Resolvable partially pairwise balanced designs and their applications in computer experiments

Transcription

1 Resolvable partially pairwise balanced designs and their applications in computer experiments Kai-Tai Fang Department of Mathematics, Hong Kong Baptist University Yu Tang, Jianxing Yin Department of Mathematics, Suzhou University Abstract The uniform design is one of major designs in computer experiments. In this paper, we develop the connection among the computer experiments, uniform designs in statistics and resolvable partially pairwise balanced designs (RPPBDs) from design theory. A series of RPPBDs with special requirements are constructed and used to produce new uniform designs under the discrete discrepancy. Keywords: resolvable partially pairwise balanced design; application; computer experiment; construction. 1 Introduction In the past decades, there has been an incredible growth in combinatorial design theory. This is not only due to a desire of the development of design theory itself, but also stimulated by the various applications of designs in This research is supported in part by a Hong Kong RGC grant RGC/HKBU 2044/02P and NSFC Grant Corresponding author: Jianxing Yin. address: jxyin@suda.edu.cn 1

2 communications, combinatorial chemistry, computer science and statistics. In this paper applications of combinatorial designs in computer experiments are discussed. Design of computer experiments is a rapidly growing area and is particularly useful in various complicated systems. It has been paid much attention in the past two decades. Computer models are often used in science, finance and engineering fields to describe complicated physical phenomena. Suppose that we have a device/process in an engineering system or in a financial system. The behavior of the device/process depends on a number of input variables x 1,..., x s. Based on the professional knowledge we can calculate the responses from the input variables by y = g(x 1,..., x s ). Due to complexity and nonlinearity of the device/process the function g has no analytic formula or the output y may be the solution of a set of differential equations, or is the output by the use of several computer softwares. Based on a set of design points and their responses, one wishes to find an approximate model that is much simpler than the true one (cf. Figure 1). This idea needs a space filling design that arranges experimental points uniformly scattered on the domain of the input variables. Many space filling designs have been suggested. See, for example, [3] and [14] for a comprehensive review on designs of computer experiments. The uniform design (UD) is one such space-filling design in computer experiments, and was motivated by three projects in system engineering in 1978 (see [6, 19]). In the past twenty years, the uniform design has been successfully applied in many areas, such as industry, system engineering, pharmaceutics, and natural sciences (see [8, 10]). Generating a uniform design is a very difficult task and is an NP hard problem. Several methods, such as the good lattice point method in quasi- Monte Carlo methods, the Latin square method, and optimization heuristic threshold accepting method, were proposed. All these methods involve a heavy computing search. We have found resolvable partially pairwise balanced designs with certain extra properties (see below for its definition) can be used directly to construct uniform designs. Therefore, in this article we propose a new method for construction of uniform designs without any computer calculation via combinatorial designs. Some constructions and existence results for such combinatorial configurations are provided. A number of new uniform designs are then obtained. The paper is organized as follows. Section 2 presents the relationship between resolvable partially pairwise balanced designs and uniform designs under the discrete discrepancy. In Section 3, various constructions for resolvable partially pairwise balanced designs are proposed. And a number of new uniform designs are listed. The last section contains some remarks. 2

3 2 Relationship between RPPBDs and uniform designs First let us review some terminology from combinatorial design theory. Definition 2.1 Let n be a positive integer and K be a set of positive integers. A group divisible design of index λ, denoted by (K, λ)-gdd is a triple (V, G, B), where (1) V is a set of n points; (2) G is a partition of V into subsets (called groups); (3) B is a family of subsets (called blocks) of V with sizes from K, such that a group and a block contain at most one common point; (4) every pair of points from distinct groups occurs in exactly λ blocks. The group-type (or type) of a GDD is the multiset { G : G G}, where G represents the cardinality of G. Usually, an exponential notation is u used to describe the type of a GDD: A GDD of type t 1 u2 u 1 t 2 t l l is a GDD which has u i groups of size t i for 1 i l. When K = {k}, the notation (k, λ)-gdd is used. A (k, λ)-gdd of type n k is referred to as a transversal design (TD), denoted by TD(k, λ; n). If for all i = 1, 2,..., l, t i = 1, then a (K, λ)-gdd of type 1 n is called a pairwise balanced design (PBD), or a (n, K, λ)-pbd. If K = {k}, then a (n, {k}, λ)-pbd is indeed a well-known balanced incomplete block design (BIBD), or a (n, k, λ)-bibd. A set of blocks in a GDD (or TD, PBD, BIBD) that partitions its point set V is called a parallel class. A parallel class is said to be uniform if every block in the class is of the same size. If the blocks of a GDD (or TD, PBD, input x 1. x s system y output approximate model Figure 1: Computer Experiments 3

