Short Message Communication (SMC) is a featured service of BeiDou Navigation Satellite System (BDS). After its successful deployment in 2003, Regional Short Message Communication (RSMC) service has been continuously serving China and its neighboring countries and regions, especially in life safety scenarios. In this paper, the architecture of the Global Short Message Communication (GSMC) system is proposed based on the medium earth orbit (MEO) constellation and the crosslinks of the global BeiDou navigation system (BDS-3). Three subtypes of GSMC service, i.e. positioning report, emergency search and rescue (SAR) and regular SMC are designed in accordance with the technical characteristic of integration of navigation and communication in BDS-3, which supports future wide applications of GSMC. The performance of the designed GSMC system is analyzed by numerical calculations. As BDS-3 was officially announced completion on July 31, 2020, GSMC has been providing initial service. First test results of the in orbit GSMC payloads are also presented in the paper to verify the designed capabilities. Preliminary results also show that the requirements of Global Maritime Distress and Safety System (GMDSS) can also be fulfilled.