Flat tori, lattices and spherical codes

The foliation of a sphere in an even number of dimensions by flat tori can be used to construct discrete spherical codes and also homogeneous curves for transmitting a continuous alphabet source over an AWGN channel. In both cases the performance of the code is related to the packing density of specific lattices and their orthogonal sublattices. In the continuous case the packing density of curves relies also on the search for projection lattices with good packing density. We present here a survey of this topic including some recent research results and perspectives.