List Decoding Random Euclidean Codes and Infinite Constellations
Abstract
We study the list decodability of different ensembles of codes over the real alphabet under the assumption of an omniscient adversary. It is a wellknown 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 ConstructionA lattices and infinite ConstructionA lattices. We then define and study a class of infinite constellations that generalize ConstructionA 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 numbertheoretic conjecture is true, then the list size grows as a polynomial function of the gaptocapacity.
 Publication:

arXiv eprints
 Pub Date:
 January 2019
 DOI:
 10.48550/arXiv.1901.03790
 arXiv:
 arXiv:1901.03790
 Bibcode:
 2019arXiv190103790Z
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Metric Geometry
 EPrint:
 Significantly revised