Pdmi Preprint — 27/1998 Coding Theory and Uniform Distributions

In the present paper we introduce and study finite point subsets of a special kind, called optimum distributions, in the n-dimensional unit cube. Such distributions are close related with known (δ, s, n)-nets of low discrepancy. It turns out that optimum distributions have a rich combinatorial structure. Namely, we show that optimum distributions can be characterized completely as maximum distance separable codes with respect to a non-Hamming metric. Weight spectra of such codes can be evaluated precisely. We also consider linear codes and distributions and study their general properties including the duality with respect to a suitable inner product. The corresponding generalized MacWilliams identies for weight enumerators are brifly discussed. Broad classes of linear maximum distance separable codes and linear optimum distributions are explicitly constructed in the paper by the Hermite interpolations over finite fields. 1991 Mathematics Subject Classification. 11K38, 11T71, 94B60