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 IRN, 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 Z(3) for which a high packing density is obtained for the lattice Lambda(p) obtained by projecting Z(3) into the plane orthogonal to (1, a, b). Two approaches are used to prove the central result of the paper.