1. Codes for Symbol-Pair Read Channels Yuval Cassuto EPFL – ALGO Formerly: Hitachi GST Research November 3, 2010 IPG Seminar

2. Data-Storage Systems Physical Media Read/Write Interface Error Correcting Codes User Data User Data

3. Error Correcting Codes Physical Media Read/Write Interface The Key Objective

4. Physical Media Read/Write Interface The Key Objective Error Correcting Codes “The redundancy must be introduced in the proper way to combat the particular noise structure involved” C.E Shannon, “A Mathematical Theory of Communication”, 1948

5. The Key Challenge Error Correcting Codes Physical Media Read/Write Interface Combinatorial Statistical Complex Physical Mechanisms Error-Control Guarantees Multiple Reliability Issues Simple Design Objectives

6. The Key Strategy Error Correcting Codes Physical Media Read/Write Interface ErrorModel

7. Conventional Magnetic Media W R bit

8. Patterned Magnetic Media W R bit1.Reversal stability 2.Better SNR

9. Background and Motivation Read resolution < Write resolution Symbol Read Head ResponseHard

10. Pair Read Simpler Background and Motivation Read resolution < Write resolution Head Response L R

11. Symbol Read Channels Channel Codeword Received word Error Models: Random, burst, symmetric, asymmetric, hard/soft decision

12. Symbol-Pair Read Channels Channel Codeword Received word

13. Example 23514 [ (2,3) (3,5) (5,1) (1,4) (4,2) ] 0 pair-errors Design objective: t pair-errors [ (2,3) (3,4) (5,1) (2,0) (4,2) ] 2 pair-errors Wrap around

14. Related Topics Multiple burst errors ISI channels

15. Notation Stored vector Received pair-vector No error if

16. Notation Pair error Pair distance (metric) Consistent pair-vector if or (or both)

17. Some Facts Proposition: Pair-distance relation to Hamming- distance Proposition: A code can correct t pair-errors if and only if the minimum pair-distance is ≥2t+1 Theorem: With consistent-only channel outputs –Sufficient: ≥2t+1 –Necessary: ≥2t

18. A Closer Look on D p L is the number of runs of differing indices 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0

19. Construction Idea 0 0 0 0 0 1 0 1 Good 0 0 0 0 0 0 1 1 1 1 0 0 Bad Interleaving

20. Interleave vs. Direct Is interleaving optimal? Not optimal! Interleaved Hamming code (optimal)Direct Corrects 2 pair errorsCorrects 3 pair errors

21. Construction: Cyclic Codes Theorem: A cyclic code whose g(x) has at least d H roots satisfies –Proof: Dual of BCH bound

22. Proof Sketch BCH Bound [BC 1960, H 1959]: g(x) with many consecutive roots codewords with few zeros BCH Dual-Transposed: g(x) with many roots codewords without many consecutive zeros Non-zeros occupy multiple consecutive subsets (L>1) BCH Dual: codewords with many consecutive zeros g(x) with few roots

23. Applications of L>1 Theorem: Cyclic Hamming codes have d p ≥5 –Note: not true for non-cyclic Hamming codes –Corollary: cyclic Hamming codes are pair-perfect (with d p =5) d p =4 Codeword:

24. Stronger Lower Bounds on L Theorem: A cyclic code whose g(x) has at least m roots and d H ≤ min(2m-n+2,m-1) satisfies –Main proof tool: Dual of Hartmann-Tzeng bound

25. Decoding by Reduction Received pair-vector Symbol Error/Erasure Decoder

26. Decoding by Reduction Is reduction optimal? NO, example: Theorem: Optimal for interleaved codes Symbol Error/Erasure Decoder D p =1 D p =2

27. Bounds on Code Sizes Hamming SpherePair Sphere Symbol vector Non-consistent pair-vector

28. Enumerate Pair-Spheres How many symbol-words at pair-distance D p ? Pair Sphere

29. Count L-Subsets is the number of subsets of {0,…,n-1} of size l that fall into L runs (with wrap around). Definition: Key:

30. L-Run Layouts Non all-around All-around Spacer of length 1 or more Run of length 1 or more Blacks sum to l Whites sum to n - l

31. Closed Form Count Non all-around All-around

32. Pair-Sphere and Pair-Ball Pair-Sphere: Pair-Ball: Sum to h

33. Closed Form Bounds A code corrects t pair-errors only if There exists a code with minimum pair- distance d if Pair-sphere packing (upper) bound: Pair Gilbert-Varshamov (lower) bound:

34. Asymptotic Bounds Combinatorial bounds are exact for any parameter set [n,d,q] Offer little insight on asymptotic behavior Need succinct (but still tight) bounds on pair-ball volumes

35. Pair Gilbert-Varshamov

36. Asymptotic Correctability: Pair vs. Symbol dpdp d H +12d H Asymptotic pair advantage VanishingDouble

37. Pair vs. Hamming GV Bound Pair-error codes: Yes Symbol-error codes: (Probably) No

38. Graph Theoretic View Complete directed graph (with self loops) Codes with large pair-distance = Closed walks on graph with small edge overlap

39. Conclusion Initial study of symbol-pair coding Can correct significantly more pair-errors than symbol-errors Open problems in –Algebraic coding theory (e.g. cyclic codes) –Codes on graphs/trellises –Coding+detection (soft decoding)

40. To Read More [YC,Blaum], ISIT 2010 [YC,Blaum], IT-Trans Submission