  Sparse superposition codes were originally proposed as a capacity-achieving
communication scheme over the gaussian channel, whose coding matrices were made
of i.i.d. gaussian entries.We extend this coding scheme to more generic
ensembles of rotational invariant coding matrices with arbitrary spectrum,
which include the gaussian ensemble as a special case. We further introduce and
analyse a decoder based on vector approximate message-passing (VAMP).Our main
findings, based on both a standard replica symmetric potential theory and state
evolution analysis, are the superiority of certain structured ensembles of
coding matrices (such as partial row-orthogonal) when compared to i.i.d.
matrices, as well as a spectrum-independent upper bound on VAMP's threshold.
Most importantly, we derive a simple "spectral criterion " for the scheme to be
at the same time capacity-achieving while having the best possible algorithmic
threshold, in the "large section size" asymptotic limit. Our results therefore
provide practical design principles for the coding matrices in this promising
communication scheme.
