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The conventional test-per-scan built-in self-test (BIST) scheme needs a number of shift cycles followed by one capture cycle. Fault effects received by the scan flip-flops are shifted out while shifting in the next test vector, like scan testing. Unlike deterministic testing, it is unnecessary to apply a complete test vector to the scan chains. A new scan-based BIST scheme is proposed by properly controlling the scan enable signals of the scan chains. Different weighted values are assigned to the scan enable signals of scan flip-flops in separate scan chains. Capture cycles can be inserted at any clock cycle if necessary. A new testability estimation procedure according to the proposed testing scheme is presented. A greedy procedure is proposed to select a weight for each scan chain. Experimental results show that the proposed method can improve test effectiveness of scan-based BIST greatly and most circuits can obtain complete fault coverage or very close to complete fault coverage.