Generalizing the Bierbrauer-Friedman bound for orthogonal arrays

TL;DR

This paper extends the Bierbrauer-Friedman bound to mixed-level orthogonal arrays, characterizes arrays attaining it as special codes in a multigraph, and explores new bounds for pure-level arrays using algebraic and combinatorial methods.

Contribution

It generalizes the BF bound to mixed-level arrays, characterizes arrays attaining the bound as completely regular codes, and introduces new bounds for pure-level arrays via polynomial generalizations.

Findings

Mixed-level arrays characterized by algebraic designs in a multigraph.

Arrays attaining the BF bound are radius-1 completely regular codes.

New bounds for pure-level arrays derived from polynomial generalizations of the Hamming graph.

Abstract

We characterize mixed-level orthogonal arrays in terms of algebraic designs in a special multigraph. We prove a mixed-level analog of the Bierbrauer-Friedman (BF) bound for pure-level orthogonal arrays and show that arrays attaining it are radius-1 completely regular codes (equivalently, intriguing sets, equitable 2-partitions, perfect 2-colorings) in the corresponding multigraph. For the case when the numbers of levels are powers of the same prime number, we characterize, in terms of multispreads, additive mixed-level orthogonal arrays attaining the BF bound. For pure-level orthogonal arrays, we consider versions of the BF bound obtained by replacing the Hamming graph by its polynomial generalization and show that in some cases this gives a new bound. Keywords: orthogonal array, algebraic t-design, completely regular code, equitable partition, intriguing set, Hamming graph, Bierbrauer-Friedman bound, additive codes.