U-Type Designs via New Generalized Partially Balanced Incomplete Block Designs with m = 4 , 5 and 7 Associated Classes

The traditional combinatorial designs can be used as basic designs for constructing designs of computer experiments which have been used successfully till now in various domains such as engineering, pharmaceutical industry, etc. In this paper, a new series of generalized partially balanced incomplete blocks PBIB designs with m associated classes (m = 4, 5 and 7) based on new generalized association schemes with number of treatments v arranged in w arrays of n rows and l columns (w ≥ 2, n ≥ 2, l ≥ 2) is defined. Some construction methods of these new PBIB are given and their parameters are specified using the Combinatory Method (s). For n or l even and s divisor of n or l, the obtained PBIB designs are resolvable PBIB designs. So the Fang RBIBD method is applied to obtain a series of particular U-type designs U (wnl; r wnl s 2 ) (r is the repetition number of each treatment in our resolvable PBIB design).


Introduction
Designs of computer experiments drew a wide attention in the previous two decades and were still being used successfully till now in various domains.Among the various construction methods of these designs, the traditional combinatorial designs can be used as basic designs (example: [1] [2]) and particularly the PBIB designs.The association schemes of two or three associated classes have been widely studied, while it is not the case of those over three associated classes.However, some association schemes with five associated classes have been studied (example: [3]).Besides, a method to obtain new association schemes by crossing or nesting other association schemes was given by Bailey [4] using the character tables and strata of the initial association schemes to find the parameters of the obtained association schemes.
In this paper, new association schemes with 4, 5 and 7 associated classes are described starting by a geometric representation.The parameter expressions of these association schemes are given.Moreover, some methods to construct the PBIB designs based on these association schemes are explained using an accessible construction method called the Combinatory Method (s) [5], which allows obtaining a series of PBIB designs by only using these association schemes.The parameters expressions of these new designs are given.In addition, for n or l even and s divisor of n or l, the obtained PBIB are resolvable PBIB.Then, a series of U-type designs U ; 2 r wnl wnl s is the repetition number of each treatment in the resolvable PBIB design) is obtained by applying the RBIBD method [6] on these designs.
The paper is organized as follows.In Section 2, we give new definitions of generalized association schemes with m (= 4, 5 and 7) associated classes, starting by geometric representation and we give their parameters as properties.Section 3 describes a series of construction method using the Combinatory Method (s) for obtaining the PBIB designs associated to our generalized association schemes.We give the series of the U-type designs associated to our constructed PBIB designs in Section 4. We achieve our paper with a Conclusion.
Recall some definitions: Definition 1.An m-association scheme (m ≥ 2) of v treatments [7] is a relation satisfying the following conditions: 1) Any two treatments are either 1 st , 2 nd , •••, or m th associates.The relation of association is symmetric, i.e., if the treatment α is an i th associate of β, then β is an i th associate of ( ) 2) Each treatment α has n i i th associates, the number n i being independent of ( )  .3) If any two treatments α and β are i th associates, then the number of treatments that are j th associates of α and k th associates of β is i jk p and is independent of the pair of i th associates α and β ( ) The numbers v, n i ( ) Definition 2. A PBIB design [7], based on an m-association scheme (m ≥ 2), with parameters v, b, r, k, λ i , 1, 2, i 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.The number λ i is independent of the pair of i th associates ( )

parallel class of PBIB is a collection of disjoint blocks from the b blocks whose union is V. A partition of the b blocks into q b r = parallel classes is called a resolution, and PBIB design is resolvable if it has at least one resolution and it denoted by RPBIB design.
The Combinatory Method (s) [5]: Let an array of n rows and l columns as follows: Consider s different elements of the same row i ( ) , and associate with them s other elements of a row i′ ( ) i i ′ ≠ , respecting the correspondence between the elements a ij and i j a ′ .Bringing together the 2s elements in the same block and making all possible combinations, we obtain a partially balanced incomplete block design of size 2 k s = .
Definition 3. Let U ( ) 1 ; , , r v q q  denote a design of v runs and r factors with respective 1 , , r q q  levels.This design corresponds to an v × r matrix ( )  such that the i th column x i takes values from a set of q i elements, say { } 1, , i q  , equally often.The set of all such designs, called U-type designs in the statistical literature, is denoted by U ( ) 1 ; , , r v q q  .Obviously, v must be a multiple of q i ( ) When all the q i ( ) are equal to q, we denote it by U ( ) ; r v q .Note that the rows and columns of X are identified with the runs and factors respectively [6].

Generalized Rectangular Right Angular Association Scheme (4)
Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way: Each treatment of V is associated with a unique triplet of the set 1 where: Let α be a treatment of coordinates ( ) corresponds to the treatments 3 rd associated to α .
This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (4) with four associated classes, to which we give the following equivalent definition: Definition 4. A generalized rectangular right angular association scheme (4) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w arrays of n rows and l columns such that, with respect to each treatment α : 1) The first associates of α are the other treatments of the same row in the same array.
2) The second associates of α are the other treatments of the same column in the same array.
3) The third associates of α are the remaining treatments in the same array.
4) The fourth associates of α are the other treatments of the other arrays.Property 1.The parameters of generalized rectangular right angular association schemes (4) are: A PBIB design based on a generalized rectangular right angular association scheme (4) is called generalized rectangular right angular GPBIB 4 design.

