We show that Hopfield neural networks with synchronous dynamics and asymmetric weights admit stable orbits that form sequences of maximal length. For units, these sequences have length ; that is, they cover the full state space. We present a mathematical proof that maximal-length orbits exist for all , and we provide a method to construct both the sequence and the weight matrix that allow its production. The orbit is relatively robust to dynamical noise, and perturbations of the optimal weights reveal other periodic orbits that are not maximal but typically still very long. We discuss how the resulting dynamics on slow time-scales can be used to generate desired output sequences.

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