MacWilliams duality and the Rosenbloom–Tsfasman metric

A new non-Hamming metric on linear spaces over finite fields has recently been introduced by Rosenbloom and Tsfasman [8]. We consider orbits of linear groups preserving the metric and show that weight enumerators suitably associated with such orbits satisfy MacWilliams-type identities for mutually dual codes. Furthermore, we show that the corresponding weight spectra of dual codes are related by transformations which involve multi-dimensional generalizations of known Krawtchouk polynomials. The relationships with recent results by Godsil [5] and Martin and Stinson [7] on MacWilliams-type theorems for association schemes and ordered orthogonal arrays are also briefly discussed in the paper.