Balanced incomplete block designs with block size 8

Abel, R. Julian R.; Bluskov, Iliya; Greig, Malcolm

doi:10.1002/jcd.1010

Cited by 15 publications

(4 citation statements)

References 26 publications

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“…I initially used this Corollary to show 21 · 87 + 1 ∈ B(7, 2); this can now be shown by other means [5], but the construction of Corollary 10.4 is still a nice example.…”

Section: Corollary 104 87 ∈ Gdmentioning

confidence: 95%

Designs from projective planes and PBD bases

Greig¹

1999

J. Combin. Designs

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“…I initially used this Corollary to show 21 · 87 + 1 ∈ B(7, 2); this can now be shown by other means [5], but the construction of Corollary 10.4 is still a nice example.…”

Section: Corollary 104 87 ∈ Gdmentioning

confidence: 95%

Designs from projective planes and PBD bases

Greig¹

1999

J. Combin. Designs

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“…Proof. This follows by PBD closure, noting that the design with a replication count of 6 is P G (2,5), and therefore known. Remark 9.3.…”

Section: Unitalsmentioning

confidence: 99%

“…Remark 10.5. I initially used this Corollary to show 21 · 87 + 1 ∈ B(7, 2); this can now be shown by other means [5], but the construction of Corollary 10.4 is still a nice example.…”

Section: The Q-x Constructionmentioning

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 (q 2 + 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 a + , where E a + 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. Remark 4.2. Useful examples where 1 < t < k − 1 are not common. Bennett [11] used this in the T [17, 1; 31] case, and filled in the groups with a point at infinity to show 528 − t ∈ B({16, 17, 17 − t * , 31, 32}, 1). Another example deletes three collinear points in a T [25, 1; 31] giving 772 ∈ B({22 * , 24, 25, 30, 31}, 1) as an intermediate step in constructing a B[6, 1; 5 · 772 + 1], a value that was not constructed in [50], but has since been constructed by other means [64].If we have the basic design embedded as an arc in a projective plane, then we also have the option of extending, or spiking, any line of the plane up to its full size in the plane. Theorem 4.3. If a projective plane of order q contains a {v; K ∪ k * }-arc, and 0Remark 4.4. Note that whether {1} ∈ K or not is not inconsequential if blocks of size 2 are to be avoided.Remark 4.5. This construction has often been used with transversal designs, and the most usual construction takes a point not on the special line to generate the groups, yieldingCorollary 4.6. 123 ∈ B({8, 9, 11 * }, 1).Proof. We may embed the 120 point Seiden design (discussed in Section 5) in PG(2, 16), and take t = 3 and k = 8.Remark 4.7. This value was unknown in [11].Usually the two constructions in Theorems 4.1 and 4.3 are not mixed, as no new v values result, and extra block sizes are generated. However, when we can stand the extra block sizes, we can sometimes achieve a specific block size with a specific v. This is useful in constructing incomplete transversal designs. Theorem 4.8. If v ∈ GD(K, λ, M ) and K ⊂ RT (s, 1) and M ⊂ T (s, λ), then v ∈ T (s, λ).Proof. See [38, Theorem 3.2]. The proof also shows that if a group of size m occurs, then the design will contain T [k, λ; m] as a subdesign.Remark 4.9. Wilson [62] gives an alternative construction which is typically better than Theorem 4.8. However, Theorem 4.8 suffices here; I was not able to show that I could satisfy Wilson's hypotheses for some of the cases I needed. Lemma 4.10. If 7 ≤ a ≤ 18, then 160 + a ∈ B({7, 8, 9, 23} ∪ 13 * ∪ a * , 1).Proof. Start with a T [7, 1, 23], embedded in PG(2, 23), and select two lines that intersect in this design, and apply Theorem 4.3 twice, once with t = 6, and once with t = a − 7. If the groups of the design were generated as in Remark 4.5, since 6 + a ≤ 24, we can en...

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“…[3,7,8]). There exist a 7-GDD of type 3 15 and 7-GDDs of type 7 n for n ≡ 1 mod 6, n ∈ {19, 115, 145, 205, 235}.…”

mentioning

confidence: 99%

Pairwise balanced designs whose block size set contains seven and thirteen

Greig¹,

Grüttmüller

Hartmann

2007

J of Combinatorial Designs

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Abstract:In this paper, we investigate the PBD-closure of sets K with {7, 13} ⊆ K ⊆ {7, 13, 19, 25, 31, 37, 43}. In particular, we show that v ≡ 1 mod 6, v ≥ 98689 implies v ∈ B({7, 13}). As an intermediate result, many new 13-GDDs of type 13 q and resolvable BIBD with block size 6 or 12 are also constructed. Furthermore, we show some elements to be not essential in a Wilson basis for the PBD-closed set {v : v ≡ 1 mod 6, v ≥ 7}.

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Balanced incomplete block designs with block size 8

Cited by 15 publications

References 26 publications

Designs from projective planes and PBD bases

Designs from projective planes and PBD bases

Designs from projective planes and PBD bases

Pairwise balanced designs whose block size set contains seven and thirteen

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