Rank Modulation for Flash Memories

We explore a novel data representation scheme for multi-level flash memory cells, in which a set of $n$ cells stores information in the permutation induced by the different charge levels of the individual cells. The only allowed charge-placement mechanism is a “push-to-the-top” operation which takes a single cell of the set and makes it the top-charged cell. The resulting scheme eliminates the need for discrete cell levels, as well as overshoot errors, when programming cells. We present unrestricted Gray codes spanning all possible $n$-cell states and using only “push-to-the-top” operations, and also construct balanced Gray codes. We also investigate optimal rewriting schemes for translating arbitrary input alphabet into $n$-cell states which minimize the number of programming operations.