Distributing hash families with few rows

TL;DR

This paper extends methods for constructing distributing hash families with few rows, enabling more efficient creation of covering arrays and improving their applicability in combinatorial designs.

Contribution

It generalizes the construction of distributing hash families from homogeneous to heterogeneous, introduces fractal hash families, and relaxes certain requirements for broader applicability.

Findings

Constructed fractal perfect hash families.

Extended methods to heterogeneous hash families.

Applications to large-strength perfect hash families.

Abstract

Column replacement techniques for creating covering arrays rely on the construction of perfect and distributing hash families with few rows, having as many columns as possible for a specified number of symbols. To construct distributing hash families in which the number of rows is less than the strength, we examine a method due to Blackburn and extend it in three ways. First, the method is generalized from homogeneous hash families (in which every row has the same number of symbols) to heterogeneous ones. Second, the extension treats distributing hash families, in which only separation into a prescribed number of parts is required, rather than perfect hash families, in which columns must be completely separated. Third, the requirements on one of the main ingredients are relaxed to permit the use of a large class of distributing hash families, which we call fractal. Constructions for fractal perfect and distributing hash families are given, and applications to the construction of perfect hash families of large strength are developed.