Codes Correcting Synchronization Errors for Symbol-Pair Read - PowerPoint PPT Presentation

Codes Correcting Synchronization Errors for Symbol-Pair Read Channels Presented by: Van Khu Vu(isevvk@nus.edu.sg) Joint work with Yeow Meng Chee National University of Singapore June 2020 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Symbol-Pair Read Channels Definition 1. [ Cassuto and Blaum 2011 ] ◮ Let x = ( x 1 , x 2 , . . . , x n ) ∈ Σ n be a sequence of length n over the alphabet Σ . We define the pair-mapping, π : Σ n �→ (Σ , Σ) n , where π ( x ) = (( x 1 , x 2 ) , . . . , ( x n − 1 , x n ) , ( x n , x 1 )) . The pair-vector π ( x ) is called the symbol-pair read vector of x . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Symbol-Pair Read Channels Definition 1. [ Cassuto and Blaum 2011 ] ◮ Let x = ( x 1 , x 2 , . . . , x n ) ∈ Σ n be a sequence of length n over the alphabet Σ . We define the pair-mapping, π : Σ n �→ (Σ , Σ) n , where π ( x ) = (( x 1 , x 2 ) , ( x 2 , x 3 ) , . . . , ( x n − 1 , x n ) , ( x n , x 1 )) . The pair-vector π ( x ) is called the symbol-pair read vector of x . ◮ A pair-vector v ∈ (Σ , Σ) n is called consistent if there exists its corresponding vector, denoted x ∈ Σ n , such that π ( x ) = v . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Symbol-Pair Read Channels Example ◮ Let x = ( 0 , 1 , 0 , 0 , 1 ) be a binary word of length 5. The pair-vector π ( x ) = (( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 )) is called the symbol-pair read vector of x . ◮ The pair-vector v = (( 0 , 1 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 )) is NOT consistent. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Racetrack Memory with Two Adjacent Heads c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 Head 2 : c 2 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Racetrack Memory with Two Adjacent Heads c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 , c 2 Head 2 : c 2 , c 3 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Racetrack Memory with Two Adjacent Heads c 3 c 4 c 7 c 8 c 1 c 2 c 5 c 6 Head 1 : c 1 , c 2 , . . . , c 7 Head 2 : c 2 , c 3 , . . . , c 8 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Racetrack Memory with Two Adjacent Heads c 3 c 4 c 7 c 8 c 1 c 2 c 5 c 6 Head 1 : c 1 , c 2 , . . . , c 7 Head 2 : c 2 , c 3 , . . . , c 8 ◮ The pair-vector π ( c ) = (( c 1 , c 2 ) , ( c 2 , c 3 ) , . . . , ( c 7 , c 8 )) is the output in two heads. ◮ The stored vector c = ( c 1 , c 2 , . . . , c 8 ) . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Under-shift Error c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 Head 2 : c 2 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Under-shift Error c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 , c 1 Head 2 : c 2 , c 2 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Under-shift Error c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 , c 1 , c 2 Head 2 : c 2 , c 2 , c 3 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Under-shift Error c 3 c 4 c 7 c 8 c 1 c 2 c 5 c 6 Head 1 : c 1 , c 1 , c 2 , . . . , c 7 Head 2 : c 2 , c 2 , c 3 , . . . , c 8 ◮ The pair-vector u = (( c 1 , c 2 ) , ( c 1 , c 2 ) , ( c 2 , c 3 ) , . . . , ( c 7 , c 8 )) is the output in two heads. ◮ The error in this case is a sticky-insertion of a pair. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Over-shift Error c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 Head 2 : c 2 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Over-shift Error c 1 c 2 c 3 c 4 c 5 c 6 c 7 c 8 Head 1 : c 1 , c 3 Head 2 : c 2 , c 4 Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Over-shift Error c 3 c 4 c 7 c 8 c 1 c 2 c 5 c 6 Head 1 : c 1 , c 3 , . . . , c 7 Head 2 : c 2 , c 4 , . . . , c 8 ◮ The pair-vector v = (( c 1 , c 2 ) , ( c 3 , c 4 ) , . . . , ( c 7 , c 8 )) is the output in two heads. ◮ The error in this case is a deletion of a pair. