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We present a method of transforming local image descriptors into a compact form of bit-sequences whose similarity is determined by Hamming distance. Following the locality-sensitive hashing approach, the descriptors are projected on a set of random directions that are learned from a set of non-matching data. The learned random projections result in high-entropy binary codes (HE2) that outperform codes based on standard random projections in match/non-match classification and nearest neighbor search. Despite of data compression and granularity of Hamming space, HE2-descriptor outperforms the original descriptor in the classification task. In nearest neighbor search task, the performance of the HE2-descriptor is asymptotic to the performance of the original descriptor. As a supporting result, we obtain another descriptor, HE2 + 1, and demonstrate that the performance of the original descriptor can be improved by adding a few bits derived from the descriptor itself.