THE EXISTENCE OF r x 4 GRID-BLOCK DESIGNS WITH r=3, 4

全文

(1)SIAM J. DISCRETE MATH. Vol. 23, No. 2, pp. 1045–1062. c 2009 Society for Industrial and Applied Mathematics . Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. THE EXISTENCE OF r × 4 GRID-BLOCK DESIGNS WITH r = 3, 4∗ RUCONG ZHANG† , GENNIAN GE‡ , ALAN C. H. LING§ , HUNG-LIN FU¶, AND YUKIYASU MUTOH Abstract. For a v-set V , let A be a collection of r × c arrays with elements in V . A pair (V, A) is called an r × c grid-block design if every two distinct elements i and j in V occur exactly once in the same row or in the same column of an array in A. This design originated from the use of DNA library screening. In this paper, we show the existence of r × 4 grid-block designs with r = 3, 4. We settle completely for the case of r = 4 and almost completely for the case of r = 3, leaving 15 orders undetermined. Key words. grid-block design, complete graph, decomposition, Cartesian product AMS subject classifications. 05B05, 05B30, 05C70 DOI. 10.1137/080737423. 1. Introduction and preliminaries. Let V be a v-set and A be a collection of r × c arrays with elements in V . Two elements of V are collinear if they are on the same grid line (row or column). A pair (V, A) is called an r × c grid-block design of order v if every pair of two distinct elements of V is collinear exactly once. Therefore, an r × c grid-block design of order v exists if and only if the complete graph of order v, Kv , can be decomposed into the Cartesian product of two complete graphs Kr and Kc , denoted by G(r, c) = Kr × Kc . For graph terms not defined here, we refer the reader to [13]. For convenience, we denote an r × c grid-block design of order v by a Dr×c (Kv ). It is not difficult to conclude the following necessary conditions for the existence of a Dr×c (Kv ). Lemma 1.1. If a Dr×c (Kv ) exists, then (a) v − 1 ≡ 0 (mod r + c − 2) and (b) v(v − 1) ≡ 0 (mod rc(r + c − 2)). The study of the existence of a Dr×c (Kv ) dates back to around 70 years ago when the case r = c = v was first considered by Yates [15]. Such a grid-block is known as a lattice square. Later, in 1971, Raghavarao [12] gave a construction of a Dr×c (Kv ) √ for the case when v is an odd prime. In 1995, it was pointed out by Hwang [8] that this design can be applied to DNA clone library screening. For practical use, in a 2-stage group test, we need an r × c grid-block design of order v with smaller r and c ∗ Received by the editors October 6, 2008; accepted for publication (in revised form) March 12, 2009; published electronically June 10, 2009. http://www.siam.org/journals/sidma/23-2/73742.html † Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People’s Republic of China (zhangre1982@yahoo.com.cn). ‡ Corresponding author. Department of Mathematics, Zhejiang University, Hangzhou 310027, Zhejiang, People’s Republic of China (gnge@zju.edu.cn). This author’s research was supported by the National Outstanding Youth Science Foundation of China under grant 10825103, National Natural Science Foundation of China under grant 10771193, Specialized Research Fund for the Doctoral Program of Higher Education, Program for New Century Excellent Talents in University, and Zhejiang Provincial Natural Science Foundation of China under grant D7080064. § Department of Computer Science, University of Vermont, Burlington, VT 05405 (aling@emba. uvm.edu). ¶ Department of Applied Mathematics, National Chiao Tung University, Hsin Chu, 30050 Taiwan (hlfu@math.nctu.edu.tw). This author’s research was supported by NSC 94-2115-M-009-017. Department of Mathematics, Keio University, Yokohama, Kanagawa, 223-8522 Japan (yukiyasu@ea.mbn.or.jp). This author’s research was supported by JSPS Research Fellow 09978.. 