1. 236601 - Coding and Algorithms for Memories Lecture 7 1

2. Class Overview What have we studied so far? – Background on memories – Flash memories: properties, structure and constraints – Rewriting codes – WOM codes – Other Rewriting Codes What’s next? – Rank modulation codes – ECC and constrained codes – Wear leveling & memory management – Coding for Storage HW 2 – due May 1 st 2

3. Flash Memory Cell 1 0 3 2 Can only add water (charge) 3

4. The Leakage Problem 1 0 3 2 Error! 4

5. The Overshooting Problem 1 0 3 2 Need to erase the whole block 5

6. Possible Solution – Iterative Programming Slow… 1 0 3 2 6

7. Relative Vs. Absolute Values 01 Less errors More retention Jiang, Mateescu, Schwartz, Bruck, “Rank modulation for Flash Memories”, 2008 7

8. The New Paradigm Rank Modulation Absolute values  Relative values Single cell  Multiple cells Physical cell  Logical cell 8

9. Rank Modulation 3 2 1 4 1234 Ordered set of n cells Assume discrete levels Relative levels define a permutation Basic operation: push-to-the-top Overshoot is not a concern Writing is much faster Increased reliability (data retention) 9

10. New Number Representation System permutation in lexicographical order [Lehmer 1906, Laisant 1888] FACTORADIC decimal a n-1 …a 3 a 2 a 1 = a n-1 ·(n-1)! + … + a 3 ·3! + a 2 ·2! + a 1 ·1! 0 ≤ a i ≤ i 0 1 2 0 2 1 1 0 2 1 2 0 2 0 1 2 1 0 0 0 1 1 0 1 2 0 2 1 012345012345 10

11. 231231 312312 123123 132132 213213 321321 Gray Codes for Rank Modulation Find cycle through n! states by push-to-the-top transitions 231231 312312 123123 132132 213213 321321 Transition graph, n=3 n=3 3 cycles 2 3 1 123 The problem: Is it possible to transition between all permutations? 11

12. Gray Codes for Arbitrary n Recursive construction: – Keep bottom cell fixed – (n-1)! transitions with others 1234 4 ~ (n-1)! 2 3 1 231231 312312 123123 132132 213213 321321 12 1 3 2 3 1 2 2 1 3 2 3 1 3 2 1 1 2 3 4 4 4 4 1 2 1 4 2 2 4 1 2 1 4 1 2 4 4 2 1 3 3 3 3 4 2 4 3 2 2 3 4 2 4 3 4 2 3 3 2 4 1 1 1

13. Gray Codes for Arbitrary n 1 3 2 3 1 2 2 1 3 2 3 1 3 2 1 1 2 3 4 4 4 4 1 2 1 4 2 2 4 1 2 1 4 1 2 4 4 2 1 3 3 3 3 4 2 4 3 2 2 3 4 2 4 3 4 2 3 3 2 4 1 1 1 4 1 3 4 3 1 1 4 1 3 4 3 3 3 4 1 1 4 2 2 2 13

14. Rewriting with Rank Modulation If we represent n! symbols then in the worst case we apply n-1 push-to-the-top operations to transfer from one permutation to another Problem: Is it possible to use less push-to-the- top operations in case less than n! symbols are represented? Rank Modulation Rewriting code (RMRC) (n,M) consists of – Update function: E: S n ×[M] -> S n – Decoding function D: S n -> [M] 14

15. Rewriting with Rank Modulation Definition: The cost of changing s 1 into s 2, α(s 1 ->s 2 ), is the min number of push-to-the-top operations needed to change s 1 to s 2 – Ex: α([123]->[213]) = 1, α([123]->[321]) = 2 The rewriting cost of a RMRC is the maximum update cost The transition graph G n =(V n,E n ) – V n = S n, E n ={(s 1,s 2 ) : α(s 1 ->s 2 )=1} The ball or radius r: B r (s)={ s’ : α(s->s’) ≤ r } The sphere or radius r: S r (s)={ s’ : α(s->s’) = r } The balls and the sphere sizes do not depend on r B r,S r 231231 312312 123123 132132 213213 321321 231231 312312 123123 132132 213213 321321 15

16. Rewriting with Rank Modulation For n,M, define r(n,M) to be the smallest integer such that B r(n,M) ≥ M Lemma (Lower Bound): For any RMRC (n,M), its rewriting cost is at least r(n,M) Upper bound on the rewriting cost is given by a construction 16