In this paper，the state-constrained multi-agent distributed convex optimization problem over a time-varying balanced network is investigated. In response to the limited network communication capability，a random quantizer is introduced in the agents'information interaction process to reduce data transmission effectively. Based on this，a distributed mirror descent algorithm with random quantization is proposed，and its convergence is analyzed under conventional assumptions，while the specific convergence rate is provided simultaneously. Finally，the feasibility of the developed algorithm is verified by using the distributed linear regression problem as a simulation example.