1045. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(2) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1046. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. comparable to v; see [11, 10] for reference. The existence of a D2×3 (Kv ) for all admissible v was shown by Carter [4] in her Ph.D. thesis. There are three other substantial works on this topic. First, in 2003, Mutoh et al. [11] proved that the necessary conditions for the existence of a D2×4 (Kv ) are also sufficient. Then, in 2004, Fu et al. [6] showed the existence of a D3×3 (Kv ). Finally, in 2007, Li et al. [9] gave a complete solution to the existence of a D2×5 (Kv ). In this paper, we shall extend their study and consider the grid-blocks G(3, 4) and G(4, 4). The methods which we adopt are mainly direct constructions. For smaller gridblock designs (ingredients), the constructions are based on the familiar difference method, where a finite group (normally an Abelian group but sometimes a nonAbelian group) will be utilized to generate the set of grid-blocks for a given design. Thus, instead of listing all the grid-blocks, we list a set of base grid-blocks and generate the others by an additive group and perhaps some further automorphisms. For example, the following G(4, 4) generates all 97 grid-blocks of a D4×4 (K97 ); the ith block is obtained by adding i (mod 97) to each element of this block. 0 5 16 46. 1 13 60 74. 3 81 26 61. 7 38 86 29. In order to combine the ingredients together nicely to obtain a larger one, we need some notation on combinatorial designs. For terminology in design theory, we refer the reader to [5]. For sets of positive integers K and M , let V be a set of v elements, let g be a partition of V such that each G ∈ g has m elements for m ∈ M , and let B be a collection of k-subsets (blocks) of V for k ∈ K. A triple (V , g , B) is called a group divisible design (GDD), denoted by GD[K, λ, M ; v], if every two distinct elements contained in different groups of g occur in exactly λ blocks of B and every two distinct elements contained in the same group occur in no blocks. In particular, a GD[{k}, λ, {m}; v] is written by GD[k, λ, m; v] and called a k-GDD for simplicity of notation. Furthermore, if g contains ti groups of size si for i = 1, 2, . . . , h, then h the k-GDD is said to be of type i=1 si ti . Now, if λ = 1, then the existence of a GD[k, 1, m; v] is equivalent to the decomposition of Kn (m) into Kk ’s, where v = mn and Kn (m) denotes the balanced complete n-partite graph with each partite set of size m. For convenience, the elements in each partite set will be called the points throughout this paper. A pairwise balanced design (PBD) with set of block sizes K is a K-uniform set system (X, A), such that every element of X2 is contained in exactly one block of A. A PBD of order v and set of block sizes K is denoted by (v, K, 1)-PBD. If an element k ∈ K is “starred” (written k ), it means that the PBD has exactly one block of size k. The