236601 - Coding and Algorithms for Memories Lecture 9

1. 236601 - Coding and Algorithms for MemoriesLecture 9

2. Constrained Codes for Memories

3. Read Cycle of Flash Memories • Compare cell levels with a threshold (or a sequence of thresholds) fixed threshold 0 1 1 0 1 0 1 0

4. Balanced Codes: Motivation • Charge Leakage  voltage drift in one direction • Fixed threshold vsdynamic threshold • Dynamic reading thresholds reduces the BER • A balanced vector satisfies #0’s = #1’s

5. Balanced Codes: Motivation • 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 fixed threshold 0 1 1 0 1 0 1 0

6. Balanced Codes: Motivation • 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 fixed threshold 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 fixed

7. Balanced Codes: Motivation • 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 fixed threshold dynamic threshold 0 1 1 0 1 0 1 0 0 0 1 0 0 0 0 0 fixed dynamic 0 1 1 0 1 0 1 0

8. Balanced Codes: Problems • Problems: • How to guarantee that at most half of the cells have value 1? • How to guarantee that exactly half of the cells have value • Problem 1 for two dimensional array

9. Memristors v Resistor Capacitor Inductor i q Memristor φ L.O. Chua, “Memristor – The Missing Circuit Element,” IEEE Trans., 1971

10. Practical Memristors • 2008 Hewlett Packard ROFF Current [mA] RON Voltage [V] D.B. Strukov et al, “The missing memristor found,” Nature, 2008

11. Crossbar Arrays

12. Vg cij cij=0 high resistance  low current sensed cij=1 low resistance  high current sensed Vo RL

13. Vg 1 1 0 1 cij=0 high resistance  low current sensed cij=1 low resistance  high current sensed 1 1 Vo RL Desired Path Sneak Path

14. Sneak Path • An array A has a sneak path of length 2k+1 affecting the (i,j) cell if • aij=0a • There exist r1,…,rkand c1,…ck such that aic1 = ar1c1 = ar1c2 = ⋯ = arkck= arkj= 1a • An array A satisfies the sneak-path constraint if it has no sneak paths and then is called a sneak-path free array

15. 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

16. 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

17. 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 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 1 0

18. 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 kelements into l nonempty subsetsaka the Strirling number of second kind

19. 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)