List Decoding Random Euclidean Codes and Infinite Constellations

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