  Grant-free access is a key enabler for connecting wireless devices with low
latency and low signaling overhead in massive machine-type communications
(mMTC). For massive grant-free access, user-specific signatures are uniquely
assigned to mMTC devices. In this paper, we first derive a sufficient condition
for the successful identification of active devices through maximum likelihood
(ML) estimation in massive grant-free access. The condition is represented by
the coherence of a signature sequence matrix containing the signatures of all
devices. Then, we present a design framework of non-orthogonal signature
sequences in a deterministic fashion. The design principle relies on unimodular
masking sequences with low correlation, which are applied as masking sequences
to the columns of the discrete Fourier transform (DFT) matrix. For example
constructions, we use four polyphase masking sequences represented by
characters over finite fields. Leveraging algebraic techniques, we show that
the signature sequence matrix of proposed non-orthogonal sequences has
theoretically bounded low coherence. Simulation results demonstrate that the
deterministic non-orthogonal signatures achieve the excellent performance of
joint activity and data detection by ML- and approximate message passing
(AMP)-based algorithms for massive grant-free access in mMTC.
