scholarly journals The Existence of FGDRP$(3,g^u)'$s

10.37236/123 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Jie Yan ◽  
Chengmin Wang
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Main Diagonal ◽  
Parallel Class ◽  
Uniform Frame ◽  
Index 2

By an FGDRP$(3,g^u)$, we mean a uniform frame $(X,\cal G,\cal A)$ of block size 3, index 2 and type $g^u$, where the blocks of $\cal{A}$ can be arranged into a $gu/3\times gu$ array. This array has the properties: (1) the main diagonal consists of $u$ empty subarrays of sizes $g/3\times g$; (2) the blocks in each column form a partial parallel class partitioning $X \setminus G$ for some $G\in \cal G$, while the blocks in each row contain every element of $X \setminus G$ $3$ times and no element of $G$ for some $G\in \cal{G}$. The obvious necessary conditions for the existence of an FGDRP$(3,g^u)$ are $u\geq 5$ and $g\equiv 0$ (mod 3). In this paper, we show that these conditions are also sufficient with the possible exceptions of $(g,u)\in \{(6,15),(9,18),(9,28),(9,34),(30,15)\}$.

scholarly journals Packing designs with block size 5 and index 2: The case υ even

1993 ◽  
Vol 63 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Ahmed M Assaf ◽  
Nabil Shalaby ◽  
L.P.S Singh
Keyword(s):  
Block Size ◽  
Index 2

scholarly journals A family of balanced ternary designs with block size four

1986 ◽  
Vol 33 (3) ◽  
pp. 321-327
Author(s):  
Diane Donovan
Keyword(s):  
Block Size ◽  
Infinite Family ◽  
Repeated Element ◽  
Index 2

This paper shows the existence of an infinite family of cyclic balanced ternary designs where the block size is 4, the index 2 and each block contains precisely one repeated element.

scholarly journals The existence of a class of Kirkman squares of index 2

1988 ◽  
Vol 44 (1) ◽  
pp. 33-41 ◽  
Author(s):  
E. R. Lamken ◽  
S. A. Vanstone
Keyword(s):  
Block Size ◽  
Existence Results ◽  
Doubly Resolvable ◽  
Index 2 ◽  
Kirkman Square

AbstractA Kirkman square with index λ, latinicity μ, block size k and ν points, KSk(v; μ, λ), is a t × t array (t = λ(ν−1)/μ(k − 1)) defined on a ν-set V such that (1) each point of V is contained in precisely μ cells of each row and column, (2) each cell of the array is either empty or contains a k-subset of V, and (3) the collection of blocks obtained from the nonempty cells of the array is a (ν, k, λ)-BIBD. For μ = 1, the existence of a KSk(ν; μ, λ) is equivalent to the existence of a doubly resolvable (ν, k, λ)-BIBD. In this case the only complete results are for k = 2. The case k = 3, λ = 1 appears to be quite difficult although some existence results are available. For k = 3, λ = 2 the problem seems to be more tractable. In this paper we prove the existence of a KS3(ν; 1, 2) for all ν ≡ 3 (mod 12).

scholarly journals GBRDs with Block Size Three over 2-Groups, Semi-Dihedral Groups and Nilpotent Groups

10.37236/519 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
R. Julian R. Abel ◽  
Diana Combe ◽  
Adrian M. Nelson ◽  
William D. Palmer
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Nilpotent Groups ◽  
Dihedral Groups ◽  
Even Order ◽  
Generalized Bhaskar Rao Design

There are well known necessary conditions for the existence of a generalized Bhaskar Rao design over a group $\mathbb{G}$, with block size $k=3$. We prove that they are sufficient for nilpotent groups $\mathbb{G}$ of even order, and in particular for $2$-groups. In addition, we prove that they are sufficient for semi-dihedral groups.

scholarly journals Group Divisible Designs with Two Associate Classes and

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Nittiya Pabhapote ◽  
Narong Punnim
Keyword(s):  
Necessary Conditions ◽  
Block Size ◽  
Group Divisible Designs ◽  
Group Divisible

The original classiffcation of PBIBDs defined group divisible designs GDD() with . In this paper, we prove that the necessary conditions are suffcient for the existence of the group divisible designs with two groups of unequal sizes and block size three with .

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