On The Dimension And Minimum Distance Of Graph Codes From Finite Geometry Based On Affine Planes

Graph codes have been extensively studied since Tanner's work from bipartite graphs in1981. In2006, Tom and Justesen obtained q-regular bipartite graphs from finite geometry on affine plane over finite fields. Using bipartite graphs and the extended Reed-Solomon codes as the component codes, they concretely constructed several graph codes. Furthermore, they determined the dimension of their codes and gave a lower bound for their minimum distance.In this thesis, firstly, we construct a (q+1,q)-regular bipartite graph derived from finite geometry on affine plane over finite fields by extending the idea of Tom and Justesen. Secondly, we construct a new graph code using the singly and doubly extended Reed-Solomon codes as the component codes. Finally, we calculate the dimension and give a lower bound for minimum distance of the graph code as well.