Generalized Rectangular Right Angular Association Scheme (5)
Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way: Each treatment of V is associated with a unique triplet (of coordinates) of the set 1 Let α be a treatment of coordinates ( ) corresponds to the treatments 2 nd associated to α .This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (5) with five associated classes, to which we give the following equivalent definition: Definition 6.A generalized rectangular right angular association scheme (5) is an arrangement of v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments in w (n × l) rectangular arrays such that, with respect to each treatment α : 1) The first associates of α are the other treatments of the same row in the same array.
2) The second associates of α are the other treatments of the same column in the same array.
3) The third associates of α are the remaining treatments in the same array.
4) The fourth associates of α are the treatments of same row as α , of the other arrays.
5) The fifth associates of α are the remaining treatments in the other arrays.Property 2. The parameters of the generalized rectangular right angular association schemes (5) are: A PBIB design based on a generalized rectangular right angular association scheme (5) is called generalized rectangular right angular GPBIB 5 design.

Generalized Rectangular Right Angular Association Scheme (7)
Let V be a set of v = wnl treatments, (w ≥ 2, n ≥ 2, l ≥ 2), to which we associate a geometrical representation in the following way: Each treatment of V is associated with a unique triplet of the set 1 where: Let α be a treatment of coordinates ( ) corresponds to the treatments 7 th associated to α .
This geometric representation describes a new association scheme, we call it for convenience, generalized rectangular right angular association scheme (7) with seven associated classes, to which we give the following equivalent definition: Definition 8.A generalized rectangular right angular association scheme (7) treatments in w arrays of n rows and l columns such that, with respect to each treatment α : 1) The first associates of α are the other treatments of the same row in the same array.
2) The second associates of α are the other treatments of the same column in the same array.
3) The third associates of α are the remaining treatments in the same array.
4) The fourth associates of α are the treatments in the same row and the same column as α , of the other arrays.
5) The fifth associates of α are the treatments of the same row as α in the other arrays, that are different from the fourth associates of α .
6) The sixth associates of α are the treatments of the same column as α in the other arrays, that are dif- ferent from the fourth associates of α .
7) The seventh associates of α are the remaining treatments in the other arrays.Property 3. the parameters of generalized rectangular right angular association schemes (7) are: A PBIB design based on a generalized rectangular right angular association scheme (7) is called generalized rectangular right angular GPBIB 7 design.

Construction Method of GPBIBm Designs (m = 4, 5 and 7 Associated Classes)
Let v = wnl (w ≥ 2, n ≥ 2, l ≥ 2) treatments be arranged in w arrays of n rows and l columns ( ) 1 A , ..., ( ) w A and written as follows 1, , g w ∀ = : ( ) are generalized rectangular right angular GPBIB 4 designs.Proof.For each array of the w arrays, we obtain a rectangular design with parameters: v nl * = , ( ) (see [5]).For the w arrays we obtain a generalized rectangular right angular GPBIB 4 design with parameters: v wnl = , ( ) 1, , g g w ′ ≠ ∈  they never appear together in the same block thus 4 0 λ = . Lemma 1.For the special case s = l, the previous method can also be used for the construction of nested group divisible designs, with parameters: For n or l even and s divisor of l or s, the GPBIB 4 design is a resolvable PBIB designs (RGPBIB 4 ) with r parallel classes where each parallel classes contain 2 nl q w s = blocks. Proof.
( ) ( ) ( ) ( ) n or l is even and s is divisor of n or l, then q ∈ N .
Example 1.Let v = 3 × 4 × 4 treatments be arranged in the three following arrays: , , , Applying the Combinatory Method ( ) on each array of the form ( ) ( ) 1, , g w =  , by only considering the combinations of s treatments that always contain a component of the vector ( ) , the set of all the obtained blocks provides a PBIP with 4 associated classes.
Theorem 3. The partially balanced incomplete block designs with the parameters: ○ On an array ( ) ( ) 1, , j l =  and 1, , h n =  , applying the procedure with the 1 l − other elements of the same row.There is   In total ( ) ( ) ( ) • b : Using the above construction method on each array of the form: ( ) ( ) So for the l arrays of the form the n permutations ( ) blocks.Considering all the arrays ( ) ( ) , then in total we have ( ) ( ) For w = 2, the GPBIB 4 design of Theorem 3 is a rectangular right angular PBIB 4 design with parameters as in Proposition 2 of [8].
Proposition 4 Let GPBIB 4 be a design with parameters:

= −
For n or l even and s divisor of l or s, the GPBIB 4 design is a resolvable PBIB designs (RGPBIB 4 ) with r parallel classes where each parallel classes contain 2 nl q w s = blocks.

