  Providing connectivity to a massive number of devices is a key challenge in
5G wireless systems. In particular, it is crucial to develop efficient methods
for active device identification and message decoding in a multi-cell network
with fading and path loss uncertainties. In this paper, we design such a scheme
using second-order Reed-Muller (RM) sequences. For given positive integer $m$,
a codebook is generated with up to $2^{m(m+3)/2}$ codewords of length $2^m$,
where each codeword is a unique RM sequence determined by a matrix-vector pair
with binary entries. This allows every device to send $m(m+3)/2$ bits of
information where an arbitrary number of these bits can be used to represent
the identity of a node, and the remaining bits represent a message. There can
be up to $2^{m(m+3)/2}$ devices in total. Using an iterative algorithm, an
access point can estimate the matrix-vector pairs of each nearby device, as
long as not too many devices transmit simultaneously. To improve the
performance, we also describe an enhanced RM coding scheme with slotting. We
show that both the computational complexity and the error performance of the
latter algorithm exceed another state-of-the-art algorithm. The device
identification and message decoding scheme developed in this work can serve as
the basis for grant-free massive access for billions of devices with hundreds
of simultaneously active devices in each cell.
