Advertisement

Almost designs and their links with balanced incomplete block designs

  • Jerod MichelEmail author
  • Qi Wang
Article
  • 34 Downloads

Abstract

Almost designs (t-adesigns) were proposed and discussed by Ding as a certain generalization of combinatorial designs related to almost difference sets. Unlike t-designs, it is not clear whether t-adesigns need also be \((t-1)\)-designs or \((t-1)\)-adesigns. In this paper we discuss a particular class of 3-adesigns, i.e., 3-adesigns coming from certain strongly regular graphs and tournaments, and find that these are also 2-designs. We construct several classes of these, and discuss some of the restrictions on the parameters of such a class. We also construct several new classes of 2-adesigns, and discuss some of their properties as well.

Keywords

Almost difference set Difference set Strongly regular graph t-adesign Tournament 

Mathematics Subject Classification

05B05 05E30 

Notes

Acknowledgements

The authors are very grateful to the three anonymous referees and to the Coordinating Editor for all of their detailed comments that greatly improved the quality and the presentation of this paper.

References

  1. 1.
    Assmus E.F., Key J.D.: Designs and Their Codes, vol. 103. Cambridge University Press, Cambridge (1992).CrossRefzbMATHGoogle Scholar
  2. 2.
    Behbahani M.: On strongly regular graphs. PhD thesis, Concordia University (2009).Google Scholar
  3. 3.
    Beth T., Jungnickel D., Lenz H.: Design Theory, vol. I, 2nd edn. Cambridge University Press, Cambridge (1999).CrossRefzbMATHGoogle Scholar
  4. 4.
    Bose R.C.: On the construction of balanced incomplete block designs. Ann. Eugen. 9, 353–399 (1939).MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cameron P.J., van Lint J.H.: Designs, Graphs, Codes and Their Links. London Mathematical Society Student Texts. LMSSTCambridge University Press, Cambridge (1991).CrossRefzbMATHGoogle Scholar
  6. 6.
    Cusick T.W., Ding C., Renvall A.: Stream Ciphers and Number Theory, vol. 55. North-Holland Mathematical LibraryNorth-Holland Publishing Co., Amsterdam (1998).CrossRefzbMATHGoogle Scholar
  7. 7.
    Davis J.A.: Almost difference sets and reversible divisible difference sets. Arch. Math. (Basel) 59(6), 595–602 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dembowski P.: Finite Geometries. Springer, New York (1968).CrossRefzbMATHGoogle Scholar
  9. 9.
    Ding C.: The differential cryptanalysis and design of natural stream ciphers. In: Fast Software Encryption, pp. 101–115. Springer, Berlin (1994).Google Scholar
  10. 10.
    Ding C.: Codes from Difference Sets. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2015).Google Scholar
  11. 11.
    Ding C., Yin J.: Constructions of almost difference families. Discret. Math. 308(21), 4941–4954 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ding C., Helleseth T., Martinsen H.: New families of binary sequences with optimal three-level autocorrelation. IEEE Trans. Inform. Theory 47(1), 428–433 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ding C., Pott A., Wang Q.: Skew Hadamard difference sets from Dickson polynomials of order 7. J. Comb. Designs 23(10), 436–461 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fisher R.A.: An examination of the different possible solutions of a problem in incomplete blocks. Ann. Eugen. 10, 52–75 (1940).MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goethals J., Seidel J.: Orthogonal matrices with zero diagonal. Can. J. Math. 19, 1001–1010 (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Golomb S.W., Gong G.: Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar. Cambridge University Press, Cambridge (2005).CrossRefzbMATHGoogle Scholar
  17. 17.
    Hirschfeld J.W.P.: Projective Geometries Over Finite Feilds, 2nd edn. Oxford University Press, Oxford (1998).zbMATHGoogle Scholar
  18. 18.
    Horsley D.: Generalizing Fisher’s inequality to coverings and packings. Combinatorica 37(4), 673–696 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003).CrossRefzbMATHGoogle Scholar
  20. 20.
    Johnson S.: A new upper bound for error-correcting codes. IRE Trans. Inform. Theory 8, 203–207 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Liu H., Ding C.: Infinite families of 2-designs from ${GA}_{1}(q)$ actions. arXiv:1707.02003v1 (2017).Google Scholar
  22. 22.
    Michel J.: New partial geometric difference sets and partial geometric difference families. Acta Math. Sin. 33(5), 591–606 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Michel J., Ding B.: A generalization of combinatorial designs and related codes. Des. Codes Cryptogr. 82(3), 511–529 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Neumaier A.: $t\frac{1}{2}$-designs. J. Comb. Theory (A) 78, 226–248 (1980).CrossRefzbMATHGoogle Scholar
  25. 25.
    Pasechnik D.: Skew-symmetric association schemes with two classes and strongly regular graphs of type $l_{2n-1}(4n-1)$. Acta Appl. Math. 29(1), 129–138 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Reid K., Brown E.: Doubly regular tournaments are equivalent to skew hadamard matrices. J. Comb. Theory (A) 12(3), 332–338 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Schonheim J.: On coverings. Pac. J. Math. 14, 1405–1411 (1964).CrossRefzbMATHGoogle Scholar
  28. 28.
    Stinson D.R.: Combinatorial Designs: Constructions and Analysis. Springer, New York (2003).zbMATHGoogle Scholar
  29. 29.
    Storer T.: Cyclotomy and Difference Sets, pp. 65–72. Markham, Chicago (1967).zbMATHGoogle Scholar
  30. 30.
    Wang X., Wu D.: The existence of almost difference families. J. Stat. Plan. Inference 139(12), 4200–4205 (2009).MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Wilson R.M.: Cyclotomy and difference families in elementary abelian groups. J. Number Theory 4, 17–47 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Zhang Y., Lei J.G., Zhang S.P.: A new family of almost difference sets and some necessary conditions. IEEE Trans. Inform. Theory 52(5), 2052–2061 (2006).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringSouthern University of Science and TechnologyShenzhenChina

Personalised recommendations