Least squares superposition codes of moderate dictionary size, reliable at rates up to capacity

Sparse superposition codes are developed for the additive white Gaussian noise channel with average codeword power constraint. Codewords are linear combinations of subsets of vectors, with the possible messages indexed by the choice of subset. Decoding is by least squares, tailored to the assumed form of linear combination. Communication is shown to be reliable with error probability exponentially small for all rates up to the Shannon capacity.