LLR computation for multistage decoding

Due to their linear and highly symmetrical structure, lattices are becoming of a great interest as potential transmission schemes. Lattice codes suggest a common view of channel and source coding and new tools for the analysis of information network problems. Several constructions have been proposed to build these lattices, some of which are based on multi-level coding and multistage decoding such as constructions D and π
_D
. Soft-decision decoders corresponding to the different nested error-correcting codes used to construct such lattices need at each stage the computation of a soft input, namely Log- Likelihood Ratios. In this paper, we give an efficient computation of LLRs based on Jacobi theta functions for three different types of constructions; Binary construction D, Quaternary construction D and Binary construction π
_D
.