Institute for Experimental Mathematics Ellernstrasse Essen - Germany On STORAGE Systems A.J. Han Vinck June 2004

University Duisburg-Essendigital communications group A.J. Han Vinck content We consider coding aspects of Write Once Memory Memory with defects Magneto-Optical memory Lesson: CODING is MORE THAN ERROR- CORRECTION !

University Duisburg-Essendigital communications group A.J. Han Vinck 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 ?

University Duisburg-Essendigital communications group A.J. Han Vinck EXAMPLE: Card with 3 positions FIRST TRANSMISSION PUNCH 1 hole  log 2 3 bits SECOND TRANSMISSION PUNCH 2 bits TOGETHER: (log 2 3+2)/3 = 1.2 bits per position > 1!!

University Duisburg-Essendigital communications group A.J. Han Vinck 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

University Duisburg-Essendigital communications group A.J. Han Vinck 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

University Duisburg-Essendigital communications group A.J. Han Vinck No-No and No-Yes No-No p/2 p/2 Defect agrees with prob. 1/2 No-Yes correct: no defect 0 known as 0 defect 1 Known as 1 defect 1 correct: no defect p/2 1-p p/2

University Duisburg-Essendigital communications group A.J. Han Vinck EXAMPLE: maximum of 1 defect in a word of length 3 STORE: for defect defect 1 defect 0 00 YES-NO

University Duisburg-Essendigital communications group A.J. Han Vinck STORE: In general: N-1 bits in N positions  Efficiency = 1 - defect N YES-NO

University Duisburg-Essendigital communications group A.J. Han Vinck 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 = N OPTIMAL!

University Duisburg-Essendigital communications group A.J. Han Vinck 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!

University Duisburg-Essendigital communications group A.J. Han Vinck How to solve t defects in N positions In the Yes-No situation?

University Duisburg-Essendigital communications group A.J. Han Vinck SOLUTION: Construct matrix C PROPERTIES: 1.Any t pattern is present in some row of C 2.Rows uniquely represented by n-k first digits … n -kk CODE C

University Duisburg-Essendigital communications group A.J. Han Vinck 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?????? information C Any t pattern is present in some row of C: thus this is possible

University Duisburg-Essendigital communications group A.J. Han Vinck RESULT: for t  n-k defects; k n n -kk R = = 1- = 1- n-k n t n decoding C  XC  XPrefix deterimines C C  X  C=X information Problem left: construction of matrix C Optimal!

University Duisburg-Essendigital communications group A.J. Han Vinck 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

University Duisburg-Essendigital communications group A.J. Han Vinck 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!)

University Duisburg-Essendigital communications group A.J. Han Vinck C = Efficiency 1/2 < 1 - 2/6 X’ = 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

University Duisburg-Essendigital communications group A.J. Han Vinck 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

University Duisburg-Essendigital communications group A.J. Han Vinck Example a = 3 2a-1 = 5  10 columns All zero row added Specifies each row uniquely Efficiency: 7/10 < 1-2/10 = 8/10

University Duisburg-Essendigital communications group A.J. Han Vinck Principle of magneto-optical disk Magnetic material Electro magnet Magnetic spot Heated and re- magnetized +-+- High power laser beam

University Duisburg-Essendigital communications group A.J. Han Vinck Read and erase cycle Low power laser beam Polarization of reflected beam depends on direction of magnetization -+-+ High power laser beam readingerase

University Duisburg-Essendigital communications group A.J. Han Vinck 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:

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

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

University Duisburg-Essendigital communications group A.J. Han Vinck 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)

University Duisburg-Essendigital communications group A.J. Han Vinck 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

University Duisburg-Essendigital communications group A.J. Han Vinck Principle of CD-RW Phase-change technology R ecording C Erase- 200 C (for a sufficient time back to the crystalline state) [Ag-Sb-In-Te]

University Duisburg-Essendigital communications group A.J. Han Vinck 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