Tsung-Ching Lin

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.

A Fast Algorithm for the Syndrome Calculation in Algebraic Decoding of Reed–Solomon Codes

In this paper, Fedorenko and Trifonovs procedure is applied to evaluate the syndrome of the received word in time-domain Reed-Solomon decoders. This application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures. Moreover, simulation results for this new syndrome method are given.

Novel Approaches to the Parametric Cubic-Spline Interpolation

The cubic-spline interpolation (CSI) scheme can be utilized to obtain a better quality reconstructed image. It is based on the least-squares method with cubic convolution interpolation (CCI) function. Within the parametric CSI scheme, it is difficult to determine the optimal parameter for various target images. In this paper, a novel method involving the concept of opportunity costs is proposed to identify the most suitable parameter for the CCI function needed in the CSI scheme. It is shown that such an optimal four-point CCI function in conjunction with the least-squares method can achieve a better performance with the same arithmetic operations in comparison with the existing CSI algorithm. In addition, experimental results show that the optimal six-point CSI scheme together with cross-zonal filter is superior in performance to the optimal four-point CSI scheme without increasing the computational complexity.

A Near Lossless Wavelet-Based Compression Scheme for Satellite Images

In this paper, a near lossless image compression algorithm is presented for high quality satellite image compression. The proposed algorithm makes use of the recommendation for image data compression from the Consultative Committee for Space Data Systems (CCSDS) and specific residue image bit-plane compensation. Comparing with the recommendation for satellite image compression from CCSDS, the proposed algorithm can reconstruct near lossless images with less bit rate than the recommendation of CCSDS does. Benefited from run-length coding and specific residue image bit-plane compensation, the proposed algorithm can obtain higher quality satellite image at similar bit rate or lower bit rate at the similar image quality. These results are valuable for reducing transmission time of high quality satellite image data. This work can be further improved by combining other binary compression techniques and the extension of this work may offer a VLSI or a DSP implementation of the proposed algorithm. Satellite image transmission and storage system can benefit by the proposed algorithm.

Simplified procedure for decoding reed-solomon codes using Euclid’s algorithm and the fast fourier transform over GF(2 m )

A simplified decoding algorithm to correct both errors and erasures is used in conjunction with Gaos algorithm for efficiently decoding Reed-Solomon codes.

Decoding of the (24, 12, 8) extended golay code up to four errors

A new decoder is proposed to decode the (24, 12, 8) binary extended Golay code up to four errors. It consists of the conventional hard decoder for correcting up to three errors, the detection algorithm for four errors and the soft decoding for four errors. For a weight-4 error in a received 24-bit word, Method 1 or 2 is developed to determine all six possible error patterns. The emblematic probability value of each error pattern is then defined as the product of four individual bit-error probabilities corresponding to the locations of the four errors. The most likely one among these six error patterns is obtained by choosing the maximum of the emblematic probability values of all possible error patterns. Finally, simulation results of this decoder in additive white Gaussian noise show that at least 93% and 99% of weight-4 error patterns that occur are corrected if the two Eb/N0 ratios are greater than 2 and 5 dB, respectively. Consequently, the proposed method can achieve a better percentage of successful decoding for four errors at variable signal-to-noise ratios than Lu et al.s algorithm in software. However, the speed of the method is slower than Lu et al.s algorithm.

Simplified 2-D Cubic Spline Interpolation Scheme Using Direct Computation Algorithm

It has been shown that the 2-D cubic spline interpolation (CSI) proposed by Truong is one of the best algorithms for image resampling or compression. Such a CSI algorithm together with the image coding standard, e.g., JPEG, can be used to obtain a modified image codec while still maintaining a good quality of the reconstructed image for higher compression ratios. In this paper, a fast direct computation algorithm is developed to improve the computational efficiency of the original FFT-based 2-D CSI methods. In fact, this algorithm computes the 2-D CSI directly without explicitly calculating the complex division usually needed in the FFT or Winograd discrete Fourier transform (WDFT) algorithm. In addition, this paper describes a novel way to derivate the 2-D CSI from the 1-D CSI by using the row-column method. This new fast 2-D CSI provides a regular and simple structure based upon linear correlations. Therefore, it can be implemented by the use of a modification of Kungs pipeline structure and is naturally suitable for VLSI implementations. Experimental results show that the proposed new fast 2-D CSI algorithm can achieve almost the same CSI performance with much fewer arithmetic operations in comparison with existing efficient algorithms.