Group divisible designs with block size 4 and group sizes 4 and 7

In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t, s)$.Comment: arXiv admin note: substantial text overlap with arXiv:2109.1122

Group divisible designs of four groups and block size five with configuration (1; 1; 1; 2)

We present constructions and results about GDDs with four groups and block size five in which each block has Configuration $(1, 1, 1, 2)$, that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD$(n, 4, 5; \lambda_1, \lambda_2)$ with Configuration $(1, 1, 1, 2)$, and show that the necessary conditions are sufficient for a GDD$(n, 4, 5; \lambda_1,$ $\lambda_2)$ with Configuration $(1, 1, 1, 2)$ if $n \not \equiv 0 ($mod $6)$, respectively. We also show that a GDD$(n, 4, 5; 2n, 6(n - 1))$ with Configuration $(1, 1, 1, 2)$ exists, and provide constructions for a GDD$(n = 2t, 4, 5; n, 3(n - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 12$, and a GDD$(n = 6t, 4, 5; 4t, 2(6t - 1))$ with Configuration $(1, 1, 1, 2)$ where $n \not= 6$ and $18$, respectively

GDD (n1, n, n + 1, 4; λ1, λ2) for n1 = 1 or 2

Paper presented at the 4th Strathmore International Mathematics Conference (SIMC 2017), 19 - 23 June 2017, Strathmore University, Nairobi, Kenya.The main subject matter for this paper is GDDs with three groups of sizes 1, n, (n≥ 2) and n + 1, respectively, and block size four. A block has Configuration (1, 1, 2), means the block has the point from the group of size 1 and one point from one of the other two groups and the remaining two points from the third group. A block has configuration (2, 2) if the block has exactly two points from each of the two groups of sizes n and n + 1. First, we prove that these GDDs do not exist if we require that the number of the blocks having Configuration (1, 1, 2) is equal to the number of block shaving Configuration (2, 2). Then we provide necessary conditions for the existence of a GDD ({1, n, n + 1}, 3, 4; λ1 , λ2) and prove that these conditions are sufficient for several families of GDDs. We also prove several nonexistence results, where these usual necessary conditions are satisfied.Mbarara University of Science and Technology,Uganda

THE ANALYSIS OF THE ADDITIVE MIXED MODEL FOR CLASSES OF NON ORTHOGONAL DESIGNS

Tests for fixed and random effects can be difficult to derive for nonorthogonal designs with mixed models. However, extensions of the intrablock and inter-block analyses of Balanced Incomplete Block Designs can often be obtained. Here we derive the extensions for the broad class of Group Divisible Designs. Decompositions of the design space are used to develop exact tests for fixed and random effects in the additive mixed model with random block effects. Conditions on the design which permit the standard use of the intra-block and inter-block test statistics are given. Important subclasses of Group Divisible Designs include Equireplicate Variance Balanced Block Designs and Group Divisible Partially Balanced Incomplete Block Designs with Two Associate Classes. These two subclasses are also examined. An example from the literature of an experiment on fruit trees is used to illustrate the methods

Switching codes and designs

AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters

Anti-Pasch optimal coverings with triples

It is shown that for $v\ne 7,8,11,12$ or $13$, there exists an optimal covering with triples on $v$ points that contains no Pasch configurations

Algorithms for classification of combinatorial objects

A recurrently occurring problem in combinatorics is the need to completely characterize a finite set of finite objects implicitly defined by a set of constraints. For example, one could ask for a list of all possible ways to schedule a football tournament for twelve teams: every team is to play against every other team during an eleven-round tournament, such that every team plays exactly one game in every round. Such a characterization is called a classification for the objects of interest. Classification is typically conducted up to a notion of structural equivalence (isomorphism) between the objects. For example, one can view two tournament schedules as having the same structure if one can be obtained from the other by renaming the teams and reordering the rounds. This thesis examines algorithms for classification of combinatorial objects up to isomorphism. The thesis consists of five articles – each devoted to a specific family of objects – together with a summary surveying related research and emphasizing the underlying common concepts and techniques, such as backtrack search, isomorphism (viewed through group actions), symmetry, isomorph rejection, and computing isomorphism. From an algorithmic viewpoint the focus of the thesis is practical, with interest on algorithms that perform well in practice and yield new classification results; theoretical properties such as the asymptotic resource usage of the algorithms are not considered. The main result of this thesis is a classification of the Steiner triple systems of order 19. The other results obtained include the nonexistence of a resolvable 2-(15, 5, 4) design, a classification of the one-factorizations of k-regular graphs of order 12 for k ≤ 6 and k = 10, 11, a classification of the near-resolutions of 2-(13, 4, 3) designs together with the associated thirteen-player whist tournaments, and a classification of the Steiner triple systems of order 21 with a nontrivial automorphism group.reviewe