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Coding for the Correction of Synchronization Errors

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

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Coding for the Correction of Synchronization Errors

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  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?

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