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A scan-based BIST scheme is presented which guarantees complete fault coverage with very low hardware overhead. A probabilistic analysis shows that the output of an LFSR which feeds a scan path has to be modified only at a few bits in order to transform the random patterns into a complete test set. These modifications may be implemented by a bit-flipping function which has the LFSR-state as an input, and flips the value shifted into the scan path at certain times. A procedure is described for synthesizing the additional bit-flipping circuitry, and the experimental results indicate that this mixed-mode BIST scheme requires less hardware for complete fault coverage than all the other scan-based BIST approaches published so far.