On the generalization of the Costas property in higher dimensions

We investigate the generalization of the Costas property in three or more dimensions, and we seek an appropriate definition; the two main complications are a) that the number of ''dots'' this multidimensional structure should have is not obvious, and b) that the notion of the multidimensional permutation needs some clarification. After proposing various alternatives for the generalization of the definition of the Costas property, based on the definitions of the Costas property in one or two dimensions, we also offer some construction methods, the main one of which is based on the idea of reshaping Costas arrays into higher-dimensional entities.