  Sparse regression codes (SPARCs) are a promising coding scheme that can
approach the Shannon limit over Additive White Gaussian Noise (AWGN) channels.
Previous works have proven the capacity-achieving property of SPARCs with
Gaussian design matrices. We generalize these results to right orthogonally
invariant ensembles that allow for more structured design matrices. With the
Vector Approximate Message Passing (VAMP) decoder, we rigorously demonstrate
the exponentially decaying error probability for design matrices that satisfy a
certain criterion with the exponentially decaying power allocation. For other
spectra, we design a new power allocation scheme to show that the information
theoretical threshold is achievable.
