Dynamical systems, curves and coding for continuous alphabet sources

Good codes for transmitting a continuous-alphabet source over an AWGN channel can be constructed using simple dynamical systems. The trajectories of the dynamical systems that we consider are curves in R^N, and we use these curves as signal sets for a modulation system. In this paper we consider the problem of choosing the parameters of the dynamical system such that the length of its trajectory is maximized subject to a constraint on the minimum distance between its "folds". We provide some general results on the construction of such curves and show how to select the parameters optimally in the case N=6. This is done by reducing the problem to one of choosing a vector (1, a, b) in Z³ for which a high packing density is obtained for the lattice Λ_p obtained by projecting Z³ into the plane orthogonal to (1, a, b). Two approaches are used to prove the central result of the paper.