Resolvable modified group divisible designs with block size three

Abstract:We use arcs, ovals, and hyperovals to construct class-uniformly resolvable structures. Many of the structures come from finite geometries, but we also use arcs from nongeometric designs. Most of the class-uniformly resolvable structures constructed here have block size sets that have not been constructed before. We construct CURDs with a variety of block sizes, including many with block sizes 2 and 4. In addition, these constructions give the first systematic way of constructing infinite families of CURDs with three block sizes. q

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“…The following is a modified CURGDD which has two orthogonal sets of groups; see [19] for further discussion of modified structures. Proof.…”

Section: Theorem 34 For All Odd Prime Powers Q There Exists a Curgmentioning

confidence: 99%

Geometrical constructions of class-uniformly resolvable structure

Danziger

Greig²,

Stevens

2011

J. Combin. Designs

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“…For

u = 13

and

m = 6 g - 9

, take a 3‐RGDD of type 6 13 , , [, Theorem IV.5.43]. Inflate each point by a factor of

g / 6

and note that, since

g \geq 42

, there exists a resolvable 3‐MGDD of type

{false(g / 6 false)}^{3}

, , [, Theorem 2.5]. Replace each inflated block of the 3‐RGDD by a 3‐MGDD of type

{false(g / 6 false)}^{3}

to create a resolvable 3‐DGDD of type

{false(g, 6^{g / 6} false)}^{13}

.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…Proof of Theorem By [, Theorem 7.1], we only have to eliminate the case

(u, m) = (39, 19 g - 3)

.Take a 3‐RGDD of type 2 39 , , [, Theorem IV.5.43] and a resolvable 3‐MGDD of type

{false(g / 2 false)}^{3}

, , [, Theorem 2.5]. Inflate each point of the 3‐RGDD by a factor of

g / 2

and overlay each block with a resolvable 3‐MGDD of type

{false(g / 2 false)}^{3}

to create a resolvable 3‐DGDD of type

{false(g, 2^{g / 2} false)}^{39}

.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

Group divisible designs with block size 4 and type

Forbes

2018

J of Combinatorial Designs

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Building upon the work of Wei and Ge (Designs, Codes, and Cryptography 74, 2015), we extend the range of positive integer parameters g, u, and m for which group divisible designs with block size 4 and type gum1 are known to exist. In particular, we show that the necessary conditions for the existence of these designs when m>0 and m≠g are sufficient in the following cases: g≡0( mod 6), g=2 with one exception, 2651, g=10, and g=11.

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“…A K ‐IGDD of type

{(g_{1}, h_{1})}^{u_{1}} {(g_{2}, h_{2})}^{u_{2}} \dots {(g_{s}, h_{s})}^{u_{s}}

is an IGDD in which every block has size from the set K and in which, when

1 \leq i \leq s

, there are

u_{i}

groups of size

g_{i}

, each intersecting the hole in

h_{i}

points. Theorem The necessary conditions for the existence of a 4‐IGDD of type

{false(g, h false)}^{u}

, namely,

g \geq 3 h

g (u - 1) \equiv 0 mod 3

(g - h) (u - 1) \equiv 0 mod 3

and

false(g^{2} - h^{2} false) u false(u - 1 false) \equiv 0 mod 12

, are also sufficient, except for

(u, g, h) \in {(4, 2, 0), (4, 6, 0), (4, 6, 1)}

and except possibly for

(u, g, h) \in {(14, 93, 27), (18, 93, 27)}

.…”

Section: Preliminariesmentioning

confidence: 99%

Group divisible covering designs with block size four

Wei

Colbourn

2017

J of Combinatorial Designs

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Group divisible covering designs (GDCDs) were introduced by Heinrich and Yin as a natural generalization of both covering designs and group divisible designs. They have applications in software testing and universal data compression. The minimum number of blocks in a k‐GDCD of type gu is a covering number denoted by C(k,gu). When k=3, the values of C(3,gu) have been determined completely for all possible pairs (g,u). When k=4, Francetić et al. constructed many families of optimal GDCDs, but the determination remained far from complete. In this paper, two specific 4‐IGDDs are constructed, thereby completing the existence problem for 4‐IGDDs of type false(g,hfalse)u. Then, additional families of optimal 4‐GDCDs are constructed. Consequently the cases for (g,u) whose status remains undetermined arise when g≡7mod12 and u≡3mod6, when g≡11,14,17,23mod24 and u≡5mod6, and in several small families for which one of g and u is fixed.

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Resolvable modified group divisible designs with block size three

Abstract: A resolvable modified group divisible design (RMGDD) is an MGDD whose blocks can be partitioned into parallel classes. In this article, we investigate the existence of RMGDDs with block size three and show that the necessary conditions are also sufficient with two exceptions. #

Cited by 14 publications

References 13 publications

Geometrical constructions of class-uniformly resolvable structure

Geometrical constructions of class-uniformly resolvable structure

Group divisible designs with block size 4 and type

Group divisible covering designs with block size four

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