Cyclic-group-ring-valued Hopfield associative memory of order three

Since a complex-valued Hopfield associative memory (CVHAM) was proposed, several models of multistate neural associative memories have been proposed. Connection weights are major costs of neural associative memories. Each model has different cost for connection weights, and performance according to the cost is required;models with more weight parameters should provide better noise tolerance. For example, rotor Hopfield associative memories (RHAMs) have double weight parameters of CVHAMs and provide much better noise tolerance than CVHAMs. Quaternion-valued Hopfield networks have half weight parameters and their noise tolerance is worse than that of CVHAMs. In the present paper, a cyclic-group-ring-valued Hopfield associative memory of order three is proposed. It has one and a half times weight parameters of CVHAM, and its noise tolerance is expected to be better than CVHAMs and worse than RHAMs. We prove our hypothesis by computer simulations.