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A class of neural network for constructing associative memories that learn the memory patterns as well as their neighbouring patterns is presented. The network is basically a layer of perceptrons with high-order polynomials as their discriminant functions. A learning algorithm is proposed for the network to learn arbitrary bipolar patterns. The simulation results show that the associative memories implemented in this way achieve a set of desirable characteristics, namely high storage capacity, nearest convergence, and existence of a 'no decision' state which attracts indistinguishable inputs. Furthermore, it is also possible to shape the attraction basin of a memory pattern under any metrics definition of distance.