4 BIBD), B, can be partitioned into parallel classes, the GDD (or TD, PBD, BIBD) is then called resolvable, and denoted by RGDD (or RTD, RPBD, RBIBD). Definition 2.2 A partially pairwise balanced design of order n, with block sizes from K, or a (n, K; λ 1, λ 2 )-PPBD is an ordered pair (V, B), where (1) V is a set of n points; (2) B is a family of subsets (called blocks) of V with sizes from K, such that every pair of distinct points occurs exactly in λ 1 or λ 2 blocks. The notion of resolvability extends in the natural way to PPBDs. As with GDDs, the notation RPPBDs stands for resolvable PPBDs. For statistical purpose, here we are mainly interested in RPPBDs which have mutually distinct parallel classes, each of them being uniform. In the sequel, we use RPPBDs to denote such RPPBDs. Similar notations RGDDs, RTDs, RPBDs and RBIBDs are also adopted. Using our notation, an RGDD (resp. RTD, RPBD and RBIBD) with index λ = 1 and with uniform parallel classes is an RGDD (resp. RTD, RPBD and RBIBD). We will make use of this fact frequently in the remainder of this paper. It should be emphasized that for given parameters n, K = {k 1, k 2,..., k s }, λ 1 and λ 2, the number of parallel classes of a (n, K; λ 1, λ 2 )- RPPBD is not a constant, but depends on its construction. In view of this, when an RPPBD contains r i parallel classes of block size k i (i = 1, 2,..., s), we say that it is of class type k r 1 1 kr 2 2 kr s s. Example 2.3 Suppose V = {1, 2,..., 8}. Let λ 1 = 2, λ 2 = 1 and K = {4, 2}. Then the following blocks form a (8, {4, 2}; 2, 1)- RPPBD with class type {1, 3, 5, 7}, {2, 4, 6, 8}, {1, 3, 6, 8}, {2, 4, 5, 7}, {1, 4, 5, 8}, {2, 3, 6, 7}, {1, 4, 6, 7}, {2, 3, 5, 8}, {1, 2}, {3, 4}, {5, 6}, {7, 8}. The following result is simple, but useful. Theorem 2.4 If there exist a (K, λ)- RGDD of type (g n/g ) and a (g, K, µ)- RPBD, then there exists a (n, K; λ, µ)- RPPBD. Now we give a brief description for the notion of uniform designs. 4

5 Definition 2.5 A U-type design U(n; q 1 q m ) of n runs and m factors with respectively q 1,..., q m levels corresponds to an n m matrix X = (x 1,..., x m ) such that the i-th column x i (i = 1, 2,..., m) takes values from a set of q i elements, say {1,..., q i }, equally often. The set of U(n; q 1 q m ) s is denoted by U(n; q 1 q m ). For a given measure of nonuniformity M, a design is called a uniform design, denoted by U n (q 1 q m ) if it has the smallest M-value over U(n; q 1 q m ). When some q j s are the same, the corresponding U-type and uniform design are denoted by U(n; q r 1 1 qr s s ) and U n (q r 1 1 qr s s ) respectively, where r r s = m. Remark: In conducting experiments, scientists and engineers often require any two distinct pair of factors not to be fully aliased, i.e., any column of the design matrix cannot be obtained by permuting levels from another one. This is the reason that we need an RPPBD, not only an RPPBD here. Various discrepancies for measuring non-uniformity have been widely used in quasi-monte Carlo methods as well as in experimental designs. In this paper we employ the so called discrete discrepancy for measuring the non-uniformity of design points. By using a reproducing kernel in Hilbert space, discrete discrepancy is directly defined on such a discrete domain. It has many good properties. Interested reader may refer to [12] and [15] for its detailed description. In what fillows, we use D(X; a, b) to denote the discrete discrepancy with parameters a and b for a U-type design X. Let X = (x 1, x 2,..., x m ) be a U-type design U(n; q 1 q m ). Define a matrix Z = (Z (1), Z (2),..., Z (m) ), where Z (j) = (z (j) lα ) is an n q j sub-matrix with z (j) lα = { 1, if factor xj takes level α in run l; 0, otherwise. Z is called the induced matrix of X. Suppose (λ ij ) = ZZ. Then λ ij (i j) represents the number of coincidences between any two distinct rows of X. The following analytical expression and lower bound of the discrete discrepancy on U(n; q 1 q m ) are implicitly contained in [7, 9]. Theorem 2.6 Let X be a U-type design U(n; q 1 q m ) and Z be its m induced matrix. Let (λ ij ) = ZZ i=1, µ = n/q i m and γ = µ where n 1 x denotes the integer part of x. Then D 2 (X; a, b) = m [ ] a+(qj 1)b q j + am n + n ( bm a ) λij n 2 b, (2) j=1 i,j=1,i j (1) 5

