Existence of (v,{5,w∗},1)-PBDs

Bennett, F. E.; Chang, Yanxun; Ge, Gennian; Greig, Malcolm

doi:10.1016/s0012-365x(03)00262-0

Cited by 20 publications

(62 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…As before, no two points in any given block of size 5 have identical first components (mod 5); thus each of these blocks generate 5 parallel classes on the non-infinite points and we can add one infinite point to each. 20 9 10 1 : ((0, 0), (0, 1), (0, 12), (0, 29), (1,16), (3,24)) , ((0, 0), (0, 3), (0, 16), (1,13), (3,10), (3,32)) , ((0, 0), (0, 2), (0, 32), (1,24), (1,34), (2,17)) , ((0, 0), (1, 1), (2,31), (3,2), (4,5)) , ((0, 0), (0, 15), (1,21), (1,26), (2, 2), (3, 1)) , ((0, 0), (1,5), (2,25), (3,33), (4,11)) . (2,29), (3,9), (4,5)) , ((0, 0), (0, 9), (1,27), (2,37), (3,21), (4,…”

Section: More Pbdsmentioning

confidence: 99%

“…(2,29), (3,9), (4,5)) , ((0, 0), (0, 9), (1,27), (2,37), (3,21), (4,11)) , ((0, 0), (1,8), (2,31), (3,25), (4, 1)) , ((0, 0), (0, 21), (1,12), (2,4), (3,11), (4,15)) , ((0, 0), (1,9), (2,25), (3,39), (4,27)) . (1,17), (2,29), (3,4), (4,53)) , ((0, 0), (0, 9), (0, 1), (2,27), (1,6), (1,45)) , (1,33), …”

Section: More Pbdsmentioning

confidence: 99%

“…(1,17), (2,29), (3,4), (4,53)) , ((0, 0), (0, 9), (0, 1), (2,27), (1,6), (1,45)) , (1,33), (2,14), (3,34), (4,56)) , ((0, 0), (0, 15), (1,26), (2,28), (3,37), (4,32)) . (1,4), (2,14)) , ((0, 0), (1, 39), (2, 48), (3,29), (4,24)) , ((0, 0), (0, 16), (1,15), (2,23), (2,35), (4,3)) , ((0, 0), (1,11), (2,4), (3,16), …”

Section: More Pbdsmentioning

confidence: 99%

“…(1, 2, 9, 13, 17, ∞ 0 ), (3,4,10,14,18, ∞ 0 ), (5,6,11,15,19, ∞ 0 ), (7,8,12,16,20, ∞ 0 ), (1, 2, 10, 15, 20, ∞ 1 ), (3,4,11,16,17, ∞ 1 ), (5, 6, 12, 13, 18, ∞ 1 ), (7,8,9,14,19, ∞ 1 ), (9,10,11,13,16,19), (9,10,12,15,16,18), (11,12,14,15,17,20), (13,14,17,18,19,20), (1,2,...…”

Section: Small Coversmentioning

confidence: 99%

See 3 more Smart Citations

Pair covering and other designs with block size 6

Abel

Bluskov

Greig³

et al. 2007

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

Section: More Pbdsmentioning

confidence: 99%

Section: More Pbdsmentioning

confidence: 99%

Section: More Pbdsmentioning

confidence: 99%

Section: Small Coversmentioning

confidence: 99%

See 2 more Smart Citations

Pair covering and other designs with block size 6

Abel

Bluskov

Greig³

et al. 2007

J of Combinatorial Designs

Self Cite

View full text Add to dashboard Cite

“…Proof: Take a (77, {5, 9 * }, 1)-PBD (the existence of such a PBD follows from [2]) and remove a point not in the single block of size 9 to obtain a {5, 9}-GDD of type 4 19 . The single block of size 9 can hit only 9 groups of the GDD.…”

Section: Lemma 418mentioning

confidence: 99%

Traffic Grooming in Unidirectional Wavelength-Division Multiplexed Rings with Grooming RatioC= 6

Bermond¹,

Colbourn²,

Coudert³

et al. 2005

SIAM J. Discrete Math.

Self Cite

View full text Add to dashboard Cite

International audienceSONET/WDM networks using wavelength add-drop multiplexing can be constructed using certain graph decompositions used to form a grooming, consisting of unions of primitive rings. The cost of such a decomposition is the sum, over all graphs in the decomposition, of the number of vertices of nonzero degree in the graph. The existence of such decompositions with minimum cost, when every pair of sites employs no more than $\frac{1}{6}$ of the wavelength capacity, is determined with a finite number of possible exceptions. Indeed, when the number N of sites satisfies $N \equiv 1 \pmod{3}$, the determination is complete, and when $N \equiv 2 \pmod{3}$, the only value left undetermined is N = 17. When $N \equiv 0 \pmod{3}$, a finite number of values of N remain, the largest being N = 2580. The techniques developed rely heavily on tools from combinatorial design theory

show abstract

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

Abel

Bennett

2004

J of Combinatorial Designs

View full text Add to dashboard Cite

In this paper, we look at the existence of ðv v; KÞ pairwise balanced designs (PBDs) for a few sets K of prime powers ! 8 and also for a number of subsets K of f5; 6; 7; 8; 9g, which contain f5g. For K ¼ f5; 7g; f5; 8g; f5; 7; 9g, we reduce the largest v v for which a ðv v; KÞ-PBD is unknown to 639, 812, and 179, respectively. When K is Q !8 , the set of all prime powers ! 8, we find several new designs for 1,180 v v 1,270, and reduce the largest unsolved case to 1,802. For K ¼ Q 0;1;5ð8Þ , the set of prime powers ! 8 and 0; 1, or 5 (mod 8) we reduce the largest unknown case from 8,108 to 2,612. We also obtain slight improvements when K is one of f8; 9g or Q 0;1ð8Þ , the set of prime powers 0 or 1 (mod 8).

show abstract

Existence of (v,{5,w∗},1)-PBDs

Cited by 20 publications

References 30 publications

Pair covering and other designs with block size 6

Pair covering and other designs with block size 6

Traffic Grooming in Unidirectional Wavelength-Division Multiplexed Rings with Grooming RatioC= 6

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

Contact Info

Product

Resources

About