1. Coding for the Correction of Synchronization Errors ASJ Helberg CUHK Oct 2010

2. 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

3. 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

4. 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

5. Effects • A single synchronization error causes a catastrophic burst of additive errors Tx: 0001010011011110 Rx: 0001100001110 • Boundaries of data blocks are unknown to receiver e.g. 001100 becomes 0000

6. Synchronizable codes • Comma-free codes • Prefix synchronised codes • Bounded synchronisation delay • Synchronisation with timing • Marker codes • Corrupted blocks are discarded!!

7. 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)

8. Binary Algebraic block codes • VarshamovTenengoltz construction: • One asymmetric error

9. 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

10. Example

11. 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.

12. Weight distance diagram 0 dmin = 2 2 dmin = 4 dmin = 3 3 dmin = 4 n-3 dmin = 4 dmin = 3 n-2 dmin = 4 dmin = 2 n

13. 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.

14. 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

15. Weight-distance diagram 0 dmin = s+1 s+1 dmin = 2(s+1) dmin = 2s+1 s+2 dmin = 2(s+1) dmin = 2(s+1) - i n-s+2 dmin = 2(s+1) dmin = 2s+1 n-s+1 dmin = 2(s+1) dmin = s+1 n

16. Bounds for the general algebraic construction • Lower bound on s correction capability

17. Upper Bound on correction capability

18. Upper bound on Cardinality • Hamming type upper bound

19. General algebraic construction

20. 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)

21. Example

22. Cardinalities

23. Problems • Very low cardinalities • Does not scale well • No decoding algorithm • Codeword boundaries assumed • Validity not proven in general

24. Cardinality bounds • Levenshtein

25. 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

26. Lower bounds on the capacity of the binary deletion channel

27. 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?