1. Institute for Experimental Mathematics Ellernstrasse 29 45326 Essen - Germany On STORAGE Systems A.J. Han Vinck January 2011

2. A.J. Han Vinck, 2011 content We consider coding aspects of Write Once Memory Memory with defects Magneto-Optical memory Lesson: CODING IS MORE THAN ERROR- CORRECTION !

3. A.J. Han Vinck, 2011 WRITE ONCE MEMORY Example: IBM punchcardpunching a hole is destructive Obvious method: Use card only once Efficiency: 1 bit/cell hole or not More complicated method: Use card T times Efficiency:  log 2 (T+1) bits per cell WHY ?

4. A.J. Han Vinck, 2011 EXAMPLE: Card with 3 positions FIRST TRANSMISSION PUNCH 1 hole  log 2 3 bits SECOND TRANSMISSION PUNCH 2 bits 00 01 10 11 TOGETHER: (log 2 3+2)/3 = 1.2 bits per position > 1!!

5. A.J. Han Vinck, 2011 Memories with known defects (ROM-type ) Problem: output fixed and cell is useless! correct stuck-at 0 stuck-at 1 Assumptions: Cell stuck-at with probability p

6. A.J. Han Vinck, 2011 Memories with known defects (ROM-type) WRITER knows? Yes Yes 1-p READER knows? Storage Capacity per cell WHY? Yes No 1-p defect cells are not used Additive Coding invented by: Kuznetsov/Tsybakov 4 situations No Yes 1-pdefect found as erasure probability(e) = p No No 1-h(p/2) defect is random error; probability(error) = p/2 the result is a BSC

7. A.J. Han Vinck, 2011 No-No and No-Yes No-No0101 0101 1-p/2 p/2 Defect agrees with prob. 1/2 No-Yes 0101 0 correct: no defect 0 known as 0 defect 1 Known as 1 defect 1 correct: no defect p/2 1-p p/2

8. A.J. Han Vinck, 2011 EXAMPLE: maximum of 1 defect in a word of length 3 STORE: for defect 1 01 10 11 defect 1 defect 0 00 YES-NO

9. A.J. Han Vinck, 2011 STORE: 01 10 11 In general: N-1 bits in N positions  Efficiency = 1 - defect 0 00 1 N YES-NO

10. A.J. Han Vinck, 2011 General strategy for one defect Given the defect value: store word X or complement X‘ of X -X‘ or X can be stored error free In general: N-1 bits in N positions  Efficiency = 1 - 1 N OPTIMAL!

11. A.J. Han Vinck, 2011 General strategy for N-1 defects Given N-1 defects in word of length N 1 position left to determine the parity of a word transmit 1:= odd parity 0:= even parity OPTIMAL! Efficiency = 1/N = 1 – (N-1)/N OPTIMAL!

12. A.J. Han Vinck, 2011 Complex cell with IN-OUT short caused by excessive Metal2 defect

13. A.J. Han Vinck, 2011 How to solve t defects in N positions the Yes-No situation

14. A.J. Han Vinck, 2011 SOLUTION:Construct matrix C PROPERTIES: Any t pattern is present in some row of C Rows uniquely represented by n-k first digits 0000000 0000001 … 1111111 n -kk CODE C

15. A.J. Han Vinck, 2011 RESULT: for t  n-k defects; k n n -kk X R = = 1- = 1- n-k n t n encoding Defects: d Add codeword C such that C  X agrees with d at defect position ??1?0 0????000?????? 00100 001110001111000 00000 information C Any t pattern is present in some row of C: thus this is possible

16. A.J. Han Vinck, 2011 RESULT: for t  n-k defects; k n n -kk R = = 1- = 1- n-k n t n decoding 00100 010101100101010 C  XC  XPrefix deterimines C C  X  C=X 00000 information Problem left: construction of matrix C Optimal!

17. A.J. Han Vinck, 2011 We store 3 bits of information in 6 locations info X written as X’ (C 1,C 2,C 3,C 4,C 5,C 6 ) (0,0, 0,X 1,X 2,X 3 ) (C 1,C 2,C 3,S 4,S 5,S 6 ) add modulo-2 the code vector C(d) = STORE X’ C(d) = R(d,X) =  Example: 2 defects in 6 positions

18. A.J. Han Vinck, 2011 PROPERTY: The components of R(d,X) are equal to the 2 given defects at the defect location for any defect pair (condition 1 on code C, covering) DECODING: Calculate C(d) R(d,X) = C(d) C(d) X’  we obtain X (condition 2 on code, uniqueness!)

19. A.J. Han Vinck, 2011 0 0 0 0 0 1 1 1 0 0 1 0 1 0 1 1 1 1 1 2 3 C = 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 1 1 4 5 6 Efficiency 1/2 < 1 - 2/6 X’ = 0 0 0 1 0 0 R(d,X) = (1 1 0, 1 0 1) or ? d = 1 _ _ _ 0 _ we decide to add row 3 In GENERAL CODES CAN BE CONSTRUCTED with EFFICIENCY  1-2/n example

20. A.J. Han Vinck, 2011 Construction for t = 2 C consists of 1. All binary (2a-1) tuples of weight a as columns 2. Additional all-zero row Properties: 1. Any two defects present in some row 2. Every row can be specified by  log2a  bits Proof left to the reader

21. A.J. Han Vinck, 2011 Proof Part 1 Two columns must have the pair (1,1) –there must be overlap since a > (2a-1)/2 Columns are different: hence the combinations (0,1) and (1,0) must be present. The all zero row takes care of the combination (0,0)

