Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

We investigate error-correcting codes for a the rank-modulation scheme with an application to flash memory devices. In this scheme, a set of $n$ cells stores information in the permutation induced by the different charge levels of the individual cells. The resulting scheme eliminates the need for discrete cell levels, overcomes overshoot errors when programming cells (a serious problem that reduces the writing speed), and mitigates the problem of asymmetric errors. In this paper, we study the properties of error-correcting codes for charge-constrained errors in the rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one—a distance measure known as Kendall's $\tau$-distance. We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall's $\tau$-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound.