Generalized Gray codes for local rank modulation

We consider the local rank-modulation scheme in which a sliding window going over a sequence of real-valued variables induces a sequence of permutations. Local rank-modulation is a generalization of the rank-modulation scheme, which has been recently suggested as a way of storing information in flash memory. We study Gray codes for the local rank-modulation scheme in order to simulate conventional multi-level flash cells while retaining the benefits of rank modulation. Unlike the limited scope of previous works, we consider code constructions for the entire range of parameters including the code length, sliding window size, and overlap between adjacent windows. We show our constructed codes have asymptotically-optimal rate. We also provide efficient encoding, decoding, and next-state algorithms.