Local rank modulation for flash memories

Local rank modulation scheme was suggested recently for representing information in flash memories in order to overcome drawbacks of rank modulation. For 0 <; s ≤ t ≤ n with s divides n, an (s, t, n)-LRM scheme is a local rank modulation scheme where the n cells are locally viewed cyclically through a sliding window of size t resulting in a sequence of small permutations which requires less comparisons and less distinct values. The gap between two such windows equals to s. In this work, encoding, decoding, and asymptotic enumeration of the (1, 3, n)-LRM scheme is studied. The techniques which are suggested have some generalizations for (1, t, n)-LRM, t > 3, but the proofs will become more complicated. The enumeration problem is presented also as a purely combinatorial problem. Finally, we prove the conjecture that the size of a constant weight (1, 2, n)-LRM Gray code with weight two is at most 2n.