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

In this internship we will investigate performance and decoding algorithms for polar codes using hard decisions.

In this thesis we will use and develop practical algorithms for lossy source coding (a.k.a. quantization or compression) to construct a probabilistic shaping scheme based on low-density graph codes. 

The aim of the thesis is to construct a hybrid ARQ coding scheme which employs probabilistic shaing. For both individual concepts, the structure of polar codes provides an amenable fundament. This makes a polarization-based coding scheme an interesting candidate to fuse HARQ with shaping.