  This paper presents an enhanced design of multi-dimensional (MD)
constellations which play a pivotal role in many communication systems such as
code-domain non-orthogonal multiple access (CD-NOMA). MD constellations are
attractive as their structural properties, if properly designed, lead to signal
space diversity and hence improved error rate performance. Unlike the existing
works which mostly focus on MD constellations with large minimum Euclidean
distance (MED), we look for new MD constellations with additional feature that
the minimum product distance (MPD) is also large. To this end, a non-convex
optimization problem is formulated and then solved by the convex-concave
procedure (CCCP). Compared with the state-of-the-art literature, our proposed
MD constellations lead to significant error performance enhancement over
Rayleigh fading channels whilst maintaining almost the same performance over
the Gaussian channels. To demonstrate their application, we also show that
these MD constellations give rise to good codebooks in sparse code multiple
access systems. All the obtained MD constellations can be found in
https://github.com/Aureliano1/Multi-dimensional-constellation.
