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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 $s\leq t\leq n$ with $s|n$, $(s,t,n)$-LRM scheme is a local rank modulation scheme where the $n$ cells are locally viewed through a sliding window of size $t$ resulting in a sequence of small permutations which requires less comparisons and less distinct values. The distance between two windows equals to $s$. To get the simplest hardware implementation the case of sliding window of size two was presented. Gray codes and constant weight Gray codes were presented in order to exploit the full representational power of the scheme. In this work, a tight upper-bound for cyclic constant weight Gray code in $(1,2,n)$-LRM scheme where the weight equals to $2$ is given. Encoding, decoding and enumeration of $(1,3,n)$-LRM scheme is studied.