Abstract
The detection of error and its correction is an important area of mathematics that is vastly constructed in all communication systems. Furthermore, combinatorial design theory has several applications like detecting or correcting errors in communication systems. Network (graph) designs (GDs) are introduced as a generalization of the symmetric balanced incomplete block designs (BIBDs) that are utilized directly in the above mentioned application. The networks (graphs) have been represented by vectors whose entries are the labels of the vertices related to the lengths of edges linked to it. Here, a general method is proposed and applied to construct new networks designs. This method of networks representation has simplified the method of constructing the network designs. In this paper, a novel representation of networks is introduced and used as a technique of constructing the group generated network designs of the complete bipartite networks and certain circulants. A technique of constructing the group generated network designs of the circulants is given with group generated graph designs (GDs) of certain circulants. In addition, the GDs are transformed into an incidence matrices, the rows and the columns of these matrices can be both viewed as a binary nonlinear code. A novel coding error detection and correction application is proposed and examined.