6 n ( a b ) λij ( n(n 1)[(γ + 1 µ) a ) γ ( b + (µ γ) a ) γ+1], b (3) i,j=1,i j and the lower bound on the right hand side of (3) can be achieved if and only if all the off-diagonal entries of ZZ take the same value γ, or take only two values γ and γ + 1. Now we are in a position to describe the connection between RPPBDs and uniform designs under the discrete discrepancy. Given a (n, {k 1, k 2,..., k s }; λ 1, λ 2 )- RPPBD with class type k r1 1 kr2 2 kr s s (V, B), where V = {1, 2,..., n} and r i = m. For each parallel class P j i=1 (j = 1, 2,..., m) of block size k i (1 i s), give a natural order 1, 2,..., n k i to the n k i blocks and construct a n k i -level column d j = (d lj ) as follows: set d lj = u, if point l is contained in the u-th block of P j of B. Then the m columns constructed from P j of B (j = 1, 2,..., m) form an experiment design with n runs and m factors. The level of j-th factor is n k i (j = 1, 2,..., m; 1 i s). Further, we can prove that Theorem 2.7 If the (n, {k 1, k 2,..., k s }; λ 1, λ 2 )- RPPBD with class type k r1 1 kr2 2 krs s used above satisfies the condition λ 1 λ 2 1, then its derived experiment design is a uniform design U n ((n/k 1 ) r1 (n/k s ) rs ) under the discrete discrepancy. Proof. Let X be the experiment design derived from a (n, K; λ 1, λ 2 )- RPPBD with class type k r 1 1 kr 2 2 krs s satisfying λ 1 λ 2 1 as above, and let Z be the induced matrix of X. Since all the parallel classes are uniform in the (n, K; λ 1, λ 2 )- RPPBD, each element appears in any column of design X the tame number of times. Hence, it is a U-type design. Now suppose ZZ = (λ ij ), by Theorem 2.6, we need to show that all the off-diagonal entries of ZZ take the same value γ, or take only two values γ and γ + 1, i.e. the numbers of coincidences between any two distinct rows of X can only take two value, whose differences don t exceed 1. In fact, the elements in rows r i and r j in X are coincident if and only if the pair (i, j) contained in the same block of the RPPBD. But by the definition of RPPBD, the pair (i, j) is contained exactly in λ 1 blocks or in λ 2 blocks. As λ 1 λ 2 1, the uniform property of the design then follows by Theorem 2.6. Finally, the property that parallel classes of the RPPBD are mutually distinct guarantees that there are no fully aliased columns in X. s, 6

7 Example 2.8 Applying the above technique to the RPPBD shown in Example 2.3, we obtain a uniform design U 8 ( ). row It is worth mentioning that if we regard each row of the uniform design U n (q 1 q m ) obtained from a (n, K; λ 1, λ 2 )- RPPBD as a codeword over alphabet I q, then the design forms a q-ary (m, n, m λ) constant weight code with size n and Hamming distance m λ, each of its codewords has length m, weight m, where q = max {q 1, q 2,..., q m } and λ = max {λ 1, λ 2 }. Furthermore, when λ 1 = λ 2, the derived code is an equidistance code. In this case, the corresponding RPPBD is an RPBD. 3 Constructions of RPPBDs In design theory, many researchers have been involved investigating into the existence of resolvable designs. Numerous existence results and techniques have been established (see, for example, [4, 11]). Most of existing resolvable designs with index λ 2 may contain identical parallel classes. Here, we will not intend to present a full survey concerning resolvable designs, instead, we will establish a number of construction methods, which can be utilized to generate (n, K; λ 1, λ 2 )- RPPBDs satisfying λ 1 λ 2 1, as well as uniform designs via Theorem Constructions from difference matrix Let F = {B 1,, B s } be a family of k-subsets of a certain additive group G. For any B F, define the developments of B and F as follows: devb = {B + g : g G}, s devf = devb i, i=1 where B + g = {b + g : b B}. If (G, devf) is a ( G, k, λ)-bibd, F is called a ( G, k, λ)-difference family (DF), or simply a ( G, k, λ)-df. The subsets B 1,, B s are called base blocks of the BIBD. 7

8 Let M = (m ij ) be a k G λ matrix with entries from G, and G = k {{i} G : i I k }. Let A = {{(1, m 1j +g),, (k, m kj +g)} : g G, 1 i=1 j G λ}. If (I k G, G, A) forms a TD(k, λ; G ), then M is called a ( G, k, λ)-difference matrix (DM). Note that the resultant TD is resolvable where each column of M corresponds to a block generating a parallel class and the number of columns of M is the number of parallel classes of the TD. Furthermore, it is an RTD if each column of the difference matrix cannot be obtained by adding a constant to another one. We refer to such a difference matrix as a column distinct difference matrix (CDDM). When λ = 1, a (g, k, 1)-DM is clearly a (g, k, 1)-CDDM. We remark that if a (g, k, λ)-dm exists, then so does a (g, k 1, λ)-dm (k 1 k). The derived difference matrix is simply obtained by deleting k k 1 certain rows from the old one. However, it is not always the case for CDDMs. The existence of a (g, k 1, λ)-cddm is guaranteed, only if the given (g, k, λ)-cddm (or DM) has a (g, s, λ)-cddm (s k 1 ) as its submatrix. Difference matrices have been instrumental in the construction of other types of combinatorial designs. For more detailed information and related results on DMs, the reader is referred to [4] and [5]. Below we will mainly focus our attention on CDDMs with index 2. The following three Lemmas are taken from [4]. Lemma 3.1 If q is a prime power, then there exists a (q, 2q, 2)-CDDM over (GF(q), +). Lemma 3.2 If there exists a (g 1 g 2, k, λ)-dm over G 1 G 2, then there exists a (g 1, k, g 2 λ)-dm over G 1. Lemma 3.3 There exists a (12, 6, 1)-DM over Z 6 Z 2. Making use of Lemmas , we have the following existence result. Lemma 3.4 The following difference matrices exist: (1) a (g, k, 2)-CDDM with k {3, 4, 5} if g 3 is a prime power or g {6, 12}; (2) a (2, k, 2)-CDDM with k {3, 4}. Proof. Start with a (12, 6, 1)-DM given in Lemma 3.3. Take two of its copies and add a non-zero constant to the first row of the second copy. This creates a (12, 6, 2)-DM whose first three rows form a (12, 3, 2)-CDDM. So, a (12, k, 2)-CDDM with k {3, 4, 5} exists. Applying Lemma 3.2 to 8

9 this (12, 6, 1)-DM we obtain a (6, 6, 2)-DM whose first three rows form a (6, 3, 2)-CDDM. Thus, we have also a (6, k, 2)-CDDM for any k {3, 4, 5}. A careful inspection of the proof of Lemma 3.1 shows that the DM provided in Lemma 3.1 contains a (q, 3, 2)-CDDM as its submatrix. Therefore, the other desired CDDMs all exist by Lemma 3.1. We will also need the following two simple results. The first one is quoted from [4] and the second is essentially Lemma of [4]. Lemma 3.5 For any prime power q, the multiplication table for the finite field GF(q) is a (q, q, 1)-DM over GF(q). Lemma 3.6 If a (g 1, k, λ 1 )-CDDM over G 1 and a (g 2, k, λ 2 )-CDDM over G 2 both exist, then so does a (g 1 g 2, k, λ 1 λ 2 )-CDDM over G 1 G 2. As an immediate consequence of Lemma 3.5 and Lemma 3.6, we have Corollary 3.7 Suppose g can be factorized into g = q 1 q 2 q s, where q i, i = 1, 2,..., s, is a prime power not less than 5, then there exists a (g, 5, 1)- DM. Now we are able to establish our main result of this subsection. Theorem 3.8 Let g 2 be an integer. Then there exist: (1) a (g, k, 2)-CDDM with k {3, 4, 5} if g 2 mod 4; (2) a (g, 5, 2)-CDDM if g 6 or 30 mod 36; (3) a (g, k, 2)-CDDM with k {3, 4} if g 2 mod 4. Proof. Write g = 2 u 3 v y, where u and v are non-negative integers, and y is 1 or a product of prime powers not less than 5. When u = v = 0, a (g, k, 2)-CDDM with k {3, 4, 5} follows from applying recursively Lemma 3.6, since for any prime power q not less than 5 both a (q, k, 2)-CDDM and (q, k, 1)-DM exist by Lemma 3.4 and Corollary 3.7. We then assume that (u, v) (0, 0). Further, in view of Lemma 3.6 and Corollary 3.7, it suffices to show a (2 u 3 v, k, 2)-CDDM exists for any stated value of k. Let x = 2 u 3 v. For case (1), we have u 1. If u = 0 or (u, v) {(2, 0), (2, 1)}, a (x, k, 2)-CDDM with k {3, 4, 5} was given in Lemma 3.4. If u 3 and v 2, Lemma 3.6 together with Lemma 3.4 and Corollary 3.7 guarantee that a (x, k, 2)-CDDM with k {3, 4, 5} exists. So, the conclusion holds. For case (2) and case (3), we have x = 2(3 v ). Employing the same vein as that in case (1), we can prove that the conclusion holds. 9

