Group divisible designs with block size 4 and type

Self Cite

We show that the necessary conditions for the existence of group divisible designs with block size four (4‐GDDs) of type g u m 1 are sufficient for g ≡ 0 (mod h), h = 39, 51, 57, 69, 87, 93, 111, 123 and 129, and for g = 13, 17, 19, 23, 25, 29, 31 and 35. More generally, we show that for g ≡ 3 (mod 6), the possible exceptions occur only when u = 8, and there are no exceptions at all if g ∕ 3 has a divisor d > 1 such that d ≡ 1 (mod 4) or d is a prime not greater than 43. Hence, there are no exceptions when g ≡ 3 (mod 12). Consequently, we are able to extend the known spectrum for g ≡ 1 and 5 (mod 6). Also, we complete the spectrum for 4‐GDDs of type ( 3 a ) 4 ( 6 a ) 1 ( 3 b ) 1.

Section: 21mentioning

confidence: 99%

Section: 15 6 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Group divisible designs with block size four and type —II

Forbes

2019

Self Cite

“…For type

4 0^{u} m^{1}

(u, m) goodbreakinfix\in {(6, 85), (9, 145)}

, use Theorem with a 4‐GDD of type

8^{u} {(m ∕ 5)}^{1}

. For type

4 0^{u} m^{1}

, where

u goodbreakinfix= 6

and

m goodbreakinfix\in {73, 79, 82, 91, 94, 97}

, or

u goodbreakinfix= 9

and

m goodbreakinfix\in {139, 142, 151, 154, 157}

, see Lemma .

bold-italicg goodbreakinfix= bold44

For type

4 4^{6} m^{1}

m goodbreakinfix\in {5, 8}

, see [, Lemma 2.8]. For type

4 4^{u} m^{1}

u goodbreakinfix\in {9, 15, 18}

m goodbreakinfix\leq u goodbreakinfix- 1

, take a 4‐DGDD of type

{(44, 2^{22})}^{u}

, adjoin

m

points and overlay the holes plus the new points with 4‐GDDs of type

2^{u} m^{1}

.…”

Section: ‐Gdds Of Type Gum1 With Ggoodbreakinfix≡2goodbreakinfix40mentioning

confidence: 99%

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

Forbes

2019

Self Cite

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.

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

Abel

Bunjamin

Combe

2020

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)