following lemma is easy to see. Lemma 1.2. (a) (n−1)m ≡ 0 (mod r+c−2) and (b) nm2 (n−1) ≡ 0 (mod rc(r+ c − 2)) are necessary conditions for the existence of a Dr×c (Kn (m)). Note here that we use a Dr×c (G) to denote a decomposition of G into G(r, c)’s. The following recursive constructions are essential to the constructions of our main results. These constructions were obtained earlier by Fu et al. [6] and Mutoh et al. [11]. Lemma 1.3 (see [6]). A Dr×c (Kst ) exists if a Dr×c (Kt ) and a Dr×c (Ks (t)) exist. Lemma 1.4 (see [6]). A Dr×c (Kst+1 ) exists if a Dr×c (Kt+1 ) and a Dr×c (Ks (t)) exist.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(3) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. 1047. Lemma 1.5 (see [11]). There exists a Dr×c (Kvt+1 ) if there exist a GD[K, 1, M ; v], a Dr×c (Kmt+1 ), and a Dr×c (Kk (t)) for any m ∈ M and k ∈ K. Lemma 1.6 (see [11]). There exists a Dr×c (Ks (tu)) if there exist a Dr×c (Ks (t)) and an OA(u, k), where k = max{r, c} and OA(u, k) is an orthogonal array of order u, degree k, and index 1. We shall also use the idea of Wilson in [14] to obtain grid-block designs of prime power orders. For a positive integer e, let q be a prime power such that q ≡ 1 (mod e). Then, the cyclic multiplicative subgroup of nonzero elements in the finite field of order q, GF (q), has a unique subgroup C0e of index e. The multiplicative cosets e of C0e : C0e , C1e , . . . , Ce−1 , are called the classes of index e. Now, let A = (aij ) be an r ×c grid-block with elements in GF (q). Then, we define the ordered list of differences = {ajl − ail |1 ≤ i < j ≤ r, 1 ≤ l ≤ c} ∪ {alj − ali |1 ≤ l ≤ r, 1 ≤ i < j ≤ c}. of A as ∂A By a similar idea as the one in [8], we have the following result. Lemma 1.7. For a prime power q ≡ 1 (mod rc(r + c − 2)), let e = rc(r + c − 2)/2. is a system of If there exists an r × c array A = (aij ) over GF (q) such that ∂A e representatives for the cosets of C0 , then a Dr×c (Kq ) exists. We remark here that A can be viewed as a base grid-block and all the grid-blocks of Dr×c (Kq ) can be obtained by calculating cA+x for c ∈ C0e /{1, −1} and x ∈ GF (q). Now, we are ready to state the main results. 2. 3 × 4 grid-block designs. By Lemma 1.1, it is a routine matter to show the following fact. Lemma 2.1. If a D3×4 (Kv ) exists, then v ≡ 1, 16, 21, 36 (mod 60). For convenience, the constructions are split into four parts. (We claim that the above necessary conditions are sufficient except for v = 16, which is impossible, as well as several unsettled cases.) 2.1. v ≡ 1 (mod 60). The following lemmas provide the constructions of the ingredients. Lemma 2.2. There exist both a D3×4 (K4 (20)) and a D3×4 (K5 (15)). Hence, there exist both a D3×4 (K4 (60)) and a D3×4 (K5 (60)). Proof. A D3×4 (K4 (20)) is constructed on the Abelian group V = Z80 which has only one base grid-block: 0 11 66. 1 16 47. 3 54 21. 10 33 60. A D3×4 (K5 (15)) is constructed on the Abelian group V = Z75 generated by the group V acting on the points. This design has one base grid-block: 0 14 43. 1 38 60. 3 66 24. 7 25 51. Since there are both an OA(3, 4) and an OA(4, 5), a D3×4 (K4 (60)) and a D3×4 (K5 (60)) can be obtained by applying Lemma 1.6. Lemma 2.3. There exists a D3×4 (K60t+1 ) for t = 1, 2, 3, 7. Proof. According to Lemma 1.7, we use a computer to find the array A for t = 1, 2, 3, 7. These A’s are listed in the following table.