Decoding in Compute-and-Forward Relaying: Real Lattices and the Flatness of Lattice Sums

Computing the theta series of an arbitrary lattice, and more specifically a related quantity known as the flatness factor, has been recently shown to be important for lattice code design in various wireless communication setups. However, the theta series is in general not known in closed form, excluding a small set of very special lattices. In this article, motivated by the practical applications as well as the mathematical problem itself, a simple approximation of the theta series of a lattice is derived. In relation to this, maximum-likelihood decoding in the context of compute-and-forward relaying is studied. Following previous work, it is shown that the related metric can exhibit a flat behavior, which can be characterized by the flatness factor of the decoding function. Contrary to common belief, we note that the decoding metric can be rewritten as a sum over a random lattice only when at most two sources are considered. Using a particular matrix decomposition, a link between the random lattice and the code lattice employed at the transmitter is established, which leads to an explicit criterion for code design, in contrast to implicit criteria derived previously. Finally, candidate lattices are examined with respect to the proposed criterion using the derived theta series approximation. Index Terms Lattices, Theta Series, Flatness Factor, Compute-and-Forward, Physical Layer Network Coding. The authors are with the Department of Mathematics and Systems Analysis, Aalto University, Finland (e-mail: [email protected]). Their work is supported by the Academy of Finland under Grants #268364, #276031, #282938, #283262 and #303819, a grant from the Finnish Foundation for Technology Promotion, as well as a grant from the Foundation for Aalto University Science and Technology. A preliminary version of the result from Theorem 1 was utilized in [1] and presented in IEEE ITNAC 2016. February 20, 2017 DRAFT ar X iv :1 60 1. 05 59 6v 2 [ cs .I T ] 1 7 Fe b 20 17