Pair covering designs with block size 5

Abel, R. Julian R.; Assaf, Ahmed M.; Bennett, F. E.; Bluskov, Iliya; Greig, Malcolm

doi:10.1016/j.disc.2006.09.026

Cited by 11 publications

(26 citation statements)

References 20 publications

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“…Theorem 7 (Abel et al [2,8], Abel and Greig [9], Hanani [29] There is one very useful way (namely, completion) of exploiting RGDDs. We state the result as follows.…”

Section: Definitionmentioning

confidence: 98%

“…Using q = 9, 11 and 13 in Lemma 35 (1), we can deal with v 45 + w for w 8 and using q = 8, 9, 11 and 13 in Lemmas 35(2) and 35 (3), we can deal with v 5w for 8 w 14, until this overlaps with the 80 + w value established above. For w = 15, truncating a TD(6, 15) as in Lemma 35 (2) covers 80 v 90 plus v = 75, 76. Taking a TD (15,16), we may truncate 9 groups to have size at most 1, to get a K 6 -GDD of type 16 6 1 t so that 1 t 9, then delete a block of size 6 to get K 5 -GDD of type 15 6 …”

Section: Lemma 35mentioning

confidence: 99%

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Resolvable balanced incomplete block designs with subdesigns of block size 4

Greig²,

Ling

et al. 2008

Discrete Mathematics

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In this paper, we look at resolvable balanced incomplete block designs on v points having blocks of size 4, briefly (v, 4, 1) RBIBDs. The problem we investigate is the existence of (v, 4, 1) RBIBDs containing a (w, 4, 1) RBIBD as a subdesign. We also require that each parallel class of the subdesign should be in a single parallel class of the containing design. Removing the subdesign gives an incomplete RBIBD, i.e., an IRB(v, w). The necessary conditions for the existence of an IRB(v, w) are that v 4w and v ≡ w ≡ 4 (mod 12). We show these conditions are sufficient with a finite number (179) of exceptions, and in particular whenever w ≡ 16 (mod 60) and whenever w 1852.We also give some results on pairwise balanced designs on v points containing (at least one) block of size w, i.e., a (v, {K, w * }, 1)-PBD.If the list of permitted block sizes, K 5 , contains all integers of size 5 or more, and v, w ∈ K 5 , then a necessary condition on this PBD is v 4w + 1. We show this condition is not sufficient for any w 5 and give the complete spectrum (in v) for 5 w 8, as well as showing the condition v 5w is sufficient with some definite exceptions for w = 5 and 6, and some possible exceptions when w = 15, namely 77 v 79. The existence of this PBD implies the existence of an IRB(12v + 4, 12w + 4).If the list of permitted block sizes, K 1(4) , contains all integers ≡ 1 (mod 4), and v, w ∈ K 1(4) , then a necessary condition on this PBD is v 4w + 1. We show this condition is sufficient with a finite number of possible exceptions, and in particular is sufficient when w 1037. The existence of this PBD implies the existence of an IRB(3v + 1, 3w + 1).

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“…Theorem 7 (Abel et al [2,8], Abel and Greig [9], Hanani [29] There is one very useful way (namely, completion) of exploiting RGDDs. We state the result as follows.…”

Section: Definitionmentioning

confidence: 98%

Section: Lemma 35mentioning

confidence: 99%

Resolvable balanced incomplete block designs with subdesigns of block size 4

Greig²,

Ling

et al. 2008

Discrete Mathematics

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“…For n ¼ v À 24, we give a f4; 5; 8g-GDD of type 4 n=4 over Z n with six base blocks of size 4, which generate four parallel classes each, since they consist of points that are distinct (mod 4). Adding an extra point to each parallel class then gives a f5; 8g-GDD of type 4 n 24 1 ; fill in the groups with one more new point.…”

Section: Some New Quintessential Pbdsmentioning

confidence: 99%

“…The value 93 2 Bðf5; 9gÞ was obtained in [10], by adding 21 infinite points, one to each parallel class of a 4-RGDDð9 8 Þ; for this RGDD, see [26]. [1], and form a ð49; f5; 6; 8gÞ-PBD on these points.…”

Section: The Case K ¼ F5; 7gmentioning

confidence: 99%

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Quintessential PBDs and PBDs with prime power block sizes ≥ 8

Abel

Bennett

2004

J of Combinatorial Designs

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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).

