Modern communication systems increasingly employ probabilistic amplitude shaping (PAS) to close the gap to the Shannon capacity by generating non-uniform input distributions (typically approximating Maxwell–Boltzmann distributions). At the core of PAS lies the distribution matcher (DM), which maps uniformly distributed information bits into shaped sequences of amplitudes.

A central challenge is designing DMs that achieve low rate loss and low complexity.

Enumerative Sphere Shaping (ESS) addresses these challenges by restricting sequences to lie within a hypersphere in the energy domain. Instead of fixing symbol composition (as in CCDM), ESS selects sequences such that their total energy does not exceed a predefined radius. This approach results in competitive performance in the short blocklength regime.

The student will explore ESS [1][2], including its theoretical foundation in information and coding theory, algorithmic implementation, storage and computational complexity, and comparisons with other shaping methods such as CCDM and shell mapping [3]. 

[1] https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8850066

[2] https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=8895789&casa_token=sZch6uYNxusAAAAA:Oq9owJ8tNEA5puQv1Bai7enCei8nYMoPEwt9jO3q-9upB20dTKJPltXrbI7mPSKnsVs_nK_G

[3] https://www.mdpi.com/1099-4300/22/5/581

Prerequisites