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

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