Codes in Permutations and Error Correction for Rank Modulation

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.Comment: Some typos corrected from the published journal versio

On the snake-in-the-box codes for rank modulation under Kendall’s $$\tau $$ τ -metric

“…Yehezkeally and Schwartz [16] proposed one of the longest snakes in S 3 of size 3, denoted by [2,3,1]) and T C = (t 3 , t 3 , t 3 ). Next, we construct a snakeĈ 4 in S 4 from C 3 T C in S 3 using the above construction, whereĈ 4 = (π 0 ,π 1 ,π 2 ,π 3 ).…”

“…To detect and/or correct such errors we need an appropriate distance measure in the permutations. There are several metrics on the permutations such as the ∞ -metric [12,15], the Ulam metric [5] and the Kendall's τ -metric [2,11]. In this paper, we will consider only the Kendall's τ -metric.…”

Permutation codes correcting a single burst deletion I: Unstable deletions

“…Suppose the two adjacent symbols 12 and 2 are deleted, then π = (2, 6,12,1,9,5,13,4,8,10,3,7,11) is received. By checking the positions of values from [1,4] and [10,13] in π, we have R = [4,7] and R = [3,6]. That is a = k j = 4, b = 8 in Algorithm 2, and there is no h j in [a, a + s − 1] = [4,6].…”

“…1. In a stable deletion, the remaining components of the permutation gives the vector (5,3,8,4,6,7,1), whereas in an unstable deletion the remaining components of the permutation gives the vector (4, 2, 7, 3, 5, 6, 1).…”

An area and latency assessment for coding for memories with stuck cells

“…Non-volatile memory technologies are stimulating research in many areas of communication theory, as not only have they become extraordinarily important in the semiconductor industry, they also possess unique characteristics that motivate new paradigms in how these memories are written to, how one treats errors and how one trades off memory lifetime with capacity; for a sample of these works see [1]- [14].…”