Codes in Permutations and Error Correction for Rank Modulation

TL;DR

This paper investigates permutation-based codes for rank modulation in flash memory, deriving bounds and establishing the optimal size scaling of such codes for error correction.

Contribution

It provides new bounds and exact scaling laws for permutation codes under Kendall tau distance, advancing error correction in rank modulation.

Findings

Derived bounds on code sizes for rank modulation

Established exact scaling laws for large n

Proved existence of near-optimal codes within constant factors

Abstract

Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations of $n$ elements equipped with the Kendall tau distance. We derive several lower and upper bounds on the size of codes. These bounds enable us to establish the exact scaling of the size of optimal codes for large values of $n$. We also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors.