  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 $ P $ and
$ N $ respectively, the list decoding capacity is equal to $
\frac{1}{2}\log\frac{P}{N} $. 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 number-theoretic conjecture is true, then the list size grows as a
polynomial function of the gap-to-capacity.
