Gaussian vector because of the impulsive nature of noise. By utilizing the sparse property of the vector, the impulsive noise can be treated as a unknown parameter and hence it can be canceled efficiently. By exploring the sparsity obtained, therefore, a joint estimation method is devised under optimization framework. It renders one to simultaneously estimate the frequency, noise, and the missing pattern. Numerical studies and an application to speech denoising indicate that the joint estimation method always offers precise and consistent performance when compared to the non-joint estimation approach.

A Future Simplification of Procedure for Decoding Nonsystematic Reed-Solomon Codes Using the Berlekamp-Massey Algorithm

It is well-known that the Euclidean algorithm can be used o find the systematic errata-locator polynomial and the errata-evaluator polynomial simultaneously in Berlekamps key equation that is needed to decode a Reed-Solomon (RS) codes. In this paper, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the Euclidean algorithm for efficiently decoding nonsystematic RS codes. In fact, this decoding algorithm is an appropriate modification to the algorithm developed by Shiozaki and Gao. Based on the ideas presented above, a fast algorithm described from Blahuts classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the Euclidean algorithm by the Berlekamp-Massey (BM) algorithm. These facts lead to significantly reduce the decoding complexity of the proposed RS decoder. In addition, computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new Euclidean-algorithm-based decoding approach proposed in this paper.

A cyclic weight algorithm of decoding the (47,24,11) quadratic residue code

In this article, a novel table lookup decoding algorithm, called the cyclic weight (CW) decoding algorithm, is developed to facilitate faster decoding of the binary systematic (47,24,11) quadratic residue (QR) code. It is based on the property of cyclic codes together with the weight of syndromes. This new algorithm requires a lookup table which consists of 20.43Kbytes. The advantage of the CW decoding algorithm over the previous table lookup method is that the memory size of the proposed lookup table is only about 1.89% of the lookup table needed in the decoding algorithm of Chens et al. These facts lead to significantly reduce the decoding complexity in terms of CPU time while maintaining the capability to correct up to five errors. Simulation results show that the decoding speed of the proposed algorithm is much faster than that of the algorithm of Chen et al.

Performance analysis and power allocation for decode‐and‐forward cooperative communications over Rician fading channel

This paper derives the asymptotic symbol error rate (SER) and outage probability of decode-and-forward (DF) cooperative communications over Rician fading channels. How to optimally allocate the total power is also addressed when the performance metric in terms of SER or outage probability is taken into consideration. Analysis reveals the insights that Rician factor has a great impact on the system performance as compared with the channel variance, and the relay–destination channel quality is of importance. In addition, the source–relay channel condition is irrelevant to the optimal power allocation design. Simulation and numerical evaluation substantiate the tightness of the asymptotic expressions in the high-SNR regions and demonstrate the accuracy of our theoretical analysis. Copyright