In 1987, Niederreiter introduced the idea of a (T,M,S)-net in base b. A (T,M,S)-net is a collection of points in the S-dimensional Euclidean unit cube satisfying certain uniformity properties which are useful for applications such as pseudo-random number generation and numerical integration.

A recent result of Schmid and Lawrence generalised the concept of an orthogonal array in order to cast (T,M,S)-nets in a combinatorial framework. An ordered orthogonal array OOA(t,s,\ell,v) i s an array with s\ell columns --- partitioned into s groups of size \ell --- having entries from an alphabet of v symbols and satisfying the following condition: in any t columns which are left-justified within their respective groups, each t-tuple of symbols occurs exactly times.

In this talk, we construct a family of association schemes similar to the Hamming schemes (but not P-polynomial) which afford a context in which we may extend standard results from the theory of orthogonal arrays and error-correcting codes to the case of ordered orthogonal arrays and ordered codes. In particular, we present a generalised Rao bound, a linear programming bound and a MacWilliams-type theorem for the linear case.

This is joint work with D.R. Stinson at the University of Waterloo.

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