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Many of the distributed video coding (DVC) systems described in the literature make use of a feedback channel from the decoder to the encoder to determine the rate. However, the number of requests through the feedback channel is often high, and as a result the overall delay of the system could be unacceptable in practical applications. As a solution, feedback-free DVC systems have been proposed, but the problem with these solutions is that they incorporate a difficult trade-off between encoder complexity and compression performance. Recognizing that a limited form of feedback may be supported in many video-streaming scenarios, in this paper we propose a method for constraining the number of feedback requests to a fixed maximum number of N requests for an entire Wyner-Ziv (WZ) frame. The proposed technique estimates the WZ rate at the decoder using information obtained from previously decoded WZ frames and defines the N requests by minimizing the expected rate overhead. Tests on eight sequences show that the rate penalty is less than 5% when only five requests are allowed per WZ frame (for a group of pictures of size four). Furthermore, due to improvements from previous work, the system is able to perform better than or similar to DISCOVER even when up to two requests per WZ frame are allowed. The practical usefulness of the proposed approach is studied by estimating end-to-end delay and encoder buffer requirements, indicating that DVC with constrained feedback can be an important solution in the context of video-streaming scenarios.