Codes and combinatorial structures from circular planar nearrings

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. © 2011 Springer-Verlag.