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On the construction of perfect deletion-correcting codes using design theory

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Abstract

We consider a way to construct perfect codes capable of correcting 2 or more deletions using design-theory. As a starting point we use an (ordered) block design to construct a perfect deletion correcting code. Using this code we are able to construct more perfect deletion correcting codes over smaller or larger alphabets by removing or adding symbols in a smart way.

In this way we are able to find all perfect 2-deletion correcting codes of length 4, and all perfect 3-deletion correcting codes of length 5 with different coordinates. The perfect 3-deletion correcting codes of length 5 with repeated symbols can be constructed for almost all possible alphabet sizesv, except forv=13, 14, 15, and 16, and forv≡7, 8 (mod 10),v≥17. For these values ofv we are neither able to prove the existence, nor the non-existence of perfect 3-deletion correcting codes of length 5 over an alphabet of sizev.

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References

  1. L. Calabi and W. E. Hartnett, A family of codes for the correction of substitution and synchronization errors,IEEE Transactions on Information Theory, Vol. IT-15, No. 1 (1969) pp. 102–106.

    Google Scholar 

  2. L. Calabi and W. E. Hartnett, Some general results of coding theory with applications to the study of codes for the correction of synchronization errors.Information and Control, Vol. 15 (1969), pp. 235–249.

    Google Scholar 

  3. M. Hall, Jr., and W. S. Conner, Jr., An embedding theorem for balanced incomplete block designs,Canadian Journal of Mathematics, Vol. 6 (1954) pp. 35–41.

    Google Scholar 

  4. H. Hanani, The existence and construction of balanced incomplete block designs,Ann. Math. Stat., Vol. 32 (1961) pp. 361–386.

    Google Scholar 

  5. H. Hanani, On balanced incomplete block designs with blocks having five elements,Journal of Combinatorial Theory (A), Vol. 12 (1972) pp. 184–201.

    Google Scholar 

  6. D. H. L. Hollmann, A relation between Levenshtein-type distances and insertion-and-deletion correcting capabilities of codes,IEEE Transactions on Information Theory, Vol. IT-22, No. 4 (1993), pp. 1424–1427.

    Google Scholar 

  7. V. I. Levenshtein, Binary codes capable of correcting deletions, insertions, and reversals (Russian),Doklady Akademii Nauk SSSR, Vol. 163, No. 4 (1965) pp. 845–848; (English),Soviet Physics Doklady, Vol. 10, No. 8 (1966) pp. 707–710.

    Google Scholar 

  8. V. I. Levenshtein, Asymptotically optimum binary code with correction for losses of one or two adjacent bits,Problems of Cybernetics, Vol. 19 (1967) pp. 298–304.

    Google Scholar 

  9. V. I. Levenshtein, On perfect codes in deletion and insertion metric (Russian),Discretnaya. Mathematika, Vol. 3, No. 1 (1991) pp. 3–20; (English),Discrete Mathematics Applied, Vol. 2, No. 3 (1992) pp. 241–258.

    Google Scholar 

  10. J. H. van Lint and R. M. Wilson,A Course in Combinatorics, Cambridge University Press (1992).

  11. F. F. Sellers, Bit loss and gain correction code,IRE Transaction on Information Theory, Vol. IT-8, No. 1 (1962) pp. 35–38.

    Google Scholar 

  12. D. J. Street and J. Seberry, All DBIBDs with block size four exist,Utilitas Mathematica, Vol. 18 (1980) pp. 27–34.

    Google Scholar 

  13. D. J. Street and W. H. Wilson, On directed balanced incomplete block designs with block size five,Utilitas Mathematica, Vol. 18 (1980) pp. 161–174.

    Google Scholar 

  14. E. Tanaka and T. Kasai, Synchronization and substitution error-correcting codes for the Levenshtein metric,IEEE Transactions on Information Theory, Vol. IT-22, No. 2 (1976) pp. 156–162.

    Google Scholar 

  15. G. M. Tenengolts, Class of codes correcting bit loss and errors in the preceding bit (Russian),Avtomatika i Telemekhanika, Vol. 37, No. 5 (1976) pp. 174–179; (English),Automation and Remote Control, Vol. 37 (1976) pp. 797–802.

    Google Scholar 

  16. G. Tenegolts, Nonbinary codes, correcting single deletion or insertion,IEEE Transactions on Information Theory, Vol. IT-30, No. 5 (1984) pp. 766–769.

    Google Scholar 

  17. J. D. Ullman, Near-optimal, single-synchronization-error-correcting code,IEEE Transactions on Information Theory, Vol. IT-12 (1966) pp. 418–424.

    Google Scholar 

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  1. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Patrick A. H. Bours

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  1. Patrick A. H. Bours
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Bours, P.A.H. On the construction of perfect deletion-correcting codes using design theory. Des Codes Crypt 6, 5–20 (1995). https://doi.org/10.1007/BF01390767

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