For rate-optimal transmission over the AWGN channel Shannon showed that a Gaussian distributed channel input is required. This poses problems for digital implementations, where symbols are necessarily quantized. One solution to this is coded modulation via lattice coding. Lattice coding creates code code books by quantizing continuous space into periodic regions.

The student's task is to understand lattice transmission schemes (see e.g. [1], [2], [3]) and in particular the schemes from [4], summarize the important concepts of lattice coding, and give an explanation of coding schemes achieving capacity over the AWGN channel.

[1] Conway, Sloane 1982 - Voronoi regions of lattices, second moments of polytopes, and quantization. DOI: 10.1109/TIT.1982.1056483
[2] Conway, Sloane 1983 - A fast encoding method for lattice codes and quantizers. DOI: 10.1109/TIT.1983.1056761
[3] Forney 1989 - Multidimensional constellations. 2. Voronoi constellations. DOI: 10.1109/49.29616
[4] Erez, Zamir 2004 - Achieving 0.5log(1+SNR) on the AWGN Channel With Lattice Encoding and Decoding. DOI: 10.1109/TIT.2004.834787

• Information Theory
• Introduction to Channel Coding
• Introduction to Coded Modulation helpful but not required

Enumerative sphere shaping [1] is an efficient method to perform energy-optimal distribution matching for short to medium block lengths. It is an alternative approximate shell mapping schemes based on CCDM [2].

The goal of this thesis is to extend enumerative sphere shaping to general target distributions using the ideas from [3] and compare the performance of this distribution matcher to CCDM-based approximate shell mapping.

1. https://doi.org/10.1109/TWC.2019.2951139
2. https://doi.org/10.1109/TIT.2015.2499181
3. https://doi.org/10.1109/LWC.2018.2890595