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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
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/
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)
Assmus Jr., E., Key, J.: Designs and their codes. Cambridge Tracts in Mathematics, vol. 103. Cambridge University Press, Cambridge (1992)
Bannai, E.: An introduction to association schemes. Quad. Mat. 5, 1–70 (1999)
Beth, T., Jungnickel, D., Lenz, H.: Design Theory. Cambridge University Press, Cambridge (1999)
Clay, J.: Nearrings: Geneses and Applications. Oxford University Press, Oxford (1992)
Clay, J.: Geometry in fields. Algebra Colloq. 1(4), 289–306 (1994)
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/
Davis, P.: Circulant Matrices. Chelsea Publishing, New York (1994)
Eggetsberger, R.: On extending codes from finite Ferrero pairs. Contributions to General Algebra 9, 151–162 (1994)
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)
Frigeri, A., Morini, F.: Circular planar nearrings: geometrical and combinatorial aspects (submitted), http://arxiv.org/
Fuchs, P.: A decoding method for planar near-ring codes. Riv. Mat. Univ. Parma 4(17), 325–331 (1991)
Fuchs, P., Hofer, G., Pilz, G.: Codes from planar near-rings. IEEE Trans. Inform. Theory 36, 647–651 (1990)
Hopper, N., von Ahn, L., Langford, J.: Provably secure steganography. IEEE Transactions on Computers 58(5), 662–676 (2009)
Lancaster, P., Timenetsky, M.: The Theory of Matrices with Applications. Academic Press, New York (1985)
McWilliams, F., Sloane, N.: The Theory of Error-correcting Codes. North-Holland, Amsterdam (1977)
Meir, O.: IP=PSPACE using Error Correcting Codes. Electronic Colloquium on Computational Complexity 17(137) (2010)
Street, A., Street, D.: Combinatorics of Experimental Design. Oxford University Press, New York (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Benini, A., Frigeri, A., Morini, F. (2011). Codes and Combinatorial Structures from Circular Planar Nearrings. In: Winkler, F. (eds) Algebraic Informatics. CAI 2011. Lecture Notes in Computer Science, vol 6742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21493-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-21493-6_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21492-9
Online ISBN: 978-3-642-21493-6
eBook Packages: Computer ScienceComputer Science (R0)