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- W2948286045 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 $12a in {168, 252, 336, 504, 1512}$, $36a < b < 42a$." @default.
- W2948286045 created "2019-06-14" @default.
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- W2948286045 date "2019-06-05" @default.
- W2948286045 modified "2023-09-27" @default.
- W2948286045 title "Group divisible designs with block size four and type $g^u b^1 (gu/2)^1$" @default.
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