Combinatorial structure of finite fields with two dimensional modulo metrics

Abstract

This paper shows the connection between the combinatorial structure of two dimensional metrics over finite fields (Shortly, Mannheim and Hexagonal metrics) and some group actions defined over them. We follow the well known approach of P. Delsarte [9] to this problem through the construction of association schemes. Association schemes based on this distances are the basic tools we propose to deal with the metric properties of codes defined over two dimensional metrics and their parameters. We note that some examples of cyclotomic association schemes (which we call M schemes and H schemes respectively) fit properly as weakly metric schemes for these metrics.