10 It is readily calculated that an RTD(k, 2; n) obtained by a CDDM contains exactly 2n parallel classes of blocks. If we regard the groups of an RTD(k, 2; n) as a parallel class of blocks, then a (kn, {k, n}; 2, 1)- RPPBD with class type k 2n n 1 follows. Therefore, from Theorem 3.8 we have the following. Theorem 3.9 Suppose n 2. Then there exist: (1) a (5n, {5, n}; 2, 1)- RPPBD with class type 5 2n n 1 and hence also a U 5n (n 2n 5 1 ) if n 2 mod 4 or n 6 or 30 mod 36; (2) a (4n, {4, n}; 2, 1)- RPPBD with class type 4 2n n 1 and hence also a U 4n (n 2n 4 1 ); (3) a (3n, {3, n}; 2, 1)- RPPBD with class type 3 2n n 1 and hence also a U 3n (n 2n 3 1 ). 3.2 Constructions using weighting Weighting constructions of GDDs date back to [20], and have been frequently used in design theory. This technique needs a master GDD to start with and also some small GDDs as input designs in order to end up with a new GDD. A modification of this technique can be used to obtain new (K, λ)- RGDDs from old ones. We state it into the following theorem whose proof is analogous to that in [20] and hence omitted. Theorem 3.10 Let (V, G, B) be a (s, λ)- RGDD of type g 1 g 2 g u. Suppose that there exists a positive integer ω such that for each block B = {x 1, x 2,..., x s } B, there exists a (k, µ)- RGDD of type ω s. Then there exists a (k, λµ)- RGDD of type (ωg 1 )(ωg 2 ) (ωg u ). As corollaries of Theorem 3.10, we have the following two working lemmas. Lemma 3.11 Suppose that a (u, k, 1)-RBIBD and an RTD(k, λ; g) exist. Then a (k, λ)- RGDD of type g u also exists. Proof. Regard a (u, k, 1)-RBIBD as a (k, 1)- RGDD of type 1 u, give weight g to every point of this GDD and apply Theorem Lemma 3.12 Suppose that a (k, λ)- RGDD of type g u and an RTD(k, 1; m) exist. Then a (k, λ)- RGDD of type (mg) u exists. Proof. Start with the given (k, λ)- RGDD, and then apply Theorem 3.10 with weight m. 10

11 To apply the above two lemmas one needs small RGDDs and RBIBDs. One way to get the suitable RGDDs is to start with a TD having index 1. Our authority for the existence of an RTD(k, 1; m) is [1]. Remember that a TD(k +1, 1; m) is equivalent to an RTD(k, 1; m) as well as k 2 mutually orthogonal Latin squares of order m. For the existence of RBIBDs we state the following two lemmas which are taken from [2, 11, 17] and [18]. Lemma 3.13 The necessary conditions for the existences of a (v, k, 1)- RBIBD are v 1 0 ( mod k 1) and v 0 ( mod k). For k {2, 3, 4, 5}, these conditions are sufficient except possibly for k = 5 and v {45, 225, 345, 465, 645}. Lemma 3.14 There exists a (v, 3, 2)- RBIBD if and only if v 0 ( mod 3) and v 9. Now we turn to establish several infinite classes of RPPBDs by applying the above method. Theorem 3.15 Let n be a positive integer and v = 6n + 3. Then there exist: (1) a (gv, {3, g}; 2, 1)- RPPBD with class type 3 g(6n+2) g 1 and hence also a uniform design U g(6n+3) ((g(2n + 1)) g(6n+2) (6n + 3) 1 ) when g 2; (2) a (gv, {3, 2}; 2, 1)- RPPBD with class type 3 g(6n+2) 2 g 1 and hence also a uniform design U g(6n+3) ((g(2n + 1)) g(6n+2) (g(6n + 3)/2) g 1 ) when g is even. Proof. Let V be a vg-set. For any g 2, we first apply Lemma 3.11 to get a (3, 2)- RGDD of type g v over V. The ingredients needed here are an RTD(3, 2; g) and a (v, 3, 1)-RBIBD, which exist by Theorem 3.8 and Lemma Then we apply Theorem 2.4 as follows. For conclusion (1), we take the groups of this GDD as a parallel class of blocks of size g over V. For conclusion (2), we replace each group of this GDD with a (g, 2, 1)- RBIBD, and then partition their blocks into g 1 parallel classes of blocks of size 2 over V. What remains is to calculate the number of parallel classes for the derived RPPBDs. This is straightforward. Theorem 3.16 Let n and g be integers not less than 3, and v = 3n. Then there exist: (1) a (gv, {3, g}; 2, 1)- RPPBD with class type 3 g(3n 1) g 1 and hence also a uniform design U 3ng ((ng) g(3n 1) (3n) 1 ) if g 6; 11

12 (2) a (gv, {3, 2}; 2, 1)- RPPBD with class type 3 g(3n 1) 2 g 1 and hence also a uniform design U 3ng ((ng) g(3n 1) (3ng/2) g 1 ) if g is even and g 6. Proof. Start with a (v, 3, 2)- RBIBD from Lemma 3.14 and regard it as a (3, 2)- RGDD of type 1 v. Then we apply Lemma 3.12 and the existence of an RTD(3, 1; g) to get a (3, 2)- RGDD of type g v. The conclusion then follows from Theorem 2.4 and Lemma Employing the same argument as that of Theorem 3.15, we also have the following two existence results. Theorem 3.17 Let n be a positive integer and v = 12n + 4. Then there exist: (1) a (gv, {4, g}; 2, 1)- RPPBD with class type 4 g(8n+2) g 1 and hence also a uniform design U g(12n+4) ((g(3n + 1)) g(8n+2) (12n + 4) 1 ) when g 2; (2) a (gv, {4, 2}; 2, 1)- RPPBD with class type 4 g(8n+2) 2 g 1 and hence also a uniform design U g(12n+4) ((g(3n + 1)) g(8n+2) (g(6n + 2)) g 1 ) when g is even. Theorem 3.18 Let n be a positive integer and v = 20n + 5, where n / {2, 11, 17, 23, 32}. Then there exist: (1) a (gv, {5, g}; 2, 1)- RPPBD with class type 5 g(10n+2) g 1 and hence also a uniform design U g(20n+5) ((g(4n + 1)) g(10n+2) (20n + 5) 1 ) when g 2 mod 4 or g 6 or 30 mod 36; (2) a (gv, {5, 2}; 2, 1)- RPPBD with class type 5 g(10n+2) 2 g 1 and hence also a uniform design U g(20n+5) ((g(4n+1)) g(10n+2) (g(20n + 5)/2) g 1 ) when g 0 mod 4 or g 6 or 30 mod 36. For the case g = 8, we further have Theorem 3.19 Let n be a non-negative integer. Then there exist: (1) a (8(6n+3), {3, 4}; 2, 3)- RPPBD with class type 3 8(6n+2) 4 7 and hence also a uniform design U 8(6n+3) ((8(2n + 1)) 8(6n+2) (12n + 6) 7 ); (2) a (8(12n + 4), 4; 2, 3)- RPPBD with class type 4 8(8n+2)+7 and hence also a uniform design U 8(12n+4) ((8(3n + 1)) 8(8n+2)+7 ); (3) a (8(20n + 5), {5, 4}; 2, 3)- RPPBD with class type 5 8(10n+2) 4 7 and hence also a uniform design U 8(20n+5) ((8(4n + 1)) 8(10n+2) (40n + 10) 7 ) if n / {2, 11, 17, 23, 32}. 12

