Coding Theory and Uniform Distributions

TL;DR

This paper introduces and analyzes optimum distributions in the unit cube, linking them to maximum distance separable codes and exploring their combinatorial structure, weight spectra, and duality properties.

Contribution

It provides a complete characterization of optimum distributions as MDS codes with respect to a non-Hamming metric and constructs broad classes of such codes using Hermite interpolation.

Findings

Optimum distributions are characterized as MDS codes with a non-Hamming metric.

Weight spectra of these codes are precisely evaluated.

Explicit constructions of linear MDS codes and distributions are provided.

Abstract

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 closely related with known (delta,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 identities for weight enumerators are briefly discussed. Broad classes of linear maximum distance separable codes and linear optimum distributions are explicitely constructed in the paper by the Hermite interpolations over finite fields.