Construction Method of GPBIB5 Designs
Let ( ) , , , . Applying the Combinatory Method (s) with s chosen { } 3, , s l ∈  , on each array in the form ( ) ( ) 1, , j l =  and 1, , j w =  by only considering the combinations of s treatments that contain a component of the vector ( ) g j C , then the set of all the blocks obtained, gives a PBIB design with 5 associated classes.
Theorem 5.The incomplete block designs with parameters:  The construction method for (s = 3), give the following generalized rectangular right angular GPBIB 5 design, with parameters: ( ) are generalized rectangular right angular GPBIB 7 designs.Proof.The design parameters are deduced from the construction method. Remark 4. • For w = 1, the GPBIB 7 design of Theorem 7 is a rectangular design with parameters as in Theorem 1 of [5].
• For w = 2, the GPBIB 7 design of Theorem 7 is a rectangular right angular PBIB 7 design with parameters as in Proposition 4 of [8].Proposition 8. Let GPBIB 7 be a design with parameters: v wnl = , ( ) For n or l even and s divisor of l or s, the GPBIB , , ,  be the j th column of the g th array and let ( ) and g, 1, , g w ′ =  ), by only considering the combinations of s treatments that always contain a component of the column ( ) g h j C (respectively the row ( ) g i R ), the set of all the obtained blocks provides a PBIP with 7 associated classes.
Theorem 9.The partially balanced incomplete block designs with the parameters: Proof.The design parameters are deduced from the construction method. Remark 5.For w = 2, the GPBIB 7 design of Theorem 9 is a rectangular right angular PBIB 7 design with parameters as in Proposition 5 of [8].
Proposition 10.Let GPBIB 7 be a design with parameters: v wnl = , ( ) ( )( ) The construction method for (s = 3), give a generalized rectangular right angular GPBIB 7 design with parameters:  In this section we apply the RBIBD method (see [6]) on our resolvable rectangular right angular RGPBIB m designs (m = 4, 5 and 7) to obtain a series of U-type designs ( ) ; r U v q .Let a resolvable GPBIB m designs (m = 4, 5 and 7) with r parallel classes 1 PC , , PC r  where each j-th class contains q b r = blocks (1 ≤ j ≤ r).Then we can construct a U-type design ( ) ; r U v q from resolvable GPBIB m designs (m = 4, 5 and 7) as follows: Algorithm RGPBIBm − UD • Step 1. Give a natural order 1, , q  to the q blocks in each parallel class PC j , 1, , j r =  .
• Step 2. For each PC j , construct a q-level column ( ) as follows: Set , n or l even and s divisor of n or l.
treatments 5 th associated to α .

1 .
Construction Method of GPBIB4 Designs 3.1.1.First Construction Method of GPBIB4 Designs Applying the Combinatory Method (s) on each of the w arrays, with chosen { } 2, , 1 s l ∈ −  , then we obtain w rectangular PBIB designs.The set of all blocks gives a PBIB design with 4 associated classes.Theorem 1.The partially balanced incomplete block designs with the parameters: v wnl = ,

1 n
right angular GPBIB 4 designs.Proof.•The v and k values are obvious.•r: For each treatment ( ) applying the procedure with the l other elements of the same row.There is 1 l s C − possibilities, each one being repeated ( ) − times, with n permutations ( ) , we have 1 w − arrays of the form ( ) of the same row i of the arrays ( ) − times.Considering all the arrays of the form ( )

−
the other hand, for 2, , h n =  , the two treatments appear together of h.Taking all the values of 2, , h n =  , we obtain ( ) times where the two treatments appear together.

then the two treatments appear together 1 C
for a given value of h , the treatment ( ) ′ and for a given value of h , the treatment( )

3 . 6 .=Example 2 .
right angular GPBIB 5 designs.Remark For w = 2, the GPBIB 5 design of Theorem 5 is a rectangular right angular PBIB 5 design with parameters as in Proposition 3 of[8].Proposition Let GPBIB 5 be a design with parameters:For n or l even and s divisor of l or s, the GPBIB 5 design is a resolvable PBIB designs (RGPBIB 5 ) with r parallel classes where each parallel classes contain 2 nl q w s = blocks.Let v = 3 × 2 × 3 treatments be arranged in the two following arrays: right angular GPBIB 7 designs.

=Example 4 .
For n or l even and s divisor of l or s, the GPBIB 7 design is a resolvable PBIB designs (RGPBIB 7 ) with r parallel classes where each parallel classes contain 2 Let v = 3 × 3 × 3 treatments be arranged in the three following arrays:

•,
if treatment α is contained in the u-th block of PC j , Step 3. The r q-level columns constructed from PC j , For v = wnl runs (w ≥ 2, n ≥ 2, l ≥ 2), a series of U-type n or l even and s divisor of n or l.
[8]orem 1 is a rectangular design with parameters as in the Theorem 1 of[5].•Forw= 2, the GPBIB 4 design of Theorem 1 is a rectangular right angular PBIB 4 design with parameters as in Proposition 1 of[8].Proposition 2. Let GPBIB 4 be a design with parameters:

Second Construction Method of GPBIB7 Designs with i
7design is a resolvable PBIB designs (RGPBIB 7 ) with r

Construction of the U-Type Designs Based on Resolvable GPBIBm Designs (m = 4, 5 and 7)
To illustrate the method, we applying the construction method for the columns and rows of the first array, where each column represents a block: