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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.