Ranking and Un-ranking Calculations for Constant-Weight Gray code on Local Rank Modulation

The local rank modulation scheme, a generalization of the rank modulation scheme, is proposed for accurate and effective data representation in flash memory. In this scheme, a sliding window traversing a sequence of real-valued variables yields a sequence of permutations. We focus on constant-weight Gray codes for the scheme. We represent two constructions for Gray codes of weights 2 and 3 proposed in [6] by grouping transition rules to create more easily calculable paths. From that we give calculations for the ranking and un-ranking on these Gray codes. Additionally, we show that the calculations are efficient by showing their running time in constant and logarithmic time for Gray codes of weights 2 and 3 respectively.