Advertisement

Codes and Combinatorial Structures from Circular Planar Nearrings

  • Anna Benini
  • Achille Frigeri
  • Fiorenza Morini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6742)

Abstract

Nearrings are generalized rings in which addition is not in general abelian and only one distributive law holds. Some interesting combinatorial structures, as tactical configurations and balanced incomplete block designs (BIBDs) naturally arise when considering the class of planar and circular nearrings. In [12] the authors define the concept of disk and prove that in the case of field-generated planar circular nearrings it yields a BIBD, called disk-design. In this paper we present a method for the construction of an association scheme which makes the disk-design, in some interesting cases, an union of partially incomplete block designs (PBIBDs). Such designs can be used in the construction of some classes of codes for which we are able to calculate the parameters and to prove that in some cases they are also cyclic.

Keywords

Binary codes Planar circular nearring PBIB design 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aichinger, E., Binder, F., Ecker, J., Mayr, P., Nöbauer, C.: SONATA - system of near-rings and their applications, GAP package, Version 2 (2003), http://www.algebra.uni-linz.ac.at/Sonata/
  2. 2.
    Aichinger, E., Ecker, J., Nöbauer, C.: The use of computers in near-rings theory. In: Fong, Y., et al. (eds.) Near-Rings and Near-Fields, pp. 35–41 (2001)Google Scholar
  3. 3.
    Assmus Jr., E., Key, J.: Designs and their codes. Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bannai, E.: An introduction to association schemes. Quad. Mat. 5, 1–70 (1999)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Clay, J.: Nearrings: Geneses and Applications. Oxford University Press, Oxford (1992)zbMATHGoogle Scholar
  7. 7.
    Clay, J.: Geometry in fields. Algebra Colloq. 1(4), 289–306 (1994)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Cramwinckel, J., Roijackers, E., Baart, R., Minkes, E., Ruscio, L., Joyner, D.: GUAVA, a GAP Package for computing with error-correcting codes (2009), http://www.opensourcemath.org/guava/
  9. 9.
    Davis, P.: Circulant Matrices. Chelsea Publishing, New York (1994)zbMATHGoogle Scholar
  10. 10.
    Eggetsberger, R.: On extending codes from finite Ferrero pairs. Contributions to General Algebra 9, 151–162 (1994)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Eggetsberger, R.: Circles and their interior points from field generated Ferrero pairs. In: Saad, G., Thomsen, M. (eds.) Nearrings, Nearfields and K-Loops, pp. 237–246 (1997)Google Scholar
  12. 12.
    Frigeri, A., Morini, F.: Circular planar nearrings: geometrical and combinatorial aspects (submitted), http://arxiv.org/
  13. 13.
    Fuchs, P.: A decoding method for planar near-ring codes. Riv. Mat. Univ. Parma 4(17), 325–331 (1991)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fuchs, P., Hofer, G., Pilz, G.: Codes from planar near-rings. IEEE Trans. Inform. Theory 36, 647–651 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hopper, N., von Ahn, L., Langford, J.: Provably secure steganography. IEEE Transactions on Computers 58(5), 662–676 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lancaster, P., Timenetsky, M.: The Theory of Matrices with Applications. Academic Press, New York (1985)Google Scholar
  17. 17.
    McWilliams, F., Sloane, N.: The Theory of Error-correcting Codes. North-Holland, Amsterdam (1977)Google Scholar
  18. 18.
    Meir, O.: IP=PSPACE using Error Correcting Codes. Electronic Colloquium on Computational Complexity 17(137) (2010)Google Scholar
  19. 19.
    Street, A., Street, D.: Combinatorics of Experimental Design. Oxford University Press, New York (1987)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Anna Benini
    • 1
  • Achille Frigeri
    • 2
  • Fiorenza Morini
    • 3
  1. 1.Università di BresciaItaly
  2. 2.Politecnico di MilanoItaly
  3. 3.Università di ParmaItaly

Personalised recommendations