We introduce PiCoDes: a very compact image descriptor which nevertheless allows high performance on object category recognition. In particular, we address novel-category recognition: the task of defining indexing structures and image representations which enable a large collection of images to be searched for an object category that was not known when the index was built. Instead, the training images defining the category are supplied at query time. We explicitly learn descriptors of a given length (from as small as 16 bytes per image) which have good object-recognition performance. In contrast to previous work in the domain of object recognition, we do not choose an arbitrary intermediate representation, but explicitly learn short codes. In contrast to previous approaches to learn compact codes, we optimize explicitly for (an upper bound on) classification performance. Optimization directly for binary features is difficult and nonconvex, but we present an alternation scheme and convex upper bound which demonstrate excellent performance in practice. PiCoDes of 256 bytes match the accuracy of the current best known classifier for the Caltech256 benchmark, but they decrease the database storage size by a factor of 100 and speed-up the training and testing of novel classes by orders of magnitude.