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Optimal difference systems of sets and partition-type cyclic difference packings

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Abstract

Difference systems of sets (DSSs) are combinatorial structures which were introduced by Levenshtein in connection with code synchronization. In this paper, we give some recursive constructions of DSSs by using partition-type cyclic difference packings, and obtain new infinite classes of optimal DSSs.

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References

  1. Arasu K.T., Ding C., Helleseth T., Kumar D.V., Martinsen H.M.: Almost difference sets and their sequences with optimal autocorrelation. IEEE Trans. Inform. Theory 47, 2934–2943 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Chang Y., Ding C.: Constructions of external difference families and disjoint difference families. Des. Codes Cryptogr. 40, 167–185 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Colbourn C.J., Dinitz J.H.: The CRC Handbook of Combinatorial Designs. CRC, Boca Raton, FL (1996)

    Book  MATH  Google Scholar 

  4. Cumming L.J.: A family of circular systematic comma-free codes. J. Comb. Math. Comb. Comput. 58, 87–96 (2006)

    Google Scholar 

  5. Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland/Elsevier, Amsterdam, The Netherlands (1998). North-Holland Mathematical Libary 55.

  6. Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Cryptogr. 17, 224–235 (1999)

    Article  MathSciNet  Google Scholar 

  7. Dillon J.F., Dobbertin H.: Cyclic difference sets with singer parameters. manuscript (1999).

  8. Ding C.: Optimal and perfect difference systems of sets. J. Comb. Theory (A) 116, 109–119 (2009)

    Article  MATH  Google Scholar 

  9. Ding C., Helleseth T., Lam K.Y.: Several classes of sequences wiht three-level autocorrelation. IEEE Trans. Inform. Theory 45, 2606–2612 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ding C., Helleseth T., Martinsen H.M.: New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Inform. Theory 47, 428–433 (1992)

    Article  MathSciNet  Google Scholar 

  11. Dinitz J., Stinson D.R.: Difference sets, in Contemporary Design Theory: A Collection of Surveys. Wiley, New York (1992)

    Google Scholar 

  12. Fan C.L., Lei J.G., Chang Y.X.: Constructions of difference systems of sets and disjoint difference families. IEEE Trans. Inform. Theory 54, 3195–3201 (2008)

    Article  MathSciNet  Google Scholar 

  13. Fuji-Hara R., Miao Y., Mishima M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inform. Theory. 50, 2408–2420 (2004)

    Article  MathSciNet  Google Scholar 

  14. Fuji-Hara R., Munemasa A., Tonchev V.D.: Hyperplane partitions and difference systems of sets. J. Comb. Theory (A) 113, 1699–1718 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ge G., Fuji-Hara R., Miao Y.: Further combinatorial constructions for optimal frequency-hopping sequences. J. Comb. Theory (A) 113, 1699–1718 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962)

    Article  MATH  MathSciNet  Google Scholar 

  17. Lempel A., Greenberger H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inform. Theory 20, 90–94 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  18. Levenshtein V.I.: One method of constructing quasilinear codes providing synchronization in the presence of errors. Prob. Inform. Transm. 7, 215–222 (1971)

    Google Scholar 

  19. Jungnickel D., Pott A. et al.: Difference sets. In: Pott, A. (eds) An Introduction to Difference Sets, Sequences and their Correlation Properties, vol. 542, pp. 259–295. Kluwer Academic Publishers, Netherlands (1999)

    Google Scholar 

  20. Muton Y., Tonchev V.D.: Difference systems of sets and cyclotomy. Discret. Math. to appear. In: Pott A., Kumar P.V., Helleseth T., Jungnickel D. (eds.) Properties, pp. 259–295. Kluwer, Amsterdam, The Netherlands (1999).

  21. Ogata W., Kurosawa K., Stinson D.R., Saido H.: New combinatorial designs and their applications to authentication codes and secret sharing schemes. Discret. Math. 279, 383–405 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  22. Pott A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, p. 1601. Springer-Verlag, Berlin, Germany (1995).

  23. Tonchev V.D.: Difference systems of sets and code synchronization. Rend. Sem. Mat. Messina. (II) 9, 217–226 (2003)

    MathSciNet  Google Scholar 

  24. Tonchev V.D.: Partitions of difference sets and code synchronization. Finite Fields. Appl. 11, 601–621 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tonchev V.D., Wang H.: An algorithm for optimal difference systems of sets. J. Com. Optim. 14, 165–175 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  26. Wang H.: A new bound for difference systems of sets. J. Comb. Math. Comb. Comput. 58, 161–168 (2006)

    MATH  Google Scholar 

  27. Xiang Q.: Recent results on difference sets with classical parameters. In: Pott, A., Kumar, P.V., Helleseth, T., Jungnickel, D. (eds) Difference Sets, Sequences and Their Correlation Properties, pp. 419–434. Kluwer, Amsterdam, The Netherlands (1999)

    Google Scholar 

  28. Zhang Y., Lei J.G., Zhang S.P.: A new family of almost difference sets and some necessary conditions. IEEE Trans. Inform. Theory 52, 2052–2061 (2006)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Beijing Jiaotong University, Beijing, 100044, China

    Jianguo Lei

  2. Institute of Mathematics, Hebei Normal University, Shijiazhuang, Hebei, 050016, China

    Jianguo Lei & Cuiling Fan

Authors
  1. Jianguo Lei
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  2. Cuiling Fan
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Correspondence to Cuiling Fan.

Additional information

Communicated by V.D. Tonchev.

Research supported in part by the NSFC Grant 10771051 and 10831002.

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Lei, J., Fan, C. Optimal difference systems of sets and partition-type cyclic difference packings. Des. Codes Cryptogr. 58, 135–153 (2011). https://doi.org/10.1007/s10623-010-9388-6

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  • DOI: https://doi.org/10.1007/s10623-010-9388-6

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