Coding for the Correction of
Synchronization Errors
ASJ Helberg
CUHK Oct 2010
Content
• Background
– Synchronization errors and their effects
• Previous approaches
– Resynchronization
– Concatenation
– Error correction
• Algebraic insertion/deletion correction
– Single error correcting
– Multiple error correcting
• Problems and applications
Synchronization errors
• Due to timing or other noise and inaccuracies,
• Manifests as the insertion or deletion of symbols
• Examples:
– PPM (Pulse position modulation) in optic fibres,
– Terabit per square inch magnetic recording
– Optic disc recording due to ISI and ITI
– Multipath effects in radio
Types of Synchronization errors
• Insertion or deletion, excluding additive errors
– Additive errors are special case of deletion and
insertion in same position of bits of opposite value
– Repetition/duplication error: copies bit
– Bit/peak shift: 01 becomes 10
– Bit/peak shift of size a: 0a1 becomes 10a
Effects
• A single synchronization error causes a catastrophic
burst of additive errors
Tx:
Rx:
0001010011011110
0001100001110
• Boundaries of data blocks are unknown to receiver
e.g. 001100 becomes 0000
Synchronizable codes
• Comma-free codes
• Prefix synchronised codes
• Bounded synchronisation delay
• Synchronisation with timing
• Marker codes
• Corrupted blocks are discarded!!
Sync error correcting codes
• Binary Algebraic block codes
• Nonbinary and perfect codes
• Bursts of sync errors
• Weak synchronization errors
• Convolutional codes
• Expurgated codes (Reed Muller/ LDPC)
Binary Algebraic block codes
• Varshamov Tenengoltz construction:
• One asymmetric error
Levenshtein codes
With 2n > m >= n + 1, s=1 correcting code
With m >= 2n, s=1 and t=1 correcting code
Partition a=0 was proven to have the maximum
cardinality
Largest common subword obtained from two valid
codewords is
Example
i =4 3 2 1
ix
ixi  a mod(n+1);
0
1
2
3
4
4
5
6
4
5
6
7
7
8
9
10
0
1
2
3
4
4
0
1
4
0
1
2
2
3
4
0
i
x =0 0 0 0
x =0 0 0 1
x =0 0 1 0
x =0 0 1 1
x =0 1 0 0
x =0 1 0 1
x =0 1 1 0
x =0 1 1 1
x =1 0 0 0
x =1 0 0 1
x =1 0 1 0
x =1 0 1 1
x =1 1 0 0
x =1 1 0 1
x =1 1 1 0
x =1 1 1 1
Hamming distance properties of Levenshtein
codes
• Proposition 1 : A Levenshtein code C has only one code word of
either weight w = 0 or weight w = 1.
• Proposition 2 : In a Levenshtein code there is a minimum Hamming
distance, dmin  2 between any two code words.
• Proposition 3 : Code words in a Levenshtein code have a dmin  4 if
they have the same weight.
• Proposition 4 : Levenshtein code words that differ in one unit of
weight have dmin  3.
Weight distance diagram
0
dmin = 2
dmin = 4
2
dmin = 3
dmin = 4
3
dmin = 4
n-3
dmin = 3
dmin = 4
n-2
dmin = 2
n
Generalised structure
Proposition 5
• Code words of weights w = 0, 1, 2, ..., s do not occur together in an
s - correcting code.
Proposition 6
• The minimum Hamming distance of an s - insertion/deletion
correcting code is dmin  s + 1.
• Again, the proof of propositions 5 and 6, is straight forward when
considering the resulting subwords after s deletions.
Proposition 7
• Any two number theoretic s - insertion/deletion correcting code
words which differ in weight by i, 0  i  s, have a Hamming distance
of d  2(s + 1) - i.
dmin
= (w2 - x) + (w1 - x)
= w2 + w1 - 2x
= w2 + w2 - w - 2x
= 2(w2 - x) - w
From Proposition 6, dmin  s + 1 corresponding to number of “1’s” by which
w2 differ from w1 i.e. (w2 - x) thus d  2(s + 1) - i
Weight-distance diagram
0
dmin = s+1
dmin = 2(s+1)
s+1
dmin = 2s+1
dmin = 2(s+1)
s+2
dmin = 2(s+1) - i
dmin = 2(s+1)
ns+2
dmin = 2s+1
dmin = 2(s+1)
ns+1
dmin = s+1
n
Bounds for the general algebraic construction
• Lower bound on s correction capability
• Upper Bound on correction capability
Upper bound on Cardinality
• Hamming type upper bound
General algebraic construction
Modified Fibonacci
•
•
•
•
•
•
•
•
S=1:
1, 2, 3, 4, 5, 6, 7, …
S=2
1, 2, 4, 7, 12, 20, 33, …
S=3:
1, 2, 4, 8, 14, 23, 38, …
S=4:
1, 2, 4, 8, 16, 31, 60, …
• Partitioning 2n into , thus in limit, cardinality bounded by
•
2n / m with
(non-empty partitions)
Example
v =7 4 2 1
vx
vx  a mod(m); m=12
x =0 0 0 0
x =0 0 0 1
x =0 0 1 0
x =0 0 1 1
x =0 1 0 0
x =0 1 0 1
x =0 1 1 0
x =0 1 1 1
x =1 0 0 0
x =1 0 0 1
x =1 0 1 0
x =1 0 1 1
x =1 1 0 0
x =1 1 0 1
x =1 1 1 0
x =1 1 1 1
0
7
4
11
2
9
6
13
1
8
5
12
3
10
7
14
0
7
4
11
2
9
6
1
1
8
5
0
3
10
7
2
Cardinalities
Word length
n
4
5
6
7
8
9
10
11
12
13
14
s=2
s=3
s=4
s=5
2
2
3
4
5
6
8
9
11
15
18
2
2
2
2
3
4
4
5
6
8
8
2
2
2
2
2
3
4
4
4
5
2
2
2
2
2
2
3
4
4
Problems
• Very low cardinalities
• Does not scale well
• No decoding algorithm
• Codeword boundaries assumed
• Validity not proven in general
Cardinality bounds
• Levenshtein
Lower bounds on the capacity of the binary
deletion channel
A Kirsch and E Drinea, “Directly lower bounding the information capacity for
channels with i.i.d. deletions and duplications, IEEE Transactions on Information
Theory, vol. 56, no. 1, January 2010, pp 86-102
Lower bounds on the capacity of the binary
deletion channel
Connection with network coding?
• Synchronization in NC environments is assumed
• Especially on physical layer NC
• “Pruned/punctured” codes may be useful ?
• Superimposed codes that are also sync error correcting?