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ISRN Discrete Mathematics
Volume 2012 (2012), Article ID 251457, 7 pages
http://dx.doi.org/10.5402/2012/251457
Research Article

Pairwise Balanced Design of Order and 2-Fold System of Order

1Department of Mathematics, University of Kalyani, Kalyani 741235, India
2Department of Mathematics, National Institute of Technology, Burdwan, Durgapur 713209, India

Received 4 September 2012; Accepted 23 October 2012

Academic Editors: Y. Hou and M. Pal

Copyright © 2012 Manjusri Basu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In the Steiner triple system, Bose (1939) constructed the design for and later on Skolem (1958) constructed the same for . In the literature we found a pairwise balanced design (PBD) for . We also found the 2-fold triple system of the orders 3n and . In this paper, we construct a PBD for and a 2-fold system of the order . The second construction completes the 2-fold system for all .

1. Introduction

A Latin square of order is an array, each cell of which contains exactly one of the symbols in , such that each row and each column of the array contain each of the symbols in exactly once. A latin square is said to be idempotent if  cell contains symbol for . A latin square of order is said to be half-idempotent if for cells and contain the symbol . A latin square is said to be commutative if cells and contain the same symbol, for all .

A quasi-group of order is a pair , where is a set of size and “” is a binary operation on such that for every pair of elements the equations and have unique solutions. As far as we are concerned a quasi-group is just a latin square with a headline and a sideline.

A design is an ordered pair where is a set of points and , called a block set, is of subsets of with the property that every subset of is contained in exactly blocks [1]. A design is defined as a Steiner system and denoted by . A Steiner triple system (STS) is an ordered pair , where is a finite set of points or symbols, and is a set of 3-element subsets of called triples, such that each pair of distinct elements of occurs together in exactly one triple of . The order of a Steiner triple system is the size of the set , denoted by .

Theorem 1.1 (see [2, Theorem ]). A Steiner triple system of order exists if and only if or .

The Bose Construction (, see [2, 3])
Let and let be an idempotent commutative quasi-group of order , where . Let , and define to contain the following two types of triples.

Type 1. For , (see Figure 2).

Type 2. For , (see Figure 2).

Then is a Steiner triple system of order .

The Skolem Construction (, see [2, 4])
Let and let be a half idempotent commutative quasi-group of order , where . Let , and define as follows.

Type 1. For .

Type 2. For .

Type 3. For (see Figure 3).

Then is a Steiner triple system of order .

For a positive integer, a -wise balanced design is an ordered pair , where is a finite nonempty set (of points) and is a finite nonempty multiset of subsets of (called blocks), such that every subset of is contained in a constant number of blocks. If and is the set of sizes of the blocks, then we call a design. If all blocks of have the same size (i.e., ), then is called a design or a design. A pairwise balanced design of order with block sizes from is a pair , is a family of subsets (blocks) of that satisfy if , then and every pair of distinct elements of occurs in exactly blocks of .

The Construction (see [2])
Let be an idempotent commutative quasi-group of order , where and let be the permutation . Let and let contain the following blocks.

Type 1. .

Type 2. ,, for .

Type 3. for .

Then is a with exactly one block of size 5 and the rest of size 3.

A -fold triple system is a pair of , where is a finite set and is a collection of -element subsets of called triples such that each pair of distinct elements of belongs to exactly triples of .

Theorem 1.2 (see [2, Theorem ]). The spectrum of -fold triple systems is precisely the set of all or .

Construction (see [2])
Let be an idempotent (not necessarily commutative) quasi-group, where . Let . We denote which contains the following two types of triples.

Type 1. occurs exactly twice in for all .

Type 2. For , the six triples, ,.

Then is a -fold triple system of order .

Construction (see [2, 4])
Let be an idempotent (not necessarily commutative) quasi-group, where . Let . We denote which contains the following two types of triples.

Type 1. The four triples , for all

Type 2. If , the six triples ,.

Then is a -fold triple system of order .

2. Main Results

Construction
Let be an idempotent commutative quasi-group of order , where . Now we construct where and contains the following blocks.

Type 1. , for .

Type 2. for .

Theorem 2.1. The ordered pair is a pairwise balanced design .

Proof. The number of points is (see Figure 1). The blocks of Type 1 are of length , whereas the blocks of Type 2 are of length . It is clear from the construction that any pair of points of occurs in exactly one block. Hence is a PBD .
The number of blocks of the PBD are .

fig1
Figure 1: Graphical representation of construction.
fig2
Figure 2: Graphical representation of construction (Type 1 and Type 2).
251457.fig.003
Figure 3: Graphical representation of construction (Type 3).

For example, , the quasi-group of order is Let and contains the following blocks.

Type 1. .

Type 2. , ,,.

So (S,B) is a PBD with a total of 35 blocks in which (=5) blocks of size 4 and (=30) blocks of size 3.

Construction
Let be an idempotent (not necessarily commutative) quasi-group, where . Let . We define as follows.

Type 1. occurs exactly twice in .

Type 2. For , , for all .

Type 3. If , the six triples ,.

Theorem 2.2. The ordered pair is a -fold system of order .

Proof. The number of points is . The Type 1 of new construction consists of block length 5. The Type 2 and Type 3 of the above construction consist of length . Any pair of points of occurs in exactly two blocks. Hence the construction is a -fold system. Also the ordered pair holds design.

For example, , the quasi-group of order is Let and contains the following blocks.

Type 1. , .

Type 2. ,, .

Type 3. ,,, .

So is a PBD with blocks of size and the rest of the blocks of size 3.

3. Conclusion

There already exist STS for and and PBD for . In this paper, we construct PBD for . The PBD for and, is still now open. The 2-fold triple system exists for or   and, in this paper, we construct 2-fold system exits for . These complete that the 2-fold system holds for all natural numbers .

Acknowledgment

Debabrata Kumar Ghosh thanks UGC-JRF for the financial support of his research work.

References

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  3. R. C. Bose, “On the construction of balanced incomplete block designs,” vol. 9, pp. 353–399, 1939. View at Zentralblatt MATH
  4. Th. Skolem, “Some remarks on the triple systems of Steiner,” Mathematica Scandinavica, vol. 6, pp. 273–280, 1958. View at Zentralblatt MATH