Research On The Simplified Decoding Algorithm For The(47, 24, 11) Quadratic Residue Code

With the continuous improvement of technology, the requirement of the efficient and reliable digital communications is increasingly enhanced. Thus, channel coding is often needed. As a subclass of the BCH code, quadratic residue?QR? codes have larger minimum distance than traditional BCH codes and RM codes with?approximately? equal code lengths. Therefore, they usually have better error-correcting performance. Considering the complexity, QR codes of lengths less than 100 have larger application potential. Nevertheless, the existing algebraic decoding algorithms are inefficient since they require complex computations when computing the unknown syndromes.Through studying the algebraic structure of the QR codes in this paper, we propose a new hard-decision?HD? decoding algorithm without determining the unknown syndromes. This algorithm obtains formulaic solution corresponding to 4 errors by eliminating the unknown syndromes in the Newton identities and simplifies the corresponding judgment condition. When 5 errors occurred, it flips over 1 error according to the reliability-search algorithm and then uses the method of decoding 4 errors to correct the remainder 4 errors. The algorithm apparently decreases the decoding complexity; especially when there exists 5 errors, the reliability-search algorithm will rapidly converge as the signal-to-noise ratio?SNR? increases. Based on the proposed simplified HD algorithm, we develop a soft-decision?SD? algorithm for the?47, 24, 11? QR code using the Chase-? algorithm together with a fast stopping condition.Simulation results of the HD algorithm indicate: when the number of errors is less than 3, the average decoding time keeps the same as the existing algorithms; when 4 errors occurred, the decoding time is decreased by 75.3%; when the Hamming weight of the error pattern is 5, the average decoding speed is faster than the existing algorithms by 80.4%. Additionally, simulation results of the SD algorithm indicate: when codewords modulated by BPSK are transmitted through the additive White Gaussian Noise channel?AWGN?, the total decoding time is reduced by 40.1% compared with the Chase-? SD algorithm based on the existing HD algorithm while maintaining the same performance. At FER of 10^-4, the SD decoding algorithm obtains 1.9dB coding gain compared to the HD decoding algorithm. Since the reliability-search algorithm is utilized in both the simplified HD and SD algorithms, the decoding efficiency can be further enhanced with a slight?or no? loss in decoding performance by setting an appropriate flipping times ?.