22. A.J. Han Vinck, 2011 Proof part 2 Take the matrix of 2 log2a rows and log 2 (2a) columns Taken out any two non-zero complementary rows until 2a rows are left. The remaining columns have weight a, and each row is unique, and thus can be used as identification for a specific row of the code

23. A.J. Han Vinck, 2011 Bound on rate or efficiency

24. A.J. Han Vinck, 2011 Construction t = 2, a = 2 Code: 0 0 00 1 1 1 0 1 =>1 0 11 1 0 row index

25. A.J. Han Vinck, 2011 Example a = 3 2a-1 = 5  10 columns 0 0 0 0 1 1 1 1 1 1 0 1 1 1 0 0 0 1 1 1 1 0 1 1 0 1 1 0 0 1 1 1 0 1 1 0 1 0 1 0 1 1 1 0 1 1 0 1 0 0 0 0 0 0 0 All zero row added Specifies each row uniquely Efficiency: 7/10 < 1-2/10 = 8/10

26. A.J. Han Vinck, 2011 THE GENERAL SOLUTION PROPERTIES: Any t pattern is present in some row of C Rows uniquely represented by n-k first digits 0000000 0000001 … 1111111 n -kk CODE C

27. A.J. Han Vinck, 2011 Bound on performance We calculate the fraction of matrices than are not good for the masking of t defects If this fraction is less than 1, there exist matrices that are masking t defects We use the general Kutznetsov method 0000000 0000001 … 1111111 n -kk CODE C

28. A.J. Han Vinck, 2011 Proof of bound 0000000 0000001 … 1111111 n -kk CODE 2 n-k rows u defects v = t-u defects 2 n-k-u rows can match the defect A F = # of incorrect matrices /total # of possible choises = # of wrong choises for A x # of remaining free choises / total #

29. A.J. Han Vinck, 2011 Proof of bound (2)

30. A.J. Han Vinck, 2011 Choose a random matrix Calculate the probability that there is a defect that cannot be matched Choose the bits in the matrix with probability ½ Add the systematic part

31. A.J. Han Vinck, 2011 Using the dual of linear block encoder G HTHT I n-k Property: For the matrix H T any d min - 1 rows are linarly independent Thus, H can be used as a code generator for the defect matching Since any pattern of t = d min - 1 defects can be generated by H

32. A.J. Han Vinck, 2011 Combining defects and random errors Ranndom errors can be included. We give some examples 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 1 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 0 0 1 0 1 0 1 0 0 0 10 1 1 1 0 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 0 1 1 3 defects1 defect and 1 error2 defects and 1 error R = 3/7R = 3/7R = 1/7

33. A.J. Han Vinck, 2011 Principle of magneto-optical disk Magnetic material Electro magnet Magnetic spot Heated and re- magnetized +-+- High power laser beam

34. A.J. Han Vinck, 2011 Read and erase cycle Low power laser beam Polarization of reflected beam depends on direction of magnetization -+-+ High power laser beam readingerase

35. A.J. Han Vinck, 2011 MAGNETO-OPTICAL MEMORY APPLICATION: MINI DISK WRITING PROCESS: first ERASE then WRITE erase write erase write   EFFICIENCY:.5 bit per cycle/cell QUESTIONS:Can we do better? How? How much? IMPROVEMENT: CHANGE WRITING STRATEGY:

36. A.J. Han Vinck, 2011 Problem: change in magnetic field direction slow Simple solution with efficiency = 1/2 Start: t 0 t odd t even t odd 0 0 0 0|0 0 0 0 1 0 1 0|1 1 1 1 0 0 0 0|1 0 1 1 0 1 0 1|1 1 1 1 write clean clean write Can we do better than ½ ?

37. A.J. Han Vinck, 2011 ONE APPROACH: S LOOK at PRESENT WORD or STATE S CHOOSEWRITE or ERASE Example: words of length N = 4, # of messages M = 7 a = 0 0 0 0 b = 0 0 0 or 0 c = 0 0 0 or 0 d = 0 0 0 or 0 e = 0 0 0 or 0 f = 0 0 or 0 0 g = S S SUPPOSE S = 0 0 Check that we can write the strings 0 0 0 0 0 0 0 0 0 STORAGE CAPACITY = bits/cell log 2 7 4 For n  C = 0.69 bits/cell/cycle < 1 !

38. A.J. Han Vinck, 2011 Example:6 messages, word length N = 5 Messages present at ERASE WRITE EXAMPLE: write erase PROPERTY: From ANY word(message) at erase we may write ANY messsage(word) and vice versa Efficiency is log 2 6/5 =.517 bits/cell! (n = 11 gives.53 b/c and M=58) ANOTHER APPROACH (a challenge!)

39. A.J. Han Vinck, 2011 Phase Change model CD-Rewritable CD has crystalline compound spot heated up to T1  spot crystalline (cool slow) spot heated up to T2>> T1  spot amorphous (cool fast) - amorphous area reflects less than crystalline area:  0 or 1 - back to crystalline by heating up to TE: T1 < TE << T2

40. A.J. Han Vinck, 2011 Principle of CD-RW Phase-change technology R ecording- 500-700C Erase- 200 C (for a sufficient time back to the crystalline state) [Ag-Sb-In-Te]

41. A.J. Han Vinck, 2011 Phase Change RecordingIn a rewritable DVD, different intensities of the laser change the bits in the phase change recording layer from a crystalline state to an amorphous state and vice versa.

42. A.J. Han Vinck, 2011 WOM write once memory (Rivest, 1983) WUM write unidirectional (Willems Vinck, 1986) WIM write inhibited memory (Cohen, 1998) WEM write efficient memory (Ahlswede, 1990) WAM write address fault memory (Fuja, 1995) Overview of research done