Codes correcting erasures and deletions for rank modulation

Error-correcting codes for permutations have received a considerable attention in the past few years, especially in applications of the rank modulation scheme for flash memories. While several metrics have been studied like the Kendall´s τ, Ulam, and Hamming distances, no recent research has been carried for erasures and deletions over permutations. The problems studied in this paper are motivated by a hardware implementation of the rank modulation codes. If the flash memory cells represent a permutation, which is modulated by their relative charge levels, then we explore the problems arise when some of the cells are either erased or deleted. In each case we study how these erasures and deletions affect the information carried by the remaining cells. In particular, the cells can either be stable and do not change their values in the permutation or unstable where the remaining cells form an induced permutation with less symbols. Yet another erasure model, called here soft erasures, assumes that all cells can be read, however the relative levels between some of the cells is not known. Our main approach in tackling these problems is to build upon the existing works of error-correcting codes in the three metrics mentioned above and leverage them in order to construct codes in each model of deletions and erasures. Lastly, we follow up on codes in the Ulam distance and improve upon the state of the art results.