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Errors for Symbol-pair Read Channels Definition 2. Given a pair-vector π ( x ) = (( x 1 , x 2 ) , . . . , ( x n , x 1 )) ∈ (Σ , Σ) n . ◮ A deletion is an event that a pair ( x i , x i + 1 ) is deleted from the pair-vector. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Errors for Symbol-pair Read Channels Definition 2. Given a pair-vector π ( x ) = (( x 1 , x 2 ) , . . . , ( x n , x 1 )) ∈ (Σ , Σ) n . ◮ A deletion is an event that a pair ( x i , x i + 1 ) is deleted from the pair-vector. ◮ Two deletions are called separate-deletions if they are not adjacent. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Errors for Symbol-pair Read Channels Definition 2. Given a pair-vector π ( x ) = (( x 1 , x 2 ) , . . . , ( x n , x 1 )) ∈ (Σ , Σ) n . ◮ A deletion is an event that a pair ( x i , x i + 1 ) is deleted from the pair-vector. ◮ Two deletions are called separate-deletions if they are not adjacent. ◮ A sticky-insertion is an event that a pair ( x i , x i + 1 ) is inserted at location i + 1, right after the original pair ( x i , x i + 1 ) in the pair-vector. It is also known as repetition or duplication. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Errors for Symbol-pair Read Channels Definition 2. Given a pair-vector π ( x ) = (( x 1 , x 2 ) , . . . , ( x n , x 1 )) ∈ (Σ , Σ) n . ◮ A deletion is an event that a pair ( x i , x i + 1 ) is deleted from the pair-vector. ◮ Two deletions are called separate-deletions if they are not adjacent. ◮ A sticky-insertion is an event that a pair ( x i , x i + 1 ) is inserted at location i + 1, right after the original pair ( x i , x i + 1 ) in the pair-vector. It is also known as repetition or duplication. ◮ A substitution is an event that a bit x i is read wrongly in one head and become x ′ i � = x i . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Errors for Symbol-pair Read Channels Example Let x = ( 0 , 1 , 0 , 0 , 1 , 0 , 0 , 0 ) be a stored sequence. The symbol-pair read sequence of x is π ( x ) = (( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 0 ) , ( 0 , 0 )) . ◮ If there are two separate-deletions at location i = 2 and j = 6, we obtain the pair-vector u = (( 0 , 1 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 0 )) . ◮ If there is a substitution at the second bit in the second pair and a deletion at location 6, we obtain the pair-vector v = (( 0 , 1 ) , ( 1 , 1 ) , ( 0 , 0 ) , ( 0 , 1 ) , ( 1 , 0 ) , ( 0 , 0 ) , ( 0 , 0 )) . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Main Contributions GOALS: ◮ Construct a good code correcting multiple deletions. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Main Contributions GOALS: ◮ Construct a good code correcting multiple deletions. ◮ Construct a good code correcting a combination of deletions and substitutions. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Main Contributions Theorem There exists an asymptotically optimal code correcting t separate-deletions in the symbol-pair read channel with redundancy at most t log n + o ( t log n ) . Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels

Main Contributions Theorem There exists an asymptotically optimal code correcting t separate-deletions in the symbol-pair read channel with redundancy at most t log n + o ( t log n ) . Lemma Given t > 1 , let C t ⊂ Σ n be a code correcting t sticky-insertions. The code C t can correct t separate-deletions in the symbol-pair read channel. Presented by: Van Khu Vu(isevvk@nus.edu.sg) Codes Correcting Synchronization Errors for Symbol-Pair Read Channels