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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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)
Chang Y., Ding C.: Constructions of external difference families and disjoint difference families. Des. Codes Cryptogr. 40, 167–185 (2006)
Colbourn C.J., Dinitz J.H.: The CRC Handbook of Combinatorial Designs. CRC, Boca Raton, FL (1996)
Cumming L.J.: A family of circular systematic comma-free codes. J. Comb. Math. Comb. Comput. 58, 87–96 (2006)
Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory. North-Holland/Elsevier, Amsterdam, The Netherlands (1998). North-Holland Mathematical Libary 55.
Dillon J.F.: Multiplicative difference sets via additive characters. Des. Codes Cryptogr. 17, 224–235 (1999)
Dillon J.F., Dobbertin H.: Cyclic difference sets with singer parameters. manuscript (1999).
Ding C.: Optimal and perfect difference systems of sets. J. Comb. Theory (A) 116, 109–119 (2009)
Ding C., Helleseth T., Lam K.Y.: Several classes of sequences wiht three-level autocorrelation. IEEE Trans. Inform. Theory 45, 2606–2612 (1999)
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)
Dinitz J., Stinson D.R.: Difference sets, in Contemporary Design Theory: A Collection of Surveys. Wiley, New York (1992)
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)
Fuji-Hara R., Miao Y., Mishima M.: Optimal frequency hopping sequences: a combinatorial approach. IEEE Trans. Inform. Theory. 50, 2408–2420 (2004)
Fuji-Hara R., Munemasa A., Tonchev V.D.: Hyperplane partitions and difference systems of sets. J. Comb. Theory (A) 113, 1699–1718 (2006)
Ge G., Fuji-Hara R., Miao Y.: Further combinatorial constructions for optimal frequency-hopping sequences. J. Comb. Theory (A) 113, 1699–1718 (2006)
Gordon B., Mills W.H., Welch L.R.: Some new difference sets. Can. J. Math. 14, 614–625 (1962)
Lempel A., Greenberger H.: Families of sequences with optimal Hamming correlation properties. IEEE Trans. Inform. Theory 20, 90–94 (1974)
Levenshtein V.I.: One method of constructing quasilinear codes providing synchronization in the presence of errors. Prob. Inform. Transm. 7, 215–222 (1971)
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)
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).
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)
Pott A.: Finite Geometry and Character Theory. Lecture Notes in Mathematics, p. 1601. Springer-Verlag, Berlin, Germany (1995).
Tonchev V.D.: Difference systems of sets and code synchronization. Rend. Sem. Mat. Messina. (II) 9, 217–226 (2003)
Tonchev V.D.: Partitions of difference sets and code synchronization. Finite Fields. Appl. 11, 601–621 (2005)
Tonchev V.D., Wang H.: An algorithm for optimal difference systems of sets. J. Com. Optim. 14, 165–175 (2007)
Wang H.: A new bound for difference systems of sets. J. Comb. Math. Comb. Comput. 58, 161–168 (2006)
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)
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)
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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
Keywords
- Difference system of sets
- Code synchronization
- Partition-type difference packing
- Paley–Hadamard difference set
- Almost difference set