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(4) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1048. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. t 1. Primitive element or polynomial 2. 2. ω2 + ω1 + 7. 3. 2. 7. 2. A 0 5 19 ω∞ ω3 ω 24 0 5 99 0 5 105. 1 25 47 ω0 ω5 ω 95 1 13 90 1 13 365. 3 56 30 ω1 ω 15 ω 97 3 23 142 3 23 281. 7 43 59 ω2 ω 98 ω 45 7 63 39 7 37 86. Now we can show the existence of the desired grid-block designs. Here, we verify the case t = 1, and the others are similar. Clearly, q = 61, e = 30, and f = (q −1)/e = 2. We denote F ∗ = GF (q) \ {0}, ω to be the primitive element of F ∗ and ε = ω e . Then the cosets of C0e are as follows: C0e = {1, ε, ε2, . . . , εf −1 } = {1, 60} C1e = {ω, ωε, ωε2, . . . , ωεf −1 } = {2, 59} C2e = {ω 2 , ω 2 ε, ω 2 ε2 , . . . , ω 2 εf −1 } = {4, 57} .. . e Ce−2 = {ω e−2 , ω e−2 ε, ω e−2 ε2 , . . . , ω e−2 εf −1 } = {15, 46} e Ce−1 = {ω e−1 , ω e−1 ε, ω e−1 ε2 , . . . , ω e−1 εf −1 } = {30, 31} and take all the 30 differences less than 31; that is, if the differFinally, we compute ∂A = (5, 14, 19, 24, 15, 22, 8, 26, ence is 41, we write it down as 20. So the differences are ∂A 27, 25, 16, 9, 1, 2, 3, 4, 6, 7, 20, 30, 10, 13, 18, 23, 28, 17, 11, 29, 12, 21). The 30 different differences correspond exactly to the elements 1 to 30. According to the cosets of C0e , we know that the elements 1 to 30 are exactly in the 30 different cosets. So, by Lemma 1.7, a D3×4 (K61 ) exists. Note that all the grid-blocks are obtained by calculating cA + x for c ∈ C0e /{1, −1} and x ∈ GF (q). So, in fact, we have 61 grid-blocks in this case. The following lemmas show that a D3×4 (K60t+1 ) exists for t = 4, 5, 6, 8, 9, 10, 11. Lemma 2.4. There exist both a D3×4 (K241 ) and a D3×4 (K301 ). Proof. By Lemmas 2.2 and 2.3, there exist both a D3×4 (K4 (60)) and a D3×4 (K61 ). Thus, there exists a D3×4 (K241 ) by Lemma 1.4. Again, by Lemmas 2.2 and 2.3, there exist both a D3×4 (K5 (60)) and a D3×4 (K61 ). Hence, we have a D3×4 (K301 ). Lemma 2.5. There exists a D3×4 (K361 ). Proof. First, we construct a D3×4 (K6 (60)). We know a 5-GDD of type 46 exists; see [7] for a reference. Then we inflate each point in each partite set to 15 points. In other words, we view each point in each partite set as a subset (each of size 15). So, we can get a D3×4 (K6 (60)) by using Lemma 2.2 to obtain 75 grid-blocks for each K5 in a K5 -design of K6 (4). Since a D3×4 (K61 ) exists (Lemma 2.3), we get a D3×4 (K60×6+1 ) by Lemma 1.4. Lemma 2.6. There exists a D3×4 (K8 (15)). Proof. Let the last partite set of K8 (15) be {∞0 , ∞1 , ∞2 , . . . , ∞13 , ∞14 } and V = Z105 ∪ {∞0 , ∞1 , ∞2 , . . . , ∞13 , ∞14 }. Then, the design is constructed on the. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(5) 1049. GRID-BLOCK DESIGNS. Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. group V generated by the group Z105 acting on the points. The 15 infinite points remain fixed under the action of Z105 . It has two base grid-blocks: 0 4 40. 1 14 87. 3 19 58. 9 31 ∞. 0 75 31. 48 86 5. 23 ∞∗ 77. 