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

Abstract

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...

Group divisible designs with block size five

“…4.6]. For 5-GDDs of type g u , a partial solution to the design spectrum problem has been achieved, [1], [2], [5], [6], [9], [10], [11], [14], [15], [16], [18], and for future reference, we quote the main result concerning 5-GDDs in the important paper of Wei and Ge, [16], which represents a considerable advance on [9, Theorem IV. 4.16] Proof.…”