  Sparse code multiple access (SCMA), as a codebook-based non-orthogonal
multiple access (NOMA) technique, has received research attention in recent
years. The codebook design problem for SCMA has also been studied to some
extent since codebook choices are highly related to the system's error rate
performance. In this paper, we approach the downlink SCMA codebook design
problem by formulating an optimization problem to maximize the minimum
Euclidean distance (MED) of superimposed codewords under power constraints.
While SCMA codebooks with a larger minimum Euclidean distance (MED) are
expected to obtain a better BER performance, no optimal SCMA codebook in terms
of MED maximization, to the authors' best knowledge, has been reported in the
SCMA literature yet. In this paper, a new iterative algorithm based on
alternating maximization with exact penalty is proposed for the MED
maximization problem. The proposed algorithm, when supplied with appropriate
initial points and parameters, achieves a set of codebooks of all users whose
MED is larger than any previously reported results. A Lagrange dual problem is
derived which provides an upper bound of MED of any set of codebooks. Even
though there is still a nonzero gap between the achieved MED and the upper
bound given by the dual problem, simulation results demonstrate clear
advantages in error rate performances of the proposed set of codebooks over all
existing ones. The correctness and accuracy of error curves in the simulation
results are further confirmed by the coincidences with the theoretical upper
bounds of error rates derived for any given set of codebooks.
