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We study the list decodability of different ensembles of codes over the realalphabet under the assumption of an omniscient adversary. It is a well-knownresult 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 listsizes, and the goal of the present paper is to obtain a better understanding ofthe smallest achievable list size as a function of the gap to capacity. We showa reduction from arbitrary codes to spherical codes, and derive a lower boundon the list size of typical random spherical codes. We also give an upper boundon the list size achievable using nested Construction-A lattices and infiniteConstruction-A lattices. We then define and study a class of infiniteconstellations that generalize Construction-A lattices and prove upper andlower bounds for the same. Other goodness properties such as packing goodnessand AWGN goodness of infinite constellations are proved along the way. Finally,we consider random lattices sampled from the Haar distribution and show that ifa certain number-theoretic conjecture is true, then the list size grows as apolynomial function of the gap-to-capacity.

List Decoding Random Euclidean Codes and Infinite Constellations