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 .
(a)
(b)
(a)
(b)
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.