Construction of Some Resolvable tdesigns
Alilah David
Department of Mathematics, Masinde Muliro University of Science and Technology, Nairobi, Kenya
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To cite this article:
Alilah David. Construction of Some Resolvable tdesigns. Science Journal of Applied Mathematics and Statistics. Vol. 5, No. 1, 2017, pp. 4953. doi: 10.11648/j.sjams.20170501.17
Received: July 20, 2016; Accepted: August 8, 2016; Published: February 22, 2016
Abstract: The A tdesign is a generation of balanced incomplete block design (BIBD) where λ is not restricted to the blocks in which a pair of treatments occurs but to the number of blocks in which any t treatments (t = 2,3...) occurs. The problem of finding all parameters (t, v, k, λ_{t}) for which t(v, k,λ_{t}) design exists is a long standing unsolved problem especially with λ=1 (Steiner System) as no Steiner tdesigns are known for t ∶ 6 when v > k. In this study tdesign is constructed by relating known BIB designs, combinatorial designs and algebraic structures with tdesigns. Additionally, an alternative approach for the construction of tdesigns that provides a unified framework is also presented.
Keywords: Block Designs, Resolvable Designs, tdesigns
1. Introduction
A design is an incidence structure of points and blocks with the following properties; there are points, each block is incident with points, any point is incident with blocks, and any points are incident to common blocks. Where and are all positive integers and . The four numbers and determine (blocks) and and four numbers themselves cannot be chosen arbitrarily [2].
The incidence structure associated with a design can be represented by a matrix. The pointblock incidence matrix , associated with a design with blocks is a matrix of rows and columns. The elements of are where is the point, is the block and
There is a generalization of Fisher’s inequality to designs which is due to RayChaudhuri [14] and Wilson [16]. If a design exists, where is even, then the number of blocks . A design in which is called Steiner system. For example a is a Steiner triple system (STS) and a design is a Steiner quadruple system (SQS). A design is called a balanced incomplete block design (BIBD). A tdesign is said to have repeated blocks if there are two blocks incident with the same set of k points. A tdesign with no repeated blocks is said to be simple [14].
A design with are known for only a few values of and . For there are several infinite families known. For instance, for any prime power and for any , there exists a design known as inversive geometry [7]. When, these designs are known as inversive planes. A Steiner quadruple system is also known to exist for all . Some simpledesigns, have been constructed for . Construction of a design remains one of the outstanding open problems in the study of tdesigns. Even for and , only a few examples of designs are known. In this study we construct some designs, with much emphasis on , by identifying BIB designs which are also designs [5].
2. Literature Review
The main problem in designs is the question of existence and the construction of those solutions, given admissible parameters. That is, finding all parameters for which design exists. There are many known Steiner designs but constructing Steiner it has proved to be much harder. In the case of , Kageyama [11] has shown thatthere is design if andonly if the necessary arithmetic conditions are satisfied. But for larger k, even , the result is far from complete. For the problem is wide open. All these constructions bear a distinct algebraic flavor in the sense that the underlying set upon which the design is constructed has a nice algebraic structure. Algebraic construction requires that a certain fixed (big) group to act as a group of automorphisms for the desired design.
Mathon and Rosa [12] came up with block spreading method for and for prime power index. Let be a positive integer and let be a prime power. Suppose that there exists a design satisfying . Then there exists a group divisible design (GDD) of group type with block size and index one, whenever . This method has application in the construction of Steiner designs.
Let and be positive integers, , and let be a prime power. Then there exists a number such that for any design satisfying _{, }there is a GDD of group type with block size and index one whenever . Let and be a positive integers . Then there exists a number such that for any design with prime power decomposition satisfying ;; there is a GDD of group type with block size and index one whenever . This generalized "block spreading" construction has several application such as constructing new Steiner designs and new group divisible designs with index one. Limitation of this method is that the bounds on are too large.
A block design is a family of subsets of a set of elements such that, for some fixed and , with ,; each subset has elements, and each pair of elements of occurs together in exactly subsets. The elements of are called varieties, and the subsets of are called the blocks. From Anderson [2], in a block design each element lies in exactly blocks, where
and(1)
The five parameters of a block design are therefore not independent, but have two restrictions as stated in the theorem. Whatever are, they must satisfy (1), but conversely if five numbers satisfy (1.1), there is no guarantee that a configuration exists as described by [15].
Mohácsy and RayChaudhuri [12] constructed designs from known wise balanced designs. In his works he showed that, given a positive integer and a design , with all blockssizes occurring in and , the construction produces a design , with . Onyango [16] on his part constructed designs with and from balanced incomplete block design.
Incidence Matrix
The incidence matrix of a BIBD satisfies the following properties: every column of contains exactly "1"s; every row of contains exactly "1"s; and two distinct rows of contain "1" in exactly columns
Theorem (Stinson (2004)). Let be a 01 matrix and let . Then is an incidence matrix of aBIBD if and only if and where denotes a unity matrix and denotes a matrix with every entry equal to 1.
Example of constructing BIB designs
Now consider , the conditions are fulfilled with;
Let the points be A, B, C, D, E, F, and G. Ordering the 3 blocks with A first and assume that BC, DE and FG are together in these blocks. The following results are obtained:
A  A  A 
B  D  F 
C  E  G 
Second step
Next B and C must occur twice more and not together (order of blocks not important). The result is:
A  A  A  B  B  C  C 
B  D  F  
C  E  G 
Third step
D and F must occur together. We can assume this happens in a B block. The E and G must be together in the other B block. Then there is only one choice for the two C blocks (because order is unimportant). Results obtained are:
A  A  A  B  B  C  C 
B  D  F  D  E  D  E 
C  E  G  F  G  G  F 
When using this design for practical experiments randomization is a must. Treatments according to the labels will be randomized as per the order of the blocks, and the order of the three treatments within a block. The complementary design is constructed by replacing each block with a block consisting of the remaining points. For this case this results in:
D  B  B  A  A  A  A 
E  C  C  C  C  B  B 
F  F  D  E  D  E  D 
G  G  E  G  F  F  G 
Results obtained are and . It is noted that the same BIB designs can be constructed by use of PG (2, 2).
3. Construction of Resolvable 3(v, k, λ_{t}) Design
In this construction, technique introduced by Adhikari [1] of using the symmetric differences of pairs of blocks of incomplete blocks designs to construct other designs and the technique of arithmetic of integers modulo nis applied.
3.1. Resolvable 3design with Parameters v=8,b=14,r=7,k=4,λ2=3,λ3=1
Consider resolvable 3design with parameters
Let be the set of equivalence classes mod 7 and and form a base for a (14, 8, 7, 4, 3) cyclic design mod 7. When2 is added to each element of and and same process is continued, blocks of the design are obtained as follows;














