Mixed group divisible designs with three groups and block size 4

AbstractA group divisible design (GDD) with three groups and block size 4 is called even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups are always even, always odd, or always mixed. It has been shown that the necessary conditions for the existence of GDDs of these three types are also sufficient except possibly for the minimal case of mixed designs for group size 5t (t>1). In this paper, we complete the undetermined families of mixed GDDs using two constructions based on idempotent self-orthogonal Latin squares and skew Room squares

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

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

Mutually orthogonal latin squares with large holes

Two latin squares are orthogonal if, when they are superimposed, every ordered pair of symbols appears exactly once. This definition extends naturally to `incomplete' latin squares each having a hole on the same rows, columns, and symbols. If an incomplete latin square of order $n$ has a hole of order $m$, then it is an easy observation that $n \ge 2m$. More generally, if a set of $t$ incomplete mutually orthogonal latin squares of order $n$ have a common hole of order $m$, then $n \ge (t+1)m$. In this article, we prove such sets of incomplete squares exist for all $n,m \gg 0$ satisfying $n \ge 8(t+1)^2 m$

Hamilton-Waterloo problem with triangle and C9 factors

The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s