Quintessential PBDs and PBDs with prime power block sizes ≥ 8

Assaf

2008

Discrete Mathematics

“…For all other odd values of t 5, there exists a (t, {5, 7, 9}, 1)-PBD [2]. Since we have (5, 2)-MGDDs of type 10 t for t = 5, 7, 9, the result follows from Lemma 2.7.…”

Section: Typementioning

confidence: 66%

“…The next lemma gives a couple of PBDs whose existence was mentioned in [4], but were not given in references quoted within, such as [2,9]. Proof.…”

Section: Constructionsmentioning

confidence: 98%

Section: G ≡ 0 1 (Mod 5)mentioning

confidence: 99%

“…, v 37 and v = 113 [2,9]. [11] by constructing a 5-GDD of type 90 7 and filling in its groups with a (5, 1)-GDD of type 10 9 .…”

Section: Constructionsmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Modified group divisible designs with block size 5 and even index

Assaf

2008

Discrete Mathematics

Super‐simple holey Steiner pentagon systems and related designs

J of Combinatorial Designs

Bennett

2007

A Steiner pentagon system of order v (SPS(v)) is said to be super-simple if its underlying (v, 5, 2)-BIBD is super-simple; that is, any two blocks of the BIBD intersect in at most two points. It is well known that the existence of a holey Steiner pentagon system (HSPS) of type T implies the existence of a (5, 2)-GDD of type T. We shall call an HSPS of type T super-simple if its underlying (5, 2)-GDD of type T is super-simple; that is, any two blocks of the GDD intersect in at most two points. The existence of HSPSs of uniform type h n has previously been investigated by the authors and others. In this article, we focus our attention on the existence of super-simple HSPSs of uniform type h n .

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

J of Combinatorial Designs

Assaf

Bluskov

et al. 2010

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...