Error-Correction in Flash Memories via Codes in the Ulam Metric

TL;DR

This paper introduces translocation rank codes for flash memories, extending error correction capabilities beyond classical models by handling arbitrary charge-drop errors through combinatorial and coding theory techniques.

Contribution

It develops the theory of translocation rank codes, deriving their asymptotic capacity, providing construction methods, and proposing simple decoding algorithms.

Findings

Derived the asymptotic capacity of translocation rank codes

Constructed asymptotically good codes for error correction

Presented simple decoding methods for a class of these codes

Abstract

We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed \emph{translocation rank codes} account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity of translocation rank codes, construction techniques for asymptotically good codes, as well as simple decoding methods for one class of constructed codes. As part of our exposition, we also highlight the close connections between the new code family and permutations with short common subsequences, deletion and insertion error-correcting codes for permutations, and permutation codes in the Hamming distance.