Sparse vector coding (SVC) is emerging as a feasible solution for short packet transmission, which can provide better reliability and lower latency. In this article, a uniquely decomposable constellation group (UDCG)-based SVC called UDCG-SVC is proposed. The important idea of UDCG-SVC is to map the bits into the nonzero elements of the sparse vector by using the sub-constellations in the UDCG. After random spreading, the resource blocks carry the unique decomposable superimposed constellations, which ensures that the minimum Euclidean distance (MED) of the superimposed constellation is large. This results in an improved block error rate (BLER) performance of SVC. Numerical results demonstrate the advanced BLER simulation of UDCG-SVC, and the proposed UDCG-SVC scheme is better than constellation rotation-based SVC (CR-SVC) scheme.