Lattices, projections, and applications to information theory

The contents of this thesis lie in the interface between Discrete Mathematics (particularly lattices) and Information Theory. The original contributions of this work are organized so that the first two chapters are devoted to theoretical results on q-ary and projection lattices, whereas the last ones are related to the construction of continuous source-channel codes. In the first chapters, we exhibit results on decoding q-ary lattices and on finding tilings associated to perfect error-correcting codes in the l_p norm. Regarding projection lattices, our contributions include the study of sequences of projections of a given n-dimensional lattice converging to any k-dimensional target lattice, as well as a convergence analysis of such sequences. These new results on projections extend and improve recent works on the topic and serve as building blocks for the applications to be developed throughout the last part of the thesis. In the last two chapters, we consider the problem of constructing mappings for the transmission of a continuous alphabet source over a Gaussian channel, when the channel dimension, n, is strictly greater than the source dimension, k. For one-dimensional sources, we exhibit codes based on curves on flat tori with performance significantly superior to the previous proposals in the literature with respect to the mean squared error achieved. For k > 1, we show how to apply projections of lattices to obtain codes whose mean squared error decays optimally with respect to the signal-to-noise ratio of the channel (referred to as asymptotically optimal codes). Through techniques from the rich theory of dissections of polyhedra, we present the first constructions of provenly asymptotically optimal codes for sources with dimension greater than 1 (AU)