68 34 ∞$. We remark here that each of the above grid-blocks has an extra property that all elements in the same row or same column with ∞ (∞∗ or ∞$ ) are in different residue classes (mod 5). Therefore, for the 21 grid-blocks generated by adding i, 5+i, . . . , 100+ i, i = 0, 1, 2, 3, 4, to every point in each base grid-block, respectively, we replace three infinite points (∞, ∞∗ , ∞$ ) with ∞i , ∞i+5 , ∞i+10 correspondingly. Then, we have the desired D3×4 (K8 (15)). Lemma 2.7. There exists a D3×4 (K481 ). Proof. By Lemma 2.6, we have a D3×4 (K8 (15)). Then we can get a D3×4 (K8 (60)) by applying Lemma 1.6 with an OA(4,4). Since a D3×4 (K61 ) exists, the desired gridblock design can be obtained by Lemma 1.4. Lemma 2.8. There exists a D3×4 (K9 (15)). Proof. The design is constructed on the Abelian group V = Z135 generated by the group V acting on the points. This design has two base grid-blocks: 0 5 15. 1 13 44. 3 24 79. 7 37 104. 0 50 98. 14 115 58. 61 89 5. 83 66 25. Lemma 2.9. There exists a D3×4 (K541 ). Proof. By the fact that a (37, {5, 9∗})-PBD exists (see [2] for a reference), we can get a {5, 9}-GDD of type 49 by deleting one point not in the block of size 9. Then, by a similar idea as in Lemma 2.5, the existence of a D3×4 (K5 (15)) and a D3×4 (K9 (15)) gives the D3×4 (K60×9+1 ) we need. Lemma 2.10. There exists a D3×4 (K601 ). Proof. By Lemma 2.2, we know that a D3×4 (K5 (15)) exists. Since an OA(8, 5) exists, we get a D3×4 (K5 (120)) by Lemma 1.6. This implies that we have a D3×4 (K60×10+1 ) (by Lemmas 1.4 and 2.3). Lemma 2.11. There exists a D3×4 (K661 ). Proof. We know that a 5-GDD of type 411 exists; see [7] for a reference. Therefore, a D3×4 (K11 (60)) can be obtained by inflating each partite set to 60 points. Since a D3×4 (K61 ) exists, we get a D3×4 (K60×11+1 ) by Lemma 1.4. Theorem 2.12. There exists a D3×4 (K60t+1 ) for each positive integer t. Proof. When t ≥ 12, we know that a GD[K,1,M ;t] exists, where K = {4, 5} and M = {1, 2, . . . , 7} by [11, Lemma 3.2]. We also know that a D3×4 (K60m+1 ) for any m ∈ M and a D3×4 (Kk (60)) for any k ∈ K exist by the above lemmas. So, by Lemma 1.5, a D3×4 (K60t+1 ) exists when t ≥ 12. Combining the results of Lemma 2.3 to Lemma 2.11, we conclude the proof. 2.2. v ≡ 21 (mod 60). By Lemma 1.4, it suffices to prove that a D3×4 (K21 ) exists and a D3×4 (K3t+1 (20)) exists for each positive integer t. Lemma 2.13. There exists a D3×4 (K21 ).. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(6) 1050. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. Proof. The design is constructed on V = Z7 × {0, 1, 2}, generated by the group Z7 acting on the points. The base grid-block is the following 3 × 4 array: (0,0) (2,1) (3,2). (1,0) (4,1) (6,2). (3,0) (1,1) (5,2). (0,1) (0,2) (6,0). Lemma 2.14. There exist both a D3×4 (K7 (20)) and a D3×4 (K10 (20)). Proof. A D3×4 (K7 (20)) is constructed on the Abelian group V = Z140 generated by the group V acting on the points. This design has two base grid-blocks: 0 4 23. 1 14 61. 3 19 90. 9 31 123. 0 37 88. 11 101 70. 45 81 113. 86 7 31. A D3×4 (K10 (20)) is constructed on the Abelian group V = Z200 generated by the group V acting on the points. This design has three base grid-blocks: 199 38 132. 74 195 181. 46 103 90. 122 141 159. 44 139 55. 69 110 38. 78 54 192. 133 52 126. 61 96 73. 53 192 74. 2 18 70. 