The modulo loss in lattice dirty-paper coding

Lattice decoding of a lattice-shaped codebook is a simple alternative for ML decoding, and it is equivalent to ML decoding after modulo-lattice reduction of the channel output. For good (high-dimensional) lattices, this modulo operation is information lossless in the presence of AWGN. At a finite shaping dimension, however, the lattice decoder is inferior to direct ML decoding from the channel output. The “modulo loss” is particularly large at low SNR, and it gets up to 4dB for scalar shaping. We consider the effect of a known interference (i.e., a dirty-paper channel) on the gap between the two decoders. We show that in the limit of a strong interference, the modulo output becomes a sufficient statistic for decoding the input. Thus, in the strong-interference regime, ML decoding suffers the same “modulo loss” as lattice decoding.