Abstract
Difference systems of sets (DSS) are combinatorial structures introduced by Levenshtein in 1971, which are a generalization of cyclic difference sets and arise in connection with code synchronization. In this paper, we describe some recursive constructions for DSSs having composite order, and present some infinite series of asymptotically optimal DSSs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fuji-Hara R., Munemasa A., Tonchev V.D.: Hyperplane partitions and difference systems of sets. J. Comb. Theory Ser. A 113(8), 1689–1698 (2006).
Fujiwara Y., Fuji-Hara R.: Frequency hopping sequences with optimal auto-and cross-correlation properties and related codes, In: Proceedings of Tenth International Workshop Algebraic Combinatorial Coding Theory, pp. 83–96 (2006).
Golomb S.W., Gordon B., Welch L.R.: Comma-free codes. Can. J. Math. 10(2), 202–209 (1958).
Lei J., Fan C.: Optimal difference systems of sets and partition-type cyclic difference packings. Des. Codes Cryptogr. 58(2), 135–153 (2011).
Levenshtein V.I.: One method of constructing quasilinear codes providing synchronization in the presence of errors. Probl. Pereda. Inform. 7(3), 215–222 (1971).
Store T.: Cyclotomy and Difference Sets. Lectures in Advanced Mathematics. Markham Publishing Company, Chicago (1967).
Tonchev V.D.: Difference systems of sets and code synchronization. Rend. Sem. Mat. Messina Ser. II(9), 217–226 (2003).
Tonchev V.D.: Partitions of difference sets and code synchronization. Finite Fields Their Appl. 11(3), 601–621 (2005).
Wang X., Wang J.: Optimal difference systems of sets and difference sets. Aequ. Math. 82(1–2), 155–164 (2011).
Acknowledgements
The authors are grateful to the referees for their careful reading of the original version of this paper. Their detailed comments and suggestions much improved the quality of this paper. The authors would like to thank Professor Shinji Kuriki, Osaka Prefecture University, for his valuable advice on Theorem 11. The work of N. Miyamoto was supported in part by JSPS under Grant-in-Aid for Scientific Research (C)15K04982.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. D. Tonchev.
Rights and permissions
About this article
Cite this article
Chisaki, S., Kimura, Y. & Miyamoto, N. A recursive construction for difference systems of sets. Des. Codes Cryptogr. 87, 1059–1068 (2019). https://doi.org/10.1007/s10623-018-0505-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0505-2