Group divisible designs with block size four and type —II

Forbes, Anthony D.

doi:10.1002/jcd.21650

Cited by 11 publications

(45 citation statements)

References 17 publications

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“…Assuming

m goodbreakinfix> 0

and

m goodbreakinfix\neq g

, the necessary conditions for the existence of a 4‐GDD of type

g^{u} m^{1}

are

({, \begin{matrix} u \geq 4, m \leq g (u - 1) ∕ 2, \\ g u \equiv g (u - 1) + m \equiv 0 (normalmod 3), and \\ g^{2} u (u - 1) + 2 g u m \equiv 0 (normalmod 12), \end{matrix})

which simplify to

u goodbreakinfix\geq 4 goodbreakinfix, m goodbreakinfix\leq g (u - 1) ∕ 2 goodbreakinfix, u goodbreakinfix\equiv m goodbreakinfix- g goodbreakinfix\equiv 0 (normalmod 3)

when

g goodbreakinfix\equiv 2

4 (normalmod 6)

. As for sufficiency, it is convenient to quote [, Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the necessary conditions for the existence of 4‐GDDs of type

g^{u}

, namely,

u goodbreakinfix\geq 4 goodbreakinfix, g (u - 1) goodbreakinfix\equiv 0 (normalmod 3) and g^{2} u (u - 1) goodbreakinfix\equiv 0 (normalmod 12),

are known to be sufficient except that there are no 4‐GDDs of types

2^{4}

and

6^{4}

[ [, Theorem IV.4.6]. The next case one would naturally consider is that of 4‐GDDs of type

g^{u} m^{1}

, and here a partial solution has been achieved in the sense that for each

g

there are at most a small number of

u

where existence remains undecided .…”

Section: Introductionmentioning

confidence: 99%

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Group divisible designs with block size four and type gum1—III

Forbes

2019

J of Combinatorial Designs

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We deal with group divisible designs (GDDs) that have block size four and group type gum1, where ggoodbreakinfix≡2 or 4 (mod 6). We show that the necessary conditions for the existence of a 4‐GDD of type gum1 are sufficient when g = 14, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 58, 62, 68, 76, 88, 92, 100, 104, 116, 124, 136, 152, 160, 176, 184, 200, 208, 224, 232, 248, 272, 304, 320, 368, 400, 448, 464 and 496. Using these results we go on to show that the necessary conditions are sufficient for ggoodbreakinfix=2tqs, q = 19, 23, 25, 29, 31, s, tgoodbreakinfix=1goodbreakinfix,2goodbreakinfix,…, as well as for ggoodbreakinfix=2tq, q = 2, 5, 7, 11, 13, 17, tgoodbreakinfix=1goodbreakinfix,2goodbreakinfix,…, with possible exceptions 569m1, 809m1 and 1129m1 for a few large values of m.

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“…Assuming

m goodbreakinfix> 0

and

m goodbreakinfix\neq g

, the necessary conditions for the existence of a 4‐GDD of type

g^{u} m^{1}

are

({, \begin{matrix} u \geq 4, m \leq g (u - 1) ∕ 2, \\ g u \equiv g (u - 1) + m \equiv 0 (normalmod 3), and \\ g^{2} u (u - 1) + 2 g u m \equiv 0 (normalmod 12), \end{matrix})

which simplify to

u goodbreakinfix\geq 4 goodbreakinfix, m goodbreakinfix\leq g (u - 1) ∕ 2 goodbreakinfix, u goodbreakinfix\equiv m goodbreakinfix- g goodbreakinfix\equiv 0 (normalmod 3)

when

g goodbreakinfix\equiv 2

4 (normalmod 6)

. As for sufficiency, it is convenient to quote [, Theorem 1.2].…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, the necessary conditions for the existence of 4‐GDDs of type

g^{u}

, namely,

u goodbreakinfix\geq 4 goodbreakinfix, g (u - 1) goodbreakinfix\equiv 0 (normalmod 3) and g^{2} u (u - 1) goodbreakinfix\equiv 0 (normalmod 12),

are known to be sufficient except that there are no 4‐GDDs of types

2^{4}

and

6^{4}

[ [, Theorem IV.4.6]. The next case one would naturally consider is that of 4‐GDDs of type

g^{u} m^{1}

, and here a partial solution has been achieved in the sense that for each

g

there are at most a small number of

u

where existence remains undecided .…”

Section: Introductionmentioning

confidence: 99%

Group divisible designs with block size four and type gum1—III

Forbes

2019

J of Combinatorial Designs

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“…The number of exceptions was then reduced to four by Wang and Shen [15]. Wei et al [16] solved two of these four cases, namely m n ( , ) = (6, 7) and (6,11). The last two cases, m n ( , ) = (3, 5) and (4,7), were found by Wang et al [14].…”

mentioning

confidence: 99%

“…Later, Ge and Wei [8] showed that for each pair g t ( , ) there is only a small range of values n for which a 4-GDD of type g n t 1 is unknown whenever the necessary conditions in Lemma 1.1 are satisfied. More recently, Forbes and Forbes [5,6] showed that the necessary conditions are sufficient for a wide class of feasible group types. However, a number of undetermined cases still remain.…”

mentioning

confidence: 99%

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Some new group divisible designs with block size 4 and two or three group sizes

Abel

Bunjamin

Combe

2020

J of Combinatorial Designs

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Group divisible designs (GDDs) with block size 4 and at most 30 points are known for all feasible group types except three, namely 2354,3562, and 2255. In this paper we provide solutions for the first two of these three 4‐GDDs without assuming any automorphisms. We also construct several other 4‐GDDs. These include classes of 4‐GDDs of types (3m)4(6m)q(3n)1 for 0≤n≤(q+1)m where q∈{2,3} and solutions for 4‐GDDs of types 3t6s for a wide range of values of s≥2,t≥4 satisfying t≡0 or 1(normalmod0.4em4), including all cases with 4≤t≤s−1. Most of the remaining unknown 4‐GDDs of type 3t6s have (s−1)

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