13 Proof. The proof is similar to that above except we apply Theorem 2.4 with a (8, 4, 3)- RBIBD, which is easily constructed as follows: point set : blocks : Z 7 { }; {, 0, 1, 5} mod 7; {2, 3, 4, 6} mod 7. Theorem 3.20 Let n be a positive integer. Then there exist: (1) a ((8(6n + 3)), {3, 4}; 4, 3)- RPPBD with class type 3 16(6n+2) 4 7 and hence also a uniform design U 8(6n+3) ((8(2n + 1)) 16(6n+2) (12n + 6) 7 ); (2) a (8(12n + 4), 4; 4, 3)- RPPBD with class type 4 16(8n+2)+7 and hence also a uniform design U 8(12n+4) ((8(3n + 1)) 16(8n+2)+7 ); (3) a (8(20n + 5), {4, 5}; 4, 3)- RPPBD with class type 5 16(10n+2) 4 7 and hence also a uniform design U 8(20n+5) ((8(4n + 1)) 16(10n+2) (40n+10) 7 ) if n / {2, 11, 17, 23, 32}. Proof. For any k {3, 4, 5}, we know that a (8, k, 1)-DM exists from Lemma 3.5. Take four copies of such a DM and add four distinct constants to the first row of these copies in turn. This yields a (8, k, 4)-CDDM, and hence an RTD(k, 4; 8) exists. Thus, we apply Lemma 3.11 with g = 8 to those RBIBDs shown in Lemma 3.13 to obtain a (3, 4)- RGDD of type 8 6n+3, a (4, 4)- RGDD of type 8 12n+4 and a (5, 4)- RGDD of type 8 20n+5, which contain 16(6n + 2), 16(8n + 2) and 16(10n + 2) parallel classes in turn. The conclusion then follows from Theorem 2.4, since a (8, 4, 3)- RBIBD exists (see the proof of Theorem 3.19). 3.3 One more construction method A method of constructing RBIBDs from DFs and DMs was given by [16]. The following construction for (K, λ)- RGDDs is an extension of construction 2.1 in [2] and Theorem in [11]. More discussions about this method can also be referred to [13]. Construction 3.21 Let R be a commutative ring of order q with a unity. Suppose that there exists a set {x i : 1 i k} {y i : 1 i λ 1} of k + λ 1 distinct units in R whose differences are all units of R, and a block-disjoint (q, k, λ)-df over the additive group of R. Then there exists a (k, λ)- RGDD of type k q. 13

14 Proof. Let B 1, B 2,..., B t be the disjoint base blocks for the given (q, k, λ)- DF. Then, for 1 i k, x i B 1, x i B 2,..., x i B t also form a block-disjoint (q, k, λ)-df. Without loss of generality, we assume that 0 B 1 and x 1 = 1, the unity of R. Define a (q, k, 1)-DM M 0 = (S 0 T 0 ) over R in such a way that (1) the first row of S 0 contains all the elements from B 1, B 2,..., B t and 0 lies in the first position, (2) the first row of T 0 contains all the elements of R\ t B i, (3) for 2 j k, the j-th row of M 0 is obtained by multiplying its first row by x j. For any j (1 j λ 1), we multiply the first row of M 0 by y j and keep the other rows unchanged to form a new k q matrix M j. Note that for each 1 i < t k, the differences list {d (j) il d (j) tl : 1 l q} is of the form (y j x t )R when i = 1 or (x i x t )R when i 1, where d (j) il is the entry in the cell (i, l) of M j. It is easy to see that M j is also a (q, k, 1)-DM over R for each j (1 j λ 1). So, these DMs produce λ RTD(k, 1; q)s over I k R, which possess an identical parallel class of the form {(1, r), (2, r),..., (k, r) : r R} based on the first columns of M j (j = 0, 1,..., λ 1). The other parallel classes of these RTDs are mutually distinct. Further, each set {i} R, i I k, can be filled in by a (q, k, λ)-bibd using a DF over R corresponding to i-th row of S 0. What remains is to arrange the blocks obtained above into pairwise distinct parallel classes. First, the base blocks on {i} R (i = 1, 2,..., k) together with the blocks of the form {(1, t 1 ),..., (k, t k )} where (t 1,..., t k ) T ranges over all columns of T 0, form a parallel class on I k R. Developing (mod(, q)) the blocks in such a parallel class produces q parallel classes. Second, we develop (mod(, q)) each block of the form {(1, s 1 ),..., (k, s k )} for (s 1,..., s k ) T, a column of S 0, to obtain λ(q 1)/(k 1) parallel classes on I k R. Finally, from the remaining (λ 1) RTD(k, 1; q) obtained from difference matrixes M j (1 j λ 1), we have (λ 1)q parallel classes. In total, we obtain λq+λ(q 1)/(k 1) parallel classes on I k R. Among these, there are exactly λ identical parallel classes of the form {(1, r),..., (k, r) : r R}. Deleting these λ parallel classes and taking their blocks as groups produces a (k, λ)- RGDD of type k q, as desired. This completes the proof. i=1 Remark: The existence of a block-disjoint (q, k, λ)-df implies λ k 1. We give an example to illustrate Construction

