Hamming Graphs and Permutation Codes

A permutation code can be represented as a graph, in which the nodes correspond to the permutation codewords and the weights on the edges are the Hamming distances between the codewords. Graphs belonging to this class are called permutation Hamming graphs. This paper explores the Maximum Permutation Code Problem (MPCP), a well-known optimization problem in coding theory, by means of a graph theoretical approach. Permutation Hamming graphs turn out to satisfy strong regularity properties, such as vertex transitivity and r-partiteness. In addition, exact formulas for the degree of the vertices and for the number of the edges are presented. Furthermore, a remarkable similarity between permutation Hamming graphs and Turán graphs is enlightened. The new link with Turán graphs might help to improve current results on the MPCP.