We introduce the data model BM, which specifies kernels of motifs by means of Boolean matrices. Different from position frequency matrices these only specify which bases can appear in which position of a motif instance. Boolean matrices describe motifs still more precisely than models based on consensus strings and Hamming distance and thus allow keeping the number of false positives low. Based on the BM model we introduce a new algorithm BMA for motif discovery that attempts to reduce the number of false positives as much as possible. The main idea is to start with small kernels of motif instances and iteratively enlarge the kernels with sets of strings which have a high expected signal to noise ratio, thus keeping the signal to noise ratio as high as possible. Only after a kernel of substantial size has been found, we use position frequency matrices and add in the final stage of the algorithm BMA strings that are close to the kernel to form the final list of potential motif instance...