Practical Encoders and Decoders for Euclidean Codes from Barnes-Wall Lattices

We address the application of Barnes-Wall (BW) lattice codes for communication over additive white Gaussian noise (AWGN) channels. We introduce Construction A^prime of complex BW lattices that makes new connection between linear codes over polynomial rings and lattices. We show that Construction A^prime of BW lattices is equivalent to the multilevel construction from Reed-Muller codes proposed by Forney. To decode the BW lattice code, we adapt the low-complexity sequential BW lattice decoder (SBWD) proposed by Micciancio and Nicolosi. First we study the error performance of SBWD for decoding the infinite lattice, and demonstrate that it is powerful in making correct decisions well beyond the packing radius. Subsequently, we use the SBWD to decode lattice codes through a novel noise trimming technique, where the received vector is appropriately scaled before applying the SBWD. We show that the noise trimming technique is most effective for decoding BW lattice codes in smaller dimensions, while the gain diminishes for decoding codes in larger dimensions.