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

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

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

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

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

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

Conventional Magnetic Media W R bit

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

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

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

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

Symbol-Pair Read Channels Channel Codeword Received word

Example [ (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

Related Topics Multiple burst errors ISI channels

Notation Stored vector Received pair-vector No error if

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

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

A Closer Look on D p L is the number of runs of differing indices

Construction Idea Good Bad Interleaving

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

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

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

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:

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

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

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

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

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

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:

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

Closed Form Count Non all-around All-around

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

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:

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

Pair Gilbert-Varshamov

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

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

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

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)

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