15 Example 3.22 Let k = 3, λ = 2 and q = 7. We take R = Z 7, the residue ring of integers modulo 7. Since Z 7 is a prime field, we may take four arbitrary nonzero residues modulo 7 as our units, say {1, 2, 3} {6}. Suitable base blocks for a block-disjoint (7, 3, 2)-DF are B 1 = {0, 1, 3} and B 2 = {2, 4, 5}. Then we have S 0 = , T 0 = 6 5, S 1 = , T 1 = Here M 0 = (S 0 T 0 ), M 1 = (S 1 T 1 ). The resulting (3, 2)- RGDD of type 3 7 is based on the point set {1, 2, 3} Z 7 with groups of the form {1, 2, 3} j (j Z 7 ). It contains 18 blocks to be cycled mod(-,7) as follows: B 1 = {(0, 0), (0, 1), (0, 3)}, B 2 = {(0, 2), (0, 4), (0, 5)}, B 3 = {(1, 0), (1, 2), (1, 6)}, B 4 = {(1, 4), (1, 1), (1, 3)}, B 5 = {(2, 0), (2, 3), (2, 2)}, B 6 = {(2, 6), (2, 5), (2, 1)}, B 7 = {(0, 6), (1, 5), (2, 4)}, B 8 = {(0, 1), (1, 2), (2, 3)}, B 9 = {(0, 3), (1, 6), (2, 2)}, B 10 = {(0, 2), (1, 4), (2, 6)}, B 11 = {(0, 4), (1, 1), (2, 5)}, B 12 = {(0, 5), (1, 3), (2, 1)}, B 13 = {(0, 6), (1, 2), (2, 3)}, B 14 = {(0, 4), (1, 6), (2, 2)}, B 15 = {(0, 5), (1, 4), (2, 6)}, B 16 = {(0, 3), (1, 1), (2, 5)}, B 17 = {(0, 2), (1, 3), (2, 1)}, B 18 = {(0, 1), (1, 5), (2, 4)}. The first 7 blocks form a parallel class; cycling mod(-,7) produces 7 parallel classes. Cycling each of the last 11 blocks mod(-,7) produces a parallel class. In total, the RGDD has 18 parallel classes. Applying Construction 3.21 we have the following existence result. Theorem 3.23 If q = 10m + 1 is a prime power, then there exist a (5q, 5; 2, 1)- RPPBD with class type 5 5(q 1)/2+1 and hence also a uniform design U 5q (q 5(q 1)/2+1 ). Proof. It is known (see, for example, Theorem in [11]) that there exists a block-disjoint (q, k, (k 1)/2)-DF over GF(q) for any prime power q 1 (mod 2k). For k = 5, a block-disjoint (q, 5, 2)-DF over GF(q) consists of the base blocks x t B(0 t m 1) where x is an arbitrary primitive element of GF(q) and B = {1, x 2m, x 4m, x 6m, x 8m }. Hence we can apply Construction 3.21 to obtain a (5, 2)- RGDD of type 5 q. Applying Theorem 2.4 with the existence of a (5, 5, 1)-RBIBD, the desired RPPBD follows. 15

16 4 Concluding remarks Uniform designs are very useful in computer experiments and other experiments for scientific research and industrial quality control. In this paper, we developed the connection between uniform designs and resolvable partially pairwise balanced designs. A number of new uniform designs were given via RPPBDs. One can then expect that many more techniques and existence results on resolvable designs in design theory can be modified to produce new uniform designs. References [1] R. J. R. Abel, A. E. Brouwer, C. J. Colbourn and J. H. Dinitz, Mutually orthogonal Latin squares, in: C. J. Colbourn and J. H. Dinitz (eds), CRC Handbook of Combinatiorial Designs, CRC Press, Boca Raton, FL, 1996, [2] R. J. R. Abel, G. Ge, M. Greig and L. Zhu, Resolvable balanced incomplete block designs with block size 5, J. Statist. Plann. Inference, 95 (2001), [3] R. A. Bayes, R. J. Buck, E. Riccomagno and H. P. Wynn, Experimental design and observation for large systems, J. Roy. Statist. Soc. Ser. B, 58 (1996), [4] T. Beth, D. Jungnickel and H. Lenz, Design Theory (second edition), Cambridge Univ. Press, Cambridge, UK, [5] C. J. Colbourn and W. de Launey, Difference Matrices, in: C. J. Colbourn and J. H. Dinitz (eds), CRC Handbook of Combinatiorial Designs, CRC Press, Boca Raton, FL, 1996, [6] K. T. Fang, The uniform design: application of number-theoretic methods in experimental design, Acta Math. Appl. Sinica, 3 (1980), [7] K. T. Fang, G. N. Ge, M. Q. Liu and H. Qin, Optimal supersaturated design and their constructions, Discrete Math., to appear. [8] K. T. Fang and D. K. J. Lin, Uniform designs and their application in industry, in: R. Khattree and C. R. Rao (eds) Handbook on Statistics 22: Statistics in Industry, Elsevier, North-Holland, 2003, [9] K. T. Fang, D. K. J. Lin and M. Q. Liu, Optimal mixed-level supersaturated design, Metrika, 58, 2003,

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