NEW CONSTRUCTION METHOD OF RECTANGULAR PARTIALLY BALANCED INCOMPLETE BLOCK DESIGNS AND SINGULAR GROUP DIVISIBLE DESIGNS

In this study, we propose a new method of construct ion of rectangular Partially Balanced Incomplete Bl ock (PBIB) designs of size k = 2s and 2 ≤ s < l and singular Group Divisible (GD) designs of size k = 2l, l is the number of columns in a rectangular matrix containin g v = n × l treatments. We give the parameters expression of the obtained designs. Furthermore, we provide two algorithms of calculation for practica l use.


INTRODUCTION
A PBIB designs, based on an m-association scheme, with parameters v, b, r, k, _i, i = 1,2,3...m, is a block design with v treatments and b blocks of size k each such that every treatment occurs in r blocks and any two distinct treatments being i th associate occur together in exactly λ i blocks. For a PBIB design based on an m-class association scheme with parameters v, b, r, k, n i it is shown by (Bose and Nair, 1938), that Equation ( Rectangular designs, introduced by (Vartak, 1955), are 3-associated PBIB designs based on rectangular association scheme of v = n×l treatments arranged in n×l rectangular array such that, with respect to each treatment, the first associates are the other l-1 (= n 1 say) treatments of the same row, the second associates are other n-1 (= n 2 say) treatments of the same column and the remaining (n-1) (l-1) (= n 3 say) treatments are the third associates (Bailey, 2004).
A Group Divisible (GD) design is a 2-associated PBIB design based on a group divisible association scheme, i.e., a set of v = n×l treatments can be divided into n groups of l treatments each such that any two treatments occur together in λ 1 blocks if they belong to the same group and in λ 2 blocks if they belong to different groups (Bailey, 2004).
The experimental rectangular designs are important subclasses of partially balanced block designs. They were introduced for the first time by (Vartak, 1955), since then, several construction methods have been proposed by (Singh et al., 2011;Kageyama and Sinha, 2003) using essentially the incidence matrices.
In our work, we propose an original method of construction from a rectangular association schemes. Block designs obtained are of size k = 2s and 2 ≤ s < l. The special case s = l, provides singular group divisible designs.

DESRIPTION OF THE COMBINATORY METHOD (S)
Let v = nl treatments arranged in an array of n rows and l columns as follows: Consider s different treatments of the same row i (2≤s≤l) and associate with them s other treatments of a row i' (i ≠ i'), respecting the correspondence between the treatments a ij and a i'j . Bringing together the 2s treatments in the same block and making all possible combinations, we obtain a partially balanced incomplete block design of size k = 2s.

Theorem 1:
The combinatory method (s) for 2≤s<l associated with a rectangular association scheme, allow obtaining a rectangular PBIB design in three associated classes of parameters: s 2 s 1 s 2 1 l 2 2 l 1 3 l 2 v ln,b n n 1 C / 2, r n 1 C ,k 2s,

Proof:
• The values of v and k are obvious In fact: For each row, there is s l C possibilities and we apply the procedure with the (n -1) other rows. So, for the array we have n (n -1) s l C / 2 blocks (we divide on 2 to avoid the repetition).

Remark 1:
The relations of existence of PBIB design (1)  The more the s increases (s < l) the more the blocks' value decreases. This would provide us with economic designs comparing with the block numbers.

Algorithm of Construction of Rectangular Design
For n and l are large enough and for practical use, we propose an algorithm describing the method of construction of designs, by the combinatory method (s): The combinatory method (s) can also be used for the construction of singular group divisible designs.

Proof:
The design parameters are easily deduced from Theorem 1.

DISCUSSION
The combinatory method (S) (2 < s ≤ l) which we propose, allows the construction of rectangular PBIB designs from only rectangular association schemes for v = nl treatments, thus the parameters of those obtained designs depend only on parameters of those association schemes. By against, the usual construction methods of rectangular designs require one hand, the existence of balanced incomplete blocks BIB designs (those designs do not necessarily exist for any parameters), on the other hand, the parameters of the rectangular PBIB designs obtained by the latter methods necessarily depend on the parameters of the initial BIB designs.
In addition, our method gives group divisible designs when the parameter s = l.

CONCLUSION
Note that our method is original, little exigent as it need only rectangular association schemes. It is also programmable and thus it is accessible to all users of experimental designs. Our combinatory method (S) can be considered as a basic tool to construct new PBIB designs associated with new rectangular right angular association schemes (m), m = 4, 5 and 7 in which these numbers of associated classes are rarely or not studied in the literature of experimental designs.