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236601 - Coding and Algorithms for

MemoriesLecture 9

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Constrained Codes for Memories

•Compare cell levels with a threshold (or a sequence of thresholds)

0 1 1 0 1 0 1 0

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fixed threshold

Read Cycle of Flash Memories

•Charge Leakage voltage drift in one direction•Fixed threshold vs dynamic threshold• Dynamic reading thresholds

reduces the BER• A balanced vector satisfies

#0’s = #1’s

Balanced Codes: Motivation

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Balanced Codes: Motivation

0 1 1 0 1 0 1 0

•In writing, half of cells store 0 and the other half store 1•In reading, the n/2 cells with lower voltages are read as 0 The other n/2 cells with higher voltages are read as 1•Relative ranking is most likely preserved

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fixed threshold

•In writing, half of cells store 0 and the other half store 1•In reading, the n/2 cells with lower voltages are read as 0 The other n/2 cells with higher voltages are read as 1•Relative ranking is most likely preserved

Balanced Codes: Motivation

0 1 1 0 1 0 1 0

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0 0 1 0 0 0 0 0fixed

fixed threshold

•In writing, half of cells store 0 and the other half store 1•In reading, the n/2 cells with lower voltages are read as 0 The other n/2 cells with higher voltages are read as 1•Relative ranking is most likely preserved

Balanced Codes: Motivation

0 1 1 0 1 0 1 0

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dynamic threshold

0 0 1 0 0 0 0 00 1 1 0 1 0 1 0

fixeddynamic

fixed threshold

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Balanced Codes: Problems• Problems:

1. How to guarantee that at most half of the cells have value 1?

2. How to guarantee that exactly half of the cells have value

3. Problem 1 for two dimensional array

Memristors

9L.O. Chua, “Memristor – The Missing Circuit Element,” IEEE Trans., 1971

( , )v M x i i

( , )dx f x idt

Resistor

v R i

Capacitor

q C v

Inductor

L i

Memristor

φ

q

v

i

M q

Practical Memristors• 2008 Hewlett Packard

10D.B. Strukov et al, “The missing memristor found,” Nature, 2008

2( ) 1 ( )v ONOFF

RM q R q t

D

RON

ROFF

Voltage [V]

Curr

ent [

mA]

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Crossbar Arrays

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Vg

RL

Vo

cijcij=0 high resistance low current sensedcij=1 low resistance high current sensed

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Vg

RL

Vo

0cij=0 high resistance low current sensedcij=1 low resistance high current sensed

1

1

1 1

Desired PathSneak Path

1

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Sneak Path• An array A has a sneak path of length 2k+1 affecting the

(i,j) cell if– aij=0

a

– There exist r1,…,rk and c1,…ck such thataic1 = ar1c1 = ar1c2 = ⋯ = arkck = arkj = 1

a

• An array A satisfies the sneak-path constraint if it has no sneak paths and then is called a sneak-path free array

1 10 1

1 1

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Characterization of Sneak Paths• An array A has an isolated zero-rectangle if it

contains a rectangle with exactly a single zero• An array satisfies the isolated zero-rectangle

constraint if it has no isolated zero-rectangles and is called an isolated zero-rectangle free array

• Theorem: The sneak path constraint and the isolated zero-rectangle constraint are equivalent

1 10 1

1 1

1 11 0 1

1 1

1 10 0 1

1 1

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Characterization of Sneak Paths• An array A has an isolated zero-rectangle if there is a rectangle with

exactly a single zero• An array satisfies the isolated zero-rectangle constraint if it has no

isolated zero-rectangles and is called an isolated zero-rectangle free array

• Theorem: The sneak path constraint and the isolated zero-rectangle constraint are equivalent

• Lemma: An array is an isolated zero-rectangle free array iff the 1s in every two rows either completely overlap or are disjoint

1 10 1

1 1

1 11 0 1

1 1

1 10 0 1

1 1

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Characterization of Sneak Paths• Theorem: The sneak path constraint and the

isolated zero-rectangle constraint are equivalent• Lemma: An array is an isolated zero-rectangle

free array iff the 1s in every two rows either completely overlap or are disjoint

1 1 1

1 1 1

0 00

0 00

1

1

0 0 0 0 0

0 0 0 0 001 0 0 0 0

00 0 0 0 0

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Number of Sneak Paths Arrays

• N(m,n) = number of mⅹn isolated 0-rectangle free arrays

• Lemma 1:

• Lemma 2:

• S(k,l) = number of ways to partition k elements into l nonempty subsetsaka the Strirling number of second kind

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Number of Sneak Paths Arrays

• N(m,n) = number of mⅹn isolated 0-rectangle free arrays

• Lemma 1:

• Lemma 2:

• N(m,n) ≈ (m+n)log(m+n)