The paper proposes an improved difference-of-syndromes decoding algorithm for decoding quadratic residue codes up to half the minimum distance. Therein, soft-decision information is not essential, but it can be used to speed up the computations. The improvement over the prior art is demonstrated by simulations. Moreover, the proposed algorithm is also compared with the classic Berlekamp-Massey algorithm by taking two BCH codes as examples.

In this paper, an efficient decoding algorithm is developed to facilitate faster decoding of the binary systematic quadratic residue (QR) codes. It is based on the difference of syndromes (DS), and hence, is called the DS algorithm hereinafter. This new method combines the advantages of the syndrome-weight algorithm and properties of the cyclic codes. Actually, it is a natural generalization of the cyclic weight (CW) algorithm for the (47, 24, 11) QR code developed by Lin et al., and its validity of decoding any binary systematic QR code is also proved. The complexity analysis and simulation results show that the DS algorithm dramatically reduces the decoding complexity without performance loss and considerably requires less memory when compared with previous ones. Utilizing the (47, 24, 11), the (71, 36, 11), the (73, 37, 13), and the (89, 45, 17) QR codes as examples, the DS algorithm not only significantly improves the decoding efficiency, but also saves the memory evidently. When the (23, 12, 7) QR code is considered, the DS algorithm performs almost as well as the currently known best algorithm in terms of decoding efficiency and memory requirements. Thus, all QR codes of lengths less than 100 can be decoded efficiently by using the proposed algorithm. Especially, when the proposed algorithm is applied to decode the (89, 45, 17) QR code, the best one among all the QR codes of lengths less than 100, the decoding speed is raised 26 times and the memory is saved up to 76.6% in comparison with the existing fastest decoding algorithm.