Hung-Peng Lee

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 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 Weight Method of Decoding the Binary BCH Code

In this paper, a weight method with using a reduced lookup table is developed to decode the three possible errors in (15, 5, 7) and (31, 16, 7) BCH code. The data in the reduced lookup table consists of syndrome patterns and corresponding error patterns which only have one and two errors occurred in the message block of the received codeword. The proposed algorithm makes use of the properties of cyclic codes, weight of syndrome, and the reduced lookup table. It often results in a significant reduction in the memory requirements comparing to the traditional lookup table or other algebraic decoding methods. This weight decoding algorithm together with a reduced lookup table makes a fast and low complexity of the table lookup decoding algorithm. Moreover, a computer simulation shows that such a novel method is a much faster algorithm in software than the traditional full lookup table searching algorithm.

A weight method of decoding the (23, 12, 7) Golay code using reduced table lookup

In this paper, a weight method with using a reduced lookup table is developed to decode the three possible errors in (23, 12, 7) Golay code. The reduced lookup table consists of syndrome patterns and corresponding error patterns which only have one and two errors occurred in the message block of the received codeword. The useful proposed algorithm makes use of the properties of cyclic codes, weight of syndrome, and reduced lookup table. It often results in a significant reduction in the memory requirements comparing to the traditional lookup table. This weight algorithm together with a reduced table lookup makes a fast and low complexity of the table lookup decoding algorithm. Moreover, a computer simulation shows that such a novel method is a much faster algorithm in software than the traditional full lookup table searching algorithm.

Decoding of the (31, 16, 7) quadratic residue code with four-error correcting capability

A high-speed algorithm for decoding the binary (31, 16, 7) quadratic residue (QR) code up to four errors is proposed. Core to the key idea lies in an innovative integration of the reliability-search procedure and the insight of the weight distribution of the code for searching the candidate codewords. Accordingly, the maximum-likelihood (ML) criterion is applied to pick the most possible codeword. Through simulation over the additive white Gaussian noise (AWGN) channel, it is concluded that the error-correcting performance of the new decoder is superior to that of a Chase-II decoder when no more than four errors occur. The overall bit error rate (BER) of the proposed decoder is close to that of the Chase-II decoder. More importantly, the new decoder results in great reductions of 72.34% and 92.37% in the signal-to-noise ratio (SNR) values of 0 and 7, respectively. The proposed algorithm suggests an alternative to enhance the error-correcting capability of the hard-decision decoder, but with a lower decoding complexity.

Decoding the (31, 16, 7) quadratic residue code in GF(2^5)

The binary QR codes are well known for their good behavior. The proposed algebraic decoding algorithm for decoding the (31, 16, 7) QR code with reducible generator polynomial is able to correct up to three errors in the finite field GF(25). The proposed algorithm is based on an application of the decoding algorithm given by Truong et al. and Chen et al. to modify the decoding algorithm proposed by Reed et al. All syndromes in the error-locator polynomial are computed in the finite field GF(25). Thus, the decoding time can be reduced. Moreover, the simulation results for comparing the proposed decoding algorithm with decoding algorithm given by Reed et al. are given. This algorithm is suitable for implementation in a programmable microprocessor or special-purpose VLSI chip.