8 91 106. Lemma 2.15. There exists a D3×4 (K19 (20)). Proof. In order to obtain a D3×4 (K19 (20)), we first construct a D3×4 (K6 (60)), and then together with a D3×4 (K4 (20)) we obtain the desired design. First, we know that a 5-GDD of type 46 exists; see [7] for a reference. Then we inflate each point in each partite set to 15 points. So, we have a D3×4 (K6 (60)) by obtaining 75 grid-blocks for each K5 in a K5 -design of K6 (4). Next, in order to construct the desired D3×4 (K19 (20)), we first partition each partite set of K6 (60) into three subsets (each of size 20). Then, we add 20 new points to K6 (60). We view these 20 points as a subset A which is different from the subsets in K6 (60). So, we have 19 subsets (each has 20 points) altogether. By Lemma 2.2, since a D3×4 (K4 (20)) exists, we have a D3×4 (K19 (20)) as desired. Lemma 2.16. For each positive integer t, a D3×4 (K3t+1 (20)) exists. Proof. It is well known that a (v, {4, 7, 10, 19}, 1)-PBD exists for each v ≡ 1( mod 3) and v ≥ 4. Then, by inflating each point to 20 points and using Lemmas 2.2, 2.14, and 2.15, respectively, we conclude the proof. Theorem 2.17. For each positive integer t, a D3×4 (K60t+21 ) exists. Proof. By Lemmas 1.4, 2.13, and 2.16, the proof is complete. 2.3. v ≡ 36 (mod 60). Lemma 2.18. There exists a D3×4 (K36 ). Proof. The design is constructed on V = Z18 × {0, 1}, generated by the group Z18 acting on the points. Note that there are two base grid-blocks. The first one is of short orbit, which goes one sixth of the cycle, and the second one is of full orbit. (0,0) (6,0) (12,0). (9,0) (15,0) (3,0). (0,1) (6,1) (12,1). (9,1) (15,1) (3,1). (0,0) (2,0) (12,1). (1,0) (12,0) (5,1). (4,0) (17,1) (9,0). (2,1) (1,1) (15,1). The following lemmas are essential for our direct constructions. Lemma 2.19. There exists a D3×4 (K7 (10)).. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(7) 1051. GRID-BLOCK DESIGNS. Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. Proof. The design is constructed on the Abelian group V = Z70 , generated by the group V acting on the points. It has one base grid-block: 1 5 23. 2 32 7. 4 55 38. 10 43 67. In what follows, if a complete multipartite graph with ti partite sets of size si for i = 1, 2, . . . , h, can be decomposed into G(r, c)’s, then we say that an r × c grid-block design of type Πhi=1 si ti exists. Lemma 2.20. There exists a 3 × 4 grid-block design of type 106 51 . Proof. Let V = Z60 ∪ {∞0 , ∞1 , ∞2 , ∞3 , ∞4 }. The design is constructed on the group V generated by the group Z60 acting on the points. The five infinite points remain fixed under the action of Z60 . It has one base grid-block: 0 16 33. 1 36 14. 3 31 10. 11 2 ∞. We remark here that the above grid-block has an extra property that all elements in the same row or same column with ∞ are in different residue classes (mod 5). Therefore, for the 12 grid-blocks generated by adding i, 5+i, . . . , 55+i, i = 0, 1, 2, 3, 4, to every point in each base grid-block, respectively, we use ∞i for ∞. Then we have the desired design. Lemma 2.21. There exists a D3×4 (K6 (10)). Proof. Let the last partite set of K6 (10) be A = {∞0 , ∞1 , ∞2 , . . . , ∞8 , ∞9 } and V = Z50 ∪ A. The design is constructed on the group V generated by the group Z50 acting on the points. The 10 infinite points remain fixed under the action of Z50 . It has one base grid-block: 0 27 19. 1 3 12. 4 ∞∗ 40. 38 21 ∞. Now, similar to Lemma 2.20, we replace ∞∗ , ∞ by ∞i , ∞i+5 , respectively, and i = 0, 1, 2, 3, 4, for the corresponding grid-blocks. Lemma 2.22. There exists a D3×4 (K12 (10)). Proof. Let the last partite set of K12 (10) be {∞1 , ∞2 , . . . , ∞10 } and V = Z110 ∪ {∞1 , ∞2 , . . . , ∞10 }. Then, the design is constructed on the group V generated by Z110 acting on the points. Furthermore, the 10 infinite points remain fixed under the action of Z110 . It has two base grid-blocks: 0 5 13. 