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New results on GDDs, covering, packing and directable designs with block size 5

Abel

Assaf

Bluskov

et al. 2010

J of Combinatorial Designs

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This article looks at (5, k) GDDs and (v, 5, k) pair packing and pair covering designs. For packing designs, we solve the (4t, 5, 3) class with two possible exceptions, solve 16 open cases with k odd, and improve the maximum number of blocks in some (v, 5, k) packings when v small (here, the Schönheim bound is not always attainable). When k = 1, we construct v = 432 and improve the spectrum for v ≡ 14, 18 (mod 20). We also extend one of Hanani's conditions under which the Schönheim bound cannot be achieved (this extension affects (20t+9, 5, 1), (20t+17, 5, 1) and (20t+13, 5, 3)) packings. For covering designs we find the covering numbers C(280, 5, 1), C(44, 5, 17) and C(44, 5, k) with k ≡ 13 (mod 20). We also know that the covering number, C(v, 5, 2), exceeds the Schönheim bound by 1 for v = 9, 13 and 15. For GDDs of type g n , we have one new design of type 30 9 when k = 1, and three new designs for k = 2, namely, types g 15 with g ∈{13, 17, 19}. If k is even and a (5, k) GDD of type g u is known, then we also have a directable (5, k) GDD of type g u . q NEW RESULTS ON GDDS, COVERING, PACKING AND DIRECTABLE DESIGNS 339 2. AUXILIARY DESIGNS Some of the terminology we will use is quite standard in design theory; see [24]. For clarification of our notation (specifically how we indicate the standard parameters), we refer to pairwise balanced designs (PBDs), (including BIBDs), as (v, K , ) designs, where K is a list of block sizes that possibly occur. A group divisible design is referred to as a (K , ) GDD of group type g t 1 1 , . . . , g t n n if there are t i groups of size g i and transversal designs of order n as TD (k, n), dropping the subscript when = 1; note that a TD (k, n) is a (k, ) GDD of group type n k . The prefix "R" will denote a resolvable design.Theorem 2.1 (Hanani [36]). Necessary conditions for the existence of a (v, k, ) BIBD are that (v −1) ≡ 0 (mod k −1), v(v −1) ≡ 0 (mod k(k −1)) and v ≥ k. These conditions are sufficient when k = 5 with the definite exception of (v, k, ) = (15, 5, 2). Theorem 2.2 (Abel et al. [5]). A TD(5, n) exists for all n ≥ 4 with the definite exception of n = 6 and the possible exception of n = 10. A TD(6, n) exists for all n ≥ 5 with the definite exception of n = 6 and the possible exception of n = 10, 14, 18 and 22.The next theorem is a combination of work by several authors, see for instance, [34,35,58]. One new design, a (5, 1)-GDD of type 30 9 will be given later in Example 9.1. [34,35], Yin et al. Theorem 2.3 (Ge and Ling[58]). The necessary conditions for the existence of a (5, 1) GDD of type g u are sufficient except for g u ∈{2 5 , 2 11 , 3 5 , 6 5 }, and possibly except for a. g u = 3 45 , 3 65 ; b. g ≡ 2, 6, 14, 18 (mod 20) and:(i) g = 2 and u ∈{15, 35, 71, 75, 95, 111, 115, 195, 215}; (ii) g = 6 and u ∈{15, 35, 75, 95}; (iii) g = 18 and u ∈{11, 15, 71, 111, 115}; (iv) g ∈{14, 22, 26, 34, 38, 46, 58, 62} and u ∈{11, 15, 71, 75, 111, 115}; (v) g ∈{42, 54} and u = 15; (vi) g = 2 with gcd(30, ) = 1 and 33 ≤ ≤ 2443, and u = 15; c. g ≡ 10 (mod 20) and: (i) g...

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Pair covering designs with block size 5

Cited by 11 publications

References 20 publications

Resolvable balanced incomplete block designs with subdesigns of block size 4

Resolvable balanced incomplete block designs with subdesigns of block size 4

Quintessential PBDs and PBDs with prime power block sizes ≥ 8

New results on GDDs, covering, packing and directable designs with block size 5

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