Gray codes over certain run-length sequences for local rank modulation

Abstract

In the local rank modulation (LRM) scheme, a sliding window produces a sequence of permutations by moving over a sequence of variables. LRM has been presented as a method of storing data in flash memory, which represents a natural generalization of the classical rank modulation scheme. In this paper, we present a study on Gray codes over certain run-length sequences for the $(1,2,n)$-LRM scheme to simulate virtual multilevel flash memory cells while maintaining the advantages of LRM. Unlike previous studies on the LRM scheme, we present Gray codes over certain run-length sequences in the $(1,2,n)$-LRM scheme. This class of Gray codes can overcome the drawback of the many distinct charge levels required in the rank modulation scheme and in certain Gray codes for LRM. Furthermore, we demonstrate that the proposed codes have an asymptotically optimal rate.