New Nonuniform Group Divisible Designs and Mixed Steiner Systems

TL;DR

This paper introduces new constructions and parameters for mixed Steiner systems and nonuniform group divisible designs, expanding the known existence of complex combinatorial designs with higher t-values.

Contribution

It presents novel constructions for mixed Steiner systems using orthogonal arrays and resolvable Steiner systems, and introduces many new nonuniform GDDs with higher t-values and parameters previously unknown.

Findings

New mixed Steiner systems based on orthogonal arrays.

Existence of nonuniform GDDs with t > 4.

Many new parameters for nonuniform GDDs derived from large sets of H-designs.

Abstract

This paper considers two closely related concepts, mixed Steiner system and nonuniform group divisible design (GDD). The distinction between the two concepts is the minimum Hamming distance, which is required for mixed Steiner systems but not required for nonuniform group divisible $t$-designs. In other words, it means that every mixed Steiner system is a nonuniform GDD, but the converse is not true. A new construction for mixed Steiner systems based on orthogonal arrays and resolvable Steiner systems is presented. Some of the new mixed Steiner systems (also GDDs) depend on the existence of Mersenne primes or Fermat primes. New parameters of nonuniform GDDs derived from large sets of H-designs (which are generalizations of GDDs) are presented, and in particular, many nonuniform group divisible $t$-designs with $t > 3$ are introduced (for which only one family was known before). Some GDDs are with $t > 4$, parameters for which no such design was known before.