List Decoding Random Euclidean Codes and Infinite Constellations

We study the list decodability of different ensembles codes over real alphabet under assumption an omniscient adversary. It is a well-known result that when source and adversary have power constraints $P $ notation="LaTeX">$N respectively, decoding capacity equal to notation="LaTeX">$\frac {1}{2}\log \frac {P}{N}$ . Random spherical achieve constant sizes, goal present paper obtain better understanding smallest achievable size as function gap capacity. show reduction from arbitrary codes, derive lower bound on typical random codes. also give upper using nested Construction-A lattices infinite lattices. then define class constellations generalize prove bounds for same. Other goodness properties such packing AWGN are proved along way. Finally, we consider sampled Haar distribution if certain conjecture originates in analytic number theory true, grows polynomial gap-to-capacity.