SISO APP Searches in Lattices with Tanner Graphs
  An efficient, low-complexity, soft-output detector for general lattices is presented, based on their Tanner graph (TG) representations. Closest-point searches in lattices can be performed as non-binary belief propagation on associated TGs; soft-information output is naturally generated in the process; the algorithm requires no backtrack (cf. classic sphere decoding), and extracts extrinsic information. A lattice's coding gain enables equivalence relations between lattice points, which can be thereby partitioned in cosets. Total and extrinsic a posteriori probabilities at the detector's output further enable the use of soft detection information in iterative schemes. The algorithm is illustrated via two scenarios that transmit a 32-point, uncoded super-orthogonal (SO) constellation for multiple-input multiple-output (MIMO) channels, carved from an 8-dimensional non-orthogonal lattice (a direct sum of two 4-dimensional checkerboard lattice): it achieves maximum likelihood performance in quasistatic fading; and, performs close to interference-free transmission, and identically to list sphere decoding, in independent fading with coordinate interleaving and iterative equalization and detection. Latter scenario outperforms former despite the absence of forward error correction coding---because the inherent lattice coding gain allows for the refining of extrinsic information. The lattice constellation is the same as the one employed in the SO space-time trellis codes first introduced for 2-by-2 MIMO by Ionescu et al., then independently by Jafarkhani and Seshadri. Complexity is log-linear in lattice dimensionality, vs. cubic in sphere decoders. 
Dumitru Mihai Ionescu, Haidong Zhu