Resolvable and Near Resolvable Designs

On a characterization of symmetric balanced incomplete block designs

“…The parametric ranges in the tables of Raghavarao [13] are v ≤ 100, k ≤ 15, λ ≤ 15; in the tables of Takeuchi [18] are v ≤ 100, k ≤ 30, λ ≤ 14; in the tables of Collins [2] are v ≤ 50, k ≤ 23, λ ≤ 11; and in the table of Mathon and Rosa [9] are r ≤ 41 and k ≤ v/2. Whereas the limits of these ranges in the present work are v ≤ 111, k ≤ 55, λ ≤ 30.…”

“…And s * indicates that the design is the complement of the design of no. s. The reference tables are denoted by R: Raghavarao [13], T: Takeuchi [18], C: Collins [2], and MR: Mathon and Rosa [9], in the last column, where "a blank" indicates that the parameters are not shown in any one of the references R, T, C, MR. (Note that MR is indicated only when any of R, T, C is not shown as existence or nonexistence of the design.) These designs are also shown because for the sake of formula of concerned b and the concerned series.…”

“…In this paper we propose a new generalized expression on parameters of SBIB designs and obtain all the possible parameters of these designs, and tabulate them, where v ≤ 111 and k ≤ 55, λ ≤ 30, in which there are some SBIB designs which are not seen elsewhere when compared to the tables of Collins [2], Kageyama [7], Mathon and Rosa [9], Raghavarao [13] and Takeuchi [18]. We may take the ranges of parameters to the higher values also if need be.…”

“…And in [6] Ionin interestingly gave a method of constructing certain SBIB designs. We refer Collins [2], Kageyama [7], Mathon and Rosa [9], Raghavarao [13] and Takeuchi [18], the parameters of the designs in these references are all in the list of parameters of designs we obtained here, but some parameters of designs are seen to be existing which are not mentioned in any one of these references [2,7,9,13,18]. Lastly Table 3.1 has been given in the third section, along with the suitable formulae for the number of blocks b i , i = 1, 2, 3, which is the main theme of this paper, with all due references needed for the sake of practical users, while constructing the other designs using these SBIB designs or constructing these SBIB designs using other designs.…”

Another complete invariant for Steiner triple systems of order 15

“…They have been studied for a long time [4], [5], [6], [7], and [12]. The description of those triple systems is based on the paper of Mathon and Rosa [8]. In this study, we have used cdd, the computer code of Komei Fukuda [2] based on the double description method of Motzkin et al [11], in order to obtain a facets description of the convex hull of the characteristic vectors of the triples for each one of the 80 Steiner triple systems.…”