1 17 40. 3 ∞1 54. 7 26 ∞6. 0 38 70. 10 68 44. 45 20 101. 73 88 27. Lemma 2.23. There exist a D3×4 (K13 (10)) and a D3×4 (K13 (30)), as well as a D3×4 (K5 (30)). Proof. A D3×4 (K13 (10)) is constructed on the Abelian group V = Z130 generated by the group V acting on the points. This design has two base grid-blocks: 29 108 65. 103 54 44. 34 40 69. 111 22 41. 82 67 89. 81 83 43. 9 94 106. 51 117 126. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(8) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1052. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. Thus, a D3×4 (K13 (30)) can be obtained by applying Lemma 1.6 and by the existence of an OA(3, 4). A D3×4 (K5 (30)) is constructed on the Abelian group V = Z150 generated by the group V acting on the points. It has two base grid-blocks: 66 118 87. 58 30 96. 140 84 103. 54 7 60. 134 123 6. 71 140 57. 42 99 55. 10 81 9. Lemma 2.24. There exists a 3 × 4 grid-block of type 304 151 . Proof. Let V = Z120 ∪ {∞0 , ∞1 , ∞2 , . . . , ∞13 , ∞14 }. The design is constructed on the group V generated by the group Z120 acting on the points. The 15 infinite points remain fixed under the action of Z120 . Then, it has two base grid-blocks: 0 5 18. 1 16 51. 3 ∞ 57. 10 39 ∞$. 0 53 90. 17 80 59. 62 7 21. 43 102 ∞∗. Now, by replacing three infinite points (∞, ∞$ , ∞∗ ) with (∞i , ∞i+5 , ∞i+10 ), respectively, as above (Lemma 2.20), we have the desired design. Lemma 2.25. There exist a D3×4 (K6 (12)), a D3×4 (K11 (12)), and a D3×4 (K16 (4)). Hence, there exist also a D3×4 (K6 (36)), a D3×4 (K11 (36)), and a D3×4 (K16 (36)). Proof. A D3×4 (K6 (12)) is constructed on the Abelian group V = Z72 generated by the group V acting on the points. The base grid-block is: 4 0 14. 38 7 5. 25 53 30. 65 50 13. A D3×4 (K11 (12)) is constructed on the Abelian group V = Z132 generated by the group V acting on the points. This design has two base grid-blocks: 120 57 44. 48 110 2. 52 78 38. 11 109 125. 70 9 16. 73 48 46. 108 113 65. 58 1 75. A D3×4 (K16 (4)) is constructed on the Abelian group V = Z64 generated by the group V acting on the points. The base grid-block is as follows: 0 5 43. 1 18 51. 3 57 21. 7 42 62. Since both an OA(3, 4) and an OA(9, 4) exist, the other desired designs can be obtained by applying Lemma 1.6. Lemma 2.26. There exist a D3×4 (K9 (5)) and a D3×4 (K9 (20)). Proof. The first design is constructed on V = Z15 ×{0, 1, 2} generated by the group Z15 acting on the points. Here, the nine partite sets are {{0+i, 3+i, 6+i, 9+i, 12+i} : i = 0, 1, 2} × {0, 1, 2}. The base grid-block is the following: (0,0) (8,0) (1,1). (1,0) (4,1) (8,1). (5,0) (2,1) (11,2). (0,1) (0,2) (10,0). Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(9) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. (0,0) (4,1) (3,2). (2,0) (10,1) (2,2). (14,2) (9,1) (10,1). 1053. (4,2) (6,2) (8,0). Now, by using an OA(4, 4), we can obtain a D3×4 (K9 (20)). Lemma 2.27. There exists a D3×4 (K12 (5)). Proof. Let V = Z55 ∪ {∞0 , ∞1 , ∞2 , ∞3 , ∞4 }. The design is constructed on the group V generated by the group Z55 acting on the points. The five infinite points remain fixed under the action of Z55 . The only base grid-block is the following: 0 5 29. 