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Local Rank Modulation for Flash Memories II

Local rank modulation scheme was suggested recently for representing information in flash memories in order to overcome drawbacks of rank modulation. For $0 < s\leq t\leq n$ 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 $2n$.