Quintessential pairwise balanced designs

“…(1, 2, 5, 4, 12), (1,3,12,9,10), (1,4,3,6,8), (1,5,11,10,6), (1,9,2,8,11), (2,3,8,7,12), (2,4,10,3,11), (2, 7, 9, 5, 10), (3,7,6,11,9), (4,6,12,10,7), (4,8,5,7,11), (5,6,9,8,12). …”

Section: Direct Constructionsmentioning

confidence: 99%

Holey Steiner pentagon systems

1999

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“…This shows the existence of the triangular PBIBDs (( n+2 2 ), n 2 − 1, 2n − 3, (n + 2)/2; (0, 1)) and (( n+1 2 ), n 2 − 1, 2n − 3, n/2; (0, 1)) is equivalent. Chang et al [22] (also in [56, p. 152]) give a list of 18 unsolved triangular PBIBDs, which we may update by noting that four (7,13,14,15) have now been constructed ( [6], [23, Designs T41, T54, T91]), including the T (10) Seiden case above, and its T (9) reduction, and that four (3,11,16,18) , 1), we record the construction in [14]; we construct a {47; 5, 7, 9}-arc in PG (2,8) by deleting a hyperoval and an external line, then line-flipping an 8-line; we then apply Lemma 12.17 with k = 5, v = 47, f = 5, and a = 1.…”

Section: Lemma 132 If T [K 1; M] Exists Then We May Construct Thementioning

confidence: 99%

Designs from projective planes and PBD bases

Greig¹

1999

J. Combin. Designs

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The essential theme of this article is the exploitation of known configurations in projective planes in order to construct pairwise balanced designs. Among the results shown are (q2 + 1)/2 ∈ B((q + 1)/2, 2) for all odd prime powers q; the resolution of 31 of the Mullin and Stinson's open cases in the spectrum of B(P7, 1), where P7 is odd prime powers ≥ 7; some constructions showing the inessential nature of values in E italica+, where E italica+ are PBD generating sets containing values equal to 1 mod(a) for 5 ≤ a ≤ 7; some constructions with block sizes being prime powers ≥ 8, yielding 7 MOLS of order 158 and 9 MOLS of order 254; and some designs on 28, 38, 42, 56, 63, 68, 72, and 156 points with interesting block sizes. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 341–374, 1999

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“…In the following lemma, we summarize the current results and the reader is referred to [3,7] for more details. (h, n) ∈ {(1, 10), (3, 6), (3, 18), (3, 28), (3, 34), (6,18), (6,19), (6, 23)}.…”

Section: Constructions By Hmolsmentioning

confidence: 99%

“…n , with the possible exception of (h, n) ∈ {(1, 6), (3,6), (3,18), (3,28), (3,34), (6,18), (6,19), (6, 23)}.…”

Section: Lemma 36 If H ≥ 1 and N ≥ 5 Then There Is A 5-hpmd Of Typmentioning

confidence: 99%