Least Squares Superposition Codes of Moderate Dictionary Size, Reliable at Rates up to Capacity

TL;DR

This paper introduces a new coding method using sparse superpositions and least squares decoding for the Gaussian noise channel, achieving reliable communication up to the Shannon capacity.

Contribution

It presents a novel coding scheme with sparse superpositions and least squares decoding that attains reliable communication at rates approaching capacity.

Findings

Error probability is exponentially small at rates up to capacity.

The coding method is effective for moderate dictionary sizes.

Decoding by least squares is tailored to the linear superposition structure.

Abstract

For the additive white Gaussian noise channel with average codeword power constraint, new coding methods are devised in which the codewords are sparse superpositions, that is, linear combinations of subsets of vectors from a given design, 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.