# Correcting Limited-Magnitude Errors in the Rank-Modulation Scheme

## TL;DR

This paper introduces error-correcting codes for permutations in flash memory storage, specifically addressing spike errors with limited magnitude, and provides constructions, bounds, and decoding methods that outperform existing bounds.

## Contribution

The paper presents new limited-magnitude rank-modulation codes with optimal constructions, recursive encoding/decoding, and bounds, advancing error correction in flash memory storage schemes.

## Key findings

- Constructed several LMRM codes with optimal properties.
- Provided bounds on code sizes for general and subgroup cases.
- Codes outperform Gilbert-Varshamov-like bounds asymptotically.

## Abstract

We study error-correcting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories call rank modulation. In this scheme, a set of $n$ flash cells are combined to create a single virtual multi-level cell. Information is stored in the permutation induced by the cell charge levels. Spike errors, which are characterized by a limited-magnitude change in cell charge levels, correspond to a low-distance change under the infinity norm. We define codes protecting against spike errors, called limited-magnitude rank-modulation codes (LMRM codes), and present several constructions for these codes, some resulting in optimal codes. These codes admit simple recursive, and sometimes direct, encoding and decoding procedures. We also provide lower and upper bounds on the maximal size of LMRM codes both in the general case, and in the case where the codes form a subgroup of the symmetric group. In the asymptotic analysis, the codes we construct out-perform the Gilbert-Varshamov-like bound estimate.