A capacity-achieving coding scheme for the AWGN channel with polynomial encoding and decoding complexity

A fundamental problem in coding theory is the design of an efficient coding scheme that achieves the capacity of the additive white Gaussian (AWGN) channel. In this article, we study a simple capacity-achieving nested lattice coding scheme whose encoding and decoding complexities are polynomial in the blocklength. Specifically, we show that by concatenating an inner nested lattice code with an outer Reed-Solomon code over an appropriate finite field, we can achieve the capacity of the AWGN channel. The main feature of this technique is that the encoding and decoding complexities grow as O(N²), while the probability of error decays exponentially in N, where N denotes the blocklength. We also show that this gives us a recipe to extend a high-complexity nested lattice code for a Gaussian channel to low-complexity concatenated code without any loss in the asymptotic rate. As examples, we describe polynomial-time coding schemes for the wiretap channel, and the compute-and-forward scheme for computing integer linear combinations of messages.