Lattice codes with the optimal decoding coefficient are capacity-achieving when dimension N → ∞. In communications systems, finite dimensional lattice codes are needed, where the optimal decoding coefficient may still fail decoding even when R−5for a 2-user CF relay using 128- and 256-dimensional lattice codes with optimized CRC length and 2 decoding trials in total.

A genie-aided decoder for finite dimensional lattice codes is considered. The decoder may exhaustively search through all possible scaling factors $\alpha \in {\mathbb{R}}$. We show that this decoder can achieve lower word error rate (WER) than the one- shot decoder using αMMSEas a scaling factor. A lower bound on the WER for the decoder is found by considering the covering sphere of the lattice Voronoi region. The proposed decoder and the bound are valid for both power-constrained lattice codes and lattices. If the genie is applied at the decoder, E8 lattice code has 0.5 dB gain and BW16 lattice code has 0.4 dB gain at WER of 10-4compared with the one-shot decoder using αMMSE. A method for estimating the WER of the decoder is provided by considering the effective sphere of the lattice Voronoi region, which shows an accurate estimate for E8 and BW16 lattice codes. In the case of per-dimension power P → ∞, an asymptotic expression of the bound is given in a closed form. A practical implementation of a simplified decoder is given by considering CRC-embedded n =128 polar code lattice.