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

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 $12a \in \{168, 252, 336, 504, 1512\}$, $36a < b < 42a$.