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This paper proposes a novel high-order associative memory system (AMS) based on the discrete Taylor series (DTS). The mathematical foundation for the new AMS scheme is derived, three training algorithms are proposed, and the convergence of learning is proved. The DTS-AMS thus developed is capable of implementing error-free approximation to multivariable polynomial functions of arbitrary order. Compared with cerebellar model articulation controllers and radial basis function neural networks, it provides higher learning precision and less memory request. Furthermore, it offers less training computation and faster convergence rate than that attainable by multilayer perceptron. Numerical simulations show that the proposed DTS-AMS is effective in higher order function approximation and has potential in practical applications.