New Nonuniform Group Divisible Designs and Mixed Steiner Systems

TL;DR

The paper addresses the construction and characterization of nonuniform group divisible designs and mixed Steiner systems, clarifying that mixed Steiner systems are a stricter subclass due to a minimum Hamming distance requirement. It develops OA-based constructions (including a correction to prior work), and large-set/LSLH-based results to produce new parameter families for $t\ge4$, plus a resolution- and OA-driven construction that yields numerous MS$(2,k,\mathbb{Z}_2^{(k-1)n} \times \mathbb{Z}_{n+1})$ variants; many parameters require $k$ and $k-1$ to be prime powers, tying some results to Mersenne/Fermat primes or $k=9$. The contributions significantly expand known parameter sets for mixed Steiner systems and nonuniform GDDs with $t\ge4$, offering new tools for coding-theoretic interpretations and combinatorial design applications. Together, the methods provide a versatile framework for generating rich families of $t$-designs with controlled minimum distance properties.

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.

Theorems & Definitions (53)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Proposition 1
• Definition 4
• Proposition 2
• Theorem 2
• proof
• Theorem 3