Lipschitz Integers for Coded Modulation and Precoding

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Signal constellations are an important ingredient for digital transmission systems, directly determining their performance. Therefore, constellations have been constructed and analyzed for many years, where for instance different constellations can be compared with the constellation figure of merit introduced by Forney and Wei. Besides two-dimensional constellations, immediately motivated by QAM signaling, already at early stages higher-dimensional approaches, in particular four-dimensional signal sets, have been of interest due to their higher flexibility. Nowadays, four-dimensional signal constellations are of increasing interest in optical communications.More recently, one of the applicants found that constellations constructed by partitioning of Lipschitz integers have a figure of merit which is up to 10 dB better than the comparable two-dimensional QAM constellations [FS15]. These remarkable gains are only observed for special subsets of Lipschitz integers and not for Lipschitz integers themselves. However, until now only some examples exist and a careful analysis and study of these constellations is necessary. Therefore, we propose to analyze novel four-dimensional constructions in this project. Noteworthy, the most important classical two-dimensional constellations can be interpreted as special subsets of Lipschitz integers which might lead to a novel theory for constellations. Furthermore, methods from coded modulation constellations might help to construct even better constellations. Coded modulation based on the new constellations can improve wired, wireless, and optical communication systems. In addition, advanced equalization and precoding techniques, in particular those based on the concepts of lattice reduction and its tightly related approach of integer forcing, are based on algebraic operations and thus Lipschitz integers and their partitioning are well suited for novel methods. Thus, we expect many interesting results for the improvement of future coding and modulation for any type of digital communication system with complex-valued signal constellations.