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The High Energy cosmic-Radiation Detection (HERD) facility is an under construction space astronomy and particle astrophysics experiment in collaboration between China and Europe, and will run on the China Space Station for more than 10 years since 2027. HERD is designed to search for dark matter with unprecedented sensitivity, investigate the century-old mystery of the origin of cosmic rays, conduct high-sensitivity surveys and monitoring of high-energy gamma rays, and explore new methods of pulsar navigation. Once operational, HERD experiment is expected to generate and distribute more than 90 PB of data over a span of 10 years. To share the data and computing resource for data processing and analysis, a distributed computing system based on grid computing technology is developed with the supports of computing and storage sites in China and Europe. HERD distributed computing infrastructure (DCI) integrates a distributed computing system managing data processing jobs based on DIRAC and dHTC, a distributed data management system distributing raw and produced data based on Rucio, and other grid computing middleware including IAM, FTS3, etc. Furthermore, HERD DCI is designed to be deeply involved with HERD Offline Software and data production workflow, providing grid computing services without normal and production users being aware of it.