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.