  We consider the problem of unsourced random access (U-RA), a grant-free
uncoordinated form of random access, in a wireless channel with a massive MIMO
base station equipped with a large number $M$ of antennas and a large number of
wireless single-antenna devices (users). We consider a block fading channel
model where the $M$-dimensional channel vector of each user remains constant
over a coherence block containing $L$ signal dimensions in time-frequency. In
the considered setting, the number of potential users $K_\text{tot}$ is much
larger than $L$ but at each time slot only $K_a \ll K_\text{tot}$ of them are
active. Previous results, based on compressed sensing, require that $K_a < L$,
which is a bottleneck in massive deployment scenarios such as
Internet-of-Things and U-RA. In the context of activity detection it is known
that such a limitation can be overcome when the number of base station antennas
$M$ is sufficiently large and a covariance based recovery algorithm is employed
at the receiver. We show that, in the context of U-RA, the same concept allows
to achieve high spectral efficiencies in the order of $\mathcal{O}(L \log L)$,
although at an exponentially growing complexity. We show also that a
concatenated coding scheme can be used to reduce the complexity to an
acceptable level while still achieving total spectral efficiencies in the order
of $\mathcal{O}(L/\log L)$.
