Compute-and-forward using nested linear codes for the Gaussian MAC

The classical modulo-lattice construction of Erez et al. has been successfully applied to several coding problems under Gaussian noise, including coding for computation over multiple-access channels (MAC). For the latter problem, an alternative construction can be developed by extending a recently proposed nested linear code to Gaussian case. In this note, it is shown that using the nested linear code with judiciously chosen input distributions, the original compute-and-forward result is recovered and larger computation rates are achievable. In particular we show that the Gaussian input distribution is not optimal in general for the computation problem over Gaussian MAC. Among other results, new achievable rates for the Gaussian two-way relay channel (TWRC) are given.