On the parameters of codes for the Lee and modular distance

Sol&, P., On the parameters of codes for the Lee and modular distance, Discrete Mathematics 89 (1991) 185-194. We introduce the concept of a weakly metric association scheme, a generalization of metric schemes. We undertake a combinatorial study of the parameters of codes in these schemes, along the lines of [9]. Applications are codes over 2, for the Lee distance and arithmetic codes for the modular distance. Our main result is an inequality which generalizes both the Delsarte upper bound on covering radius, and the MacWilliams lower bound on the external distance, yielding a strong necessary existence condition on completely regular codes. The external distance (in the Lee metric) of some self-dual codes of moderate length over Z, is computed.