  In massive connectivity scenarios with short packets, of interest is the
regime where users share wireless resources in a non-orthogonal fashion. Small
payloads combined with sporadic user activation call for approaches that
jointly address the users access to the shared resources and the design of the
channel code. In this paper, we propose a transmission scheme that combines
sparse signatures with finite-length forward error correction (FEC) coding for
non-orthogonal massive access. Our signature design is based on Euler squares,
which are special instances of quasi-cyclic partial geometries that yield
sparse graphs with favorable decoding properties. Following a graph-theoretic
approach, we explicate the benefits of the coding scheme for the receiver
processing that involves joint user detection and decoding. The proposed
construction is flexible and can be explicitly characterized for a large number
of combinations of system parameters, suitable for both grant-based and
grant-free massive access. Finally, unlike common existing schemes, our scheme
can be applied to unsourced random access (U-RA). We numerically characterize
the trade-off between system parameters such as number of users, load and
channel coding rate. The performance evaluation against the state of the art
illustrates the potential of the scheme to provide an energy-efficient solution
for U-RA.