Replacing residues with integers and with 8, the following results are obtained;
Computing the differences modulo 7 and from pairs of distinct elements in , the following values of the block designs are obtained:
31 = 2  13 = 5  20 = 2  02 = 5 
41 = 3  14 = 4  50 = 5  05 = 2 
∞1 = ∞  1∞ = ∞  60 = 6  06 = 1 
43 = 1  34 = 6  52 = 3  25 = 4 
∞3 = ∞  3∞ = ∞  62 = 4  26 = 3 
∞4 = ∞  4∞ = ∞  65 = 1  56 = 6 
This results in ∞, ∞ and each non zero residue mod 7 exactly thrice as a difference of two elements in . This design is a (14,8,7,4,3) BIBD which would result into a 3(8,4,1) design.
3.2. Construction of tdesign with Parameters v=12,b=22,r=11,k=6,λ_{2}=5,λ_{3}=2
Affine 3design with parameters
Let is the set of equivalence classes mod 11 and and then is a base for the design.
Replacing residues with integers and with 12, the following results are obtained;
Case of
Construction of where
If is the set of equivalence classes mod 7 and , and
Following the same procedure, design below with integers as elements is obtained;
Construction of design where
If is the set of equivalence classes mod 7 and and
Using the same procedure, the design below with integers as elements is obtained.










































Case of
Construction of where
Let is the equivalence classes’ mod 11 and and then is the base for the design below with integers as elements:












































This construction is equivalent to "sum construction ", of BIBDs, but in this case a BIBD is added to a BIBD that is automorphic to it. Therefore new BIBDs can be formed by the collection of a BIBD with its automorphic BIBDs.
4. Conclusion
The study has presented an alternative method that is simpler and unified for the construction of BIBDs that are very important in the experimental designs. As it provides designs for different values of , unlike many methods that provide designs for a single value of . The construction framework designed provides a platform at which new BIBDs can be formed by the collection of a BIBD with its automorphic BIBDs. In order to obtain combinatorial constructions of unique block designs, different kind of combinatorial designs are very effective.
Recommendations
Although this study has provided a technique for the construction of designs, it is still clear that construction method of designs is not known in general. In fact, it is not clear how one might construct designs with arbitrary block size.
References