Variations of Rank Modulation for Flash Memories

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1 Variations of Rank Modulation for Flash Memories Zhiying Wang Joint work with Anxiao(Andrew) Jiang Jehoshua Bruck

2 Flash Memory Control Gate Floating Gate Source Drain Substrate Block erasure X

3 Flash Memory Iterative writing Writing speed

4 Flash Memory Charge leakage 4 4

5 Flash Memory Charge leakage Data reliability 4 5

6 Rank Modulation* cell Indices 4 5 permutation: (5 4 ) Writing speed Data reliability *A. Jiang, R. Mateescu, M. Schwartz, J. Bruck, Rank Modulation for Flash Memories, 008 6

7 Decoding Rank Modulation Maximal level identification; maximal level removal For m cells, need (m-) iterations Capacity: log(5!)/5 (5 4 ) 4 5 7

8 Variation I: Partial Rank Modulation Reduce the complexity of decoding Only k iterations (k=) Capacity: log(5*4)/5 k-partial rank modulation (k-permutations): top k cells; total of m cells (5 ) (5 4 ) 4 5 8

9 Partial Rank Modulation: Updating Push-to-the-top operation

10 Gray Code for Partial Rank Modulation A cycle of all k-permutations Transition: push-to-the-top 0

11 Gray Code for Rank Modulation* m=k+= *A. Jiang, R. Mateescu, M. Schwartz, J. Bruck, Rank Modulation for Flash Memories, 008

12 Gray Code for Rank Modulation* *A. Jiang, R. Mateescu, M. Schwartz, J. Bruck, Rank Modulation for Flash Memories, 008 m=k+=

13 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter 0 Rank Modulation Counter

14 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter Rank Modulation Counter 4

15 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter Rank Modulation Counter 5

16 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter Rank Modulation Counter 6

17 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter 4 Rank Modulation Counter 7

18 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter 5 Rank Modulation Counter 8

19 Gray Code as a Counter Counter from 0 to 5 m=k+= Absolute Value Counter 0 Rank Modulation Counter 9

m=k+= Universal cycle Universal Cycles for k-permutations* A sequence (u,u,,u N ), N= m!/(m-k)! Each k-permutation is represented by exactly one (u i+,u i+,,u i+k ) * F. Chung, P. Diaconis, and R.

20 m=k+= Universal cycle Universal Cycles for k-permutations* A sequence (u,u,,u N ), N= m!/(m-k)! Each k-permutation is represented by exactly one (u i+,u i+,,u i+k ) * F. Chung, P. Diaconis, and R. Graham, Universal cycles for combinatorial structures, 99 B.W. Jackson, Universal cycles for k-subsets and k-permutations, 99 J. Robert Johnson, Universal cycles for permutations, 008 F. Ruskey and A. Williams, An explicit universal cycle for the (n )-permutations of an n-set, 009 0

21 Universal Cycles for k-permutations m=k+=

22 Example: m=4, k=

23 Example: m=4, k=

24 Construction of Gray Code Theorem: There exists a Gray code for k-partial rank modulation, for all 0<k<m. For a given permutation, the next pushto-the-top operation can be decided in time O(k log(k)) on average. 4

25 Partial Rank Modulation: Summary Reduce decoding complexity k-permutations out of m cells Gray code of partial rank modulation Given permutation size m, what can we do if we have more than m cell levels? 5

26 Variation II: Bounded Rank Modulation Permutation size m Maximum cell level D>m 6

27 Bounded Rank Modulation: Example 8 cells; cell levels {,,,4,5,6}; permutation size 4; , , 4 Only 4 levels needed Can we do better? , 4, 4 overlap All 6 levels needed 7

28 Computing the Capacity Cell levels {,,,4,5,6,7,8,9,0} ; 8 cells; permutation size cap = log(4! ) / 8 = cap = log(4! (4 ) ) / 8 = cap = log(4! 4 )/ 8 = 57. 8

29 Overlap Increases Capacity Theorem: cap(m,d,v=)>cap(m,d,v=0), for given m>, D>m+ m : permutation size D: maximum cell level v: overlap 9

30 Encoding Max level D=4, size m=, overlap v= message permutation cell level ()=, ()=0 00/000 Convolutional Rate = :4 000/0 00/00 00/00 0/00 0/00 /0 000/000 00/00 00/00 0/00 0/00 /0 000/000 00/00 00/00 0/00 0/00 /000 II I III 00/000 0/0 00/000 0/0 0/0 0

31 Decoding Max level D=4, size m=, overlap v= cell level permutation message First Two Permutation Bits Permutation Sequence Information Sequence Equal Not equal

32 Summary Partial rank modulation Less sorting iterations Smaller permutation size Gray code Bounded rank modulation Given permutation size & maximal cell level Overlap and capacity Encoder and decoder

33 Open Problems Partial rank modulation Error-correcting codes using lower levels as redundancy Efficient mapping between permutations and values Gap distributions after push-to-the-top Bounded rank modulation Exact optimal overlap values for m<d< Error-correcting codes Efficient encoding/decoding

34 Thank you! 4

Error-Correcting Codes for Rank Modulation

ISIT 008, Toronto, Canada, July 6-11, 008 Error-Correcting Codes for Rank Modulation Anxiao (Andrew) Jiang Computer Science Department Texas A&M University College Station, TX 77843, U.S.A. ajiang@cs.tamu.edu