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In this study, an efficient and fast algebraic decoding algorithm (ADA) for the binary systematic quadratic residue (QR) code of length 73 with the reducible generator polynomial to correct up to six errors is proposed. The S(I, J) matrix method given by He et al. (2001) is utilised to compute the unknown syndromes S5. A technique called swap base is proposed to correct the weight-4 error patterns. To correct the weight-5 error patterns, the new error-locator polynomials for decoding the five errors are derived. Finally, the modified shift-search algorithm (SSA) developed by Lin et al. (2010) is applied to correct the weight-6 error patterns. Moreover, the computations of all syndromes are achieved in a small finite field. Simulation results show that the proposed ADA is practical.