The error exponent of sparse regression codes with AMP decoding

Sparse regression codes (SPARCs) are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic results by deriving a large deviations bound on the probability of AMP decoding error. This bound shows that for an appropriate choice of code parameters and any fixed rate smaller than the AWGN capacity, the probability of decoding error decays exponentially in n/(log n)
^2T
where T is the number of AMP iterations required for successful decoding. The number of iterations T is inversely proportional to the logarithm of the ratio of channel capacity to rate. For the above choice of code parameters, the complexity of the AMP decoder scales as a low-order polynomial in the block length n.