236601 coding and algorithms for memories lecture 7
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236601 - Coding and Algorithms for Memories Lecture 7. 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

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236601 - Coding and Algorithms for Memories Lecture 7

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236601 coding and algorithms for memories lecture 7

236601 - Coding and Algorithms for MemoriesLecture 7


Class overview

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 1st


Flash memory cell

Flash Memory Cell

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Can only add water (charge)

0


The leakage problem

The Leakage Problem

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Error!

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The overshooting p roblem

The Overshooting Problem

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Need to erase the whole block


Possible solution iterative programming

Possible Solution – Iterative Programming

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Slow…

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Relative vs absolute values

Relative Vs. Absolute Values

Less errors

More retention

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


The new paradigm rank modulation

The New Paradigm Rank Modulation

Absolute values Relative values

Single cell Multiple cells

Physical cell  Logical cell


Rank modulation

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Rank Modulation

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)


New number representation system

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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New Number Representation System

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


Gray codes for rank modulation

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Gray Codes for Rank Modulation

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


Gray codes for arbitrary n

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~ (n-1)!

Gray Codes for Arbitrary n

  • 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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Gray codes for arbitrary n1

Gray Codes for Arbitrary n

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


Rewriting with rank modulation

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: Sn×[M] -> Sn

    • Decoding function D: Sn-> [M]


Rewriting with rank modulation1

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Rewriting with Rank Modulation

  • 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


Rewriting with rank modulation2

Rewriting with Rank Modulation

  • 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


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