Group divisible designs with block size four and type $g^u b^1 (gu/2)^1$

TL;DR

This paper characterizes the existence of specific group divisible designs with block size four and certain types, providing necessary and sufficient conditions for their existence across various parameter sets.

Contribution

It establishes new existence criteria for 4-group divisible designs with block size four and specified types, expanding the known classifications in combinatorial design theory.

Findings

Provides necessary and sufficient conditions for the existence of certain 4-GDDs.

Identifies exceptions and special cases in the existence criteria.

Extends the classification of group divisible designs with block size four.

Abstract

We discuss group divisible designs with block size four and type $g^u b^1 (gu/2)^1$, where $u = 5$, 6 and 7. For integers $a$ and $b$, we prove the following. (i) A 4-GDD of type $(4a)^5 b^1 (10a)^1$ exists if and only if $a \ge 1$, $b \equiv a$ (mod 3) and $4a \le b \le 10a$. (ii) A 4-GDD of type $(6a+3)^6 b^1 (18a+9)^1$ exists if and only if $a \ge 0$, $b \equiv 3$ (mod 6) and $6a+3 \le b \le 18a + 9$. (iii) A 4-GDD of type $(6a)^6 b^1 (18a)^1$ exists if and only if $a \ge 1$, $b \equiv 0$ (mod 3) and $6a \le b \le 18a$. (iv) A 4-GDD of type $(12a)^7 b^1 (42a)^1$ exists if and only if $a \ge 1$, $b \equiv 0$ (mod 3) and $12a \le b \le 42a$, except possibly for $12a \in \{120, 180, 240, 360, 420, 720, 840\}$, $24a < b < 42a$, for $12a \in \{144, 1008\}$, $30a < b < 42a$, and for $12a \in \{168, 252, 336, 504, 1512\}$, $36a < b < 42a$.