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236601 - Coding and Algorithms for Memories Lecture 7

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236601 - Coding and Algorithms for MemoriesLecture 7

- 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 1st

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Can only add water (charge)

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Error!

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Need to erase the whole block

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Slow…

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Less errors

More retention

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Jiang, Mateescu, Schwartz, Bruck, “Rank modulation for Flash Memories”, 2008

Absolute values Relative values

Single cell Multiple cells

Physical cell Logical cell

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Ordered set of n cells

Assume discrete levels

Relative levels define a permutation

Basic operation: push-to-the-top

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Overshoot is not a concern

Writing is much faster

Increased reliability (data retention)

permutation in

lexicographical order

FACTORADIC

decimal

[Lehmer 1906, Laisant 1888]

0 1 2

0 2 1

1 0 2

1 2 0

2 0 1

2 1 0

0 0

0 1

1 0

1 1

2 0

2 1

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an-1…a3a2a1 = an-1·(n-1)! + … +a3·3!+ a2·2!+a1·1! 0 ≤ai ≤i

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The problem: Is it possible to transition between all permutations?

Find cycle through n! states

by push-to-the-top transitions

n=3

3 cycles

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Transition graph, n=3

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~ (n-1)!

- Recursive construction:
- Keep bottom cell fixed
- (n-1)! transitions with others

1 3 2 3 1 2

2 1 3 2 3 1

3 2 1 1 2 3

444444

4 1 2 1 4 2

2 4 1 2 1 4

1 2 4 4 2 1

333333

3 4 2 4 3 2

2 3 4 2 4 3

4 2 3 3 2 4

1 1 1 1 1 1

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1 3 2 3 1 2

2 1 3 2 3 1

3 2 1 1 2 3

444444

4 1 2 1 4 2

2 4 1 2 1 4

1 2 4 4 2 1

333333

3 4 2 4 3 2

2 3 4 2 4 3

4 2 3 3 2 4

111111

4 1 3 4 3 1

1 4 1 3 4 3

3 3 4 1 1 4

222222

- 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: Sn×[M] -> Sn
- Decoding function D: Sn-> [M]

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- Definition: The cost of changing s1 into s2,α(s1->s2), is the min number of push-to-the-top operations needed to change s1 to s2
- Ex: α([123]->[213]) = 1, α([123]->[321]) = 2

- The rewriting cost of a RMRC is the maximum update cost
- The transition graph Gn=(Vn,En)
- Vn = Sn, En ={(s1,s2) : α(s1->s2)=1}

- The ballor radius r: Br(s)={ s’ : α(s->s’) ≤ r }
- The sphereor radius r: Sr(s)={ s’ : α(s->s’) = r }
- The balls and the sphere sizes do not depend on rBr,Sr

- For n,M, define r(n,M) to be the smallest integer such that Br(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