A new algorithm is developed to facilitate faster decoding of the (47,24,11) quadratic residue (QR) code. This decoder, based on the idea first developed by Reed in a 1959 MIT Lincoln Laboratory Report, uses real channel data to estimate the individual bit-error probabilities in a received word. The algorithm then sequentially inverts the bits with the highest probability of error until one of the errors is canceled. The remaining errors are then corrected by the use of algebraic decoding techniques. This new algorithm, called the reliability-search algorithm, is a complete decoder that significantly reduces the decoding complexity in terms of CPU time while maintaining the same bit-error rate (BER) performance. In fact, this algorithm is an appropriate modification to the algorithm developed by Chase.
A new algorithm is developed to facilitate faster decoding of the (47,24,11) quadratic residue (QR) code. This decoder, based on the idea first developed by Reed in a 1959 MIT Lincoln Laboratory Report, uses real channel data to estimate the individual bit-error probabilities in a received word. The algorithm then sequentially inverts the bits with the highest probability of error until one of the errors is canceled. The remaining errors are then corrected by the use of algebraic decoding techniques. This new algorithm, called the reliability-search algorithm, is a complete decoder that significantly reduces the decoding complexity in terms of CPU time while maintaining the same bit-error rate (BER) performance. In fact, this algorithm is an appropriate modification to the algorithm developed by Chase.
In this paper, Three decoding methods of the (89, 45, 17) binary quadratic residue (QR) code to be presented are hard, soft and linear programming decoding algorithms. Firstly, a new hybrid algebraic decoding algorithm for the (89, 45, 17) QR code is proposed. It uses the Laplace formula to obtain the primary unknown syndromes, as done in Lin et al.'s algorithm when the number of errors v is less than or equal to 5, whereas Gaussian elimination is adopted to compute the unknown syndromes when v ≥ 6. Secondly, an appropriate modification to the algorithm developed by Chase is also given in this paper. Therefore, combining the proposed algebraic decoding algorithm with the modified Chase-II algorithm, called a new soft-decision decoding algorithm, becomes a complete soft decoding of QR codes. Thirdly, in order to further improve the error-correcting performance of the code, linear programming (LP) is utilized to decode the (89, 45, 17) QR code. Simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with Lin et al.'s hard decoding algorithm, and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate (BER) performance. Moreover, the LP-based decoding improves the error-rate performance almost without increasing the decoding complexity, when compared with the new soft-decision decoding algorithm. It provides a coding gain of 0.2 dB at BER = 2 × 10-6.
The (23,12,7) Golay code is a perfect linear error-correcting code that can correct all patterns of three or fewer errors in 23 bit positions. A simple BCH decoding algorithm, given in E. Berlekamp (1968), can decode the (23,12,7) Golay code provided there are no more than two errors. The shift-search algorithm, developed by Reed et a. (1990), sequentially inverts the information bits until the third error is canceled. It then utilizes the BCH decoding algorithm to correct the remaining two errors. In this paper a simplified decoding algorithm, called the reliability-search algorithm, is proposed. This algorithm uses bit-error probability estimates to cancel the third error and then uses the BCH decoding algorithm to correct the remaining two errors. Simulation results show that this new algorithm significantly reduces the decoding complexity for correcting the third error while maintaining the same BER performance.
In this paper, a method to search the set of syndromes' indices needed in computing the unknown syndromes for the (73, 37, 13) quadratic residue (QR) code is proposed. According to the resulting index sets, one computes the unknown syndromes and thus finds the corresponding error-locator polynomial by using an inverse-free Berlekamp-Massey (BM) algorithm. Based on the modified Chase-II algorithm, the performance of soft-decision decoding for the (73, 37, 13) QR code is given. This result is new. Moreover, the error-rate performance of linear programming (LP) decoding for the (73, 37, 13) QR code is also investigated, and LP-based decoding is shown to be significantly superior in performance to the algebraic soft-decision decoding while requiring almost the same computational complexity. In fact, the algebraic hard-decision and soft-decision decoding of the (89, 45, 17) QR code outperforms that of the (73, 37, 13) QR code because the former has a larger minimal distance. However, experimental results indicate that the (73, 37, 13) QR code outperforms the (89, 45, 17) QR code with much fewer arithmetic operations when using the LP-based decoding algorithms. The pseudocodewords analysis partially explains this seemingly strange phenomenon.
In this paper, the performance of quadratic residue (QR) codes of lengths within 100 is given and analyzed when the hard decoding, soft decoding, and linear programming decoding algorithms are utilized. We develop a simple method to estimate the soft decoding performance, which avoids extensive simulations. Also, a simulation-based algorithm is proposed to obtain the maximum likelihood decoding performance of QR codes of lengths within 100. Moreover, four important theorems are proposed to predict the performance of the hard decoding and the maximum-likelihood decoding in which they can explore some internal properties of QR codes. It is shown that such four theorems can be applied to the QR codes with lengths less than 100 for predicting the decoding performance. In contrast, they can be straightforwardly generalized to longer QR codes. The result is never seen in the literature, to our knowledge. Simulation results show that the estimated hard decoding performance is very accurate in the whole signal-to-noise ratio (SNR) regimes, whereas the derived upper bounds of the maximum likelihood decoding are only tight for moderate to high SNR regions. For each of the considered QR codes, the soft decoding is approximately 1.5 dB better than the hard decoding. By using powerful redundant parity-check cuts, the linear programming-based decoding algorithm, i.e., the ACG-ALP decoding algorithm performs very well for any QR code. Sometimes, it is even superior to the Chase-based soft decoding algorithm significantly, and hence is only a few tenths of dB away from the maximum likelihood decoding.
In this letter, we study turbo product codes with quadratic residue codes (called QR-TPCs) as the component codes. We propose an efficient decoder based on Chase-II algorithm with two convergence conditions for the iterative decoding of QR-TPCs. For each row and column, the Chase-II decoder will stop immediately when one of the conditions is met. The simulation results show that the proposed algorithm has a lower computational complexity compared with existing decoding methods. Moreover, a comparison with 5G low-density parity-check codes shows that the proposed turbo product codes have better performance for short code lengths.
In this paper, a low-complexity decoder based on a neural network is proposed to decode binary quadratic residue (QR) codes. The proposed decoder is based on the neural min-sum algorithm and the modified random redundant decoder (mRRD) algorithm. This new method has the same asymptotic time complexity as the min-sum algorithm, which is much lower than the difference on syndromes (DS) algorithm. Simulation results show that the proposed algorithm achieves a gain of more than 0.4 dB when compared to the DS algorithm. Furthermore, a simplified approach based on trapping sets is applied to reduce the complexity of the mRRD. This simplification leads to a slight loss in error performance and a reduction in implementation complexity.