1 20 11. 3 28 45. 7 48 ∞. The above grid-block has the extra property that the all elements in the same row or same column with ∞ are in different residue classes (mod 5). Then, similar to Lemma 2.20, we have the construction. In order to properly combine the above lemmas together, we also need the following crucial theorem. Theorem 2.28 (see [2, 5, 7]). 1. A 5-GDD of type 4n+1 exists for n ≡ 0, 4 (mod 5). 2. A 5-GDD of type 4n 81 exists for n ≡ 0, 2 (mod 5) and n ≥ 7, except possibly when n = 10. 3. A 5-GDD of type 4n 121 exists for n ≡ 0 (mod 5) and n ≥ 10. 4. A 5-GDD of type 4n 161 exists for n ≡ 0, 3 (mod 5) and n ≥ 13, except possibly when n = 15. 5. A 5-GDD of type 4n 201 exists for n ≡ 0, 1 (mod 5) and n ≥ 16. 6. A 5-GDD of type 4n 241 exists for n ≡ 0, 4 (mod 5) and n ≥ 19. 7. There exist 5-GDDs of types 85 41 and 83 47 . 8. A 7-GDD of type n7 exists when n ≥ 7 is a prime power and n = 12. 9. There exist 7-GDDs of types 68 , 615 , 628 , and 629 . Lemma 2.29. There exists a D3×4 (K60n+36 ) for each odd positive integer n, where n ∈ {1, 5}. Proof. We start with the constructions of small 3 × 4 grid-block designs. First, if n = 3, then K60n+36 is the union of K6 (36) and six K36 ’s. By Lemmas 2.18 and 2.25, we have a D3×4 (K216 ). Similarly, for n = 9, since K576 is the union of K16 (36) and 16 K36 ’s, we have the construction by Lemmas 2.18 and 2.25. For n = 15, 21, a D3×4 (K60n+36 ) can also be obtained by a similar argument. Here, we need a D3×4 (K5 (180)) and a D3×4 (K7 (180)), which can be obtained by inflating a D3×4 (K5 (15)) (Lemma 2.2) and a D3×4 (K7 (20)) (Lemma 2.14) with an OA(12, 4) and an OA(9, 4), respectively. Now we consider the case n = 7. By Theorem 2.28, a 7-GDD of type 68 exists. Therefore, we may apply a basic technique of combinatorial designs to this GDD by assigning weights to the points (If a point is of weight h, then we inflate the point into h points.). Let each point in the first seven groups of size 6 be assigned weight 10, and let the weights of the points in the last group be (5, 5, 5, 5, 5, 10). Then, a 3 × 4 grid-block design of type 607 351 exists. This is by the fact that a D3×4 (K7 (10)) exists (Lemma 2.19) and that a 3 × 4 grid-block design of type 106 51 exists (Lemma 2.20). By using Lemma 1.4 and the existence of a D3×4 (K61 ) and a D3×4 (K36 ), we have the construction of a D3×4 (K60×7+36 ).. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(10) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1054. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. For n = 11, take a 7-GDD of type 117 (Theorem 2.28) and remove a point. Use the deleted point to redefine the groups and adjoin an extra point to obtain a {7, 12}GDD of type 611 121 , where the blocks of size 12 meet exactly one single point x in the group of size 12. Give weight 10 to points in the group of size 6 and the point x, and give weights 0, 5, or 10 to the remaining points of the group of size 12. Since we have 3 × 4 grid-block designs of types 106 (Lemma 2.21), 106 51 (Lemma 2.20), 107 (Lemma 2.19), and 1012 (Lemma 2.22), we get a 3 × 4 grid-block design of type 6011 351 . Add an extra point to obtain the desired design. Similarly, by using a 7GDD of type 137 (Theorem 2.28), we are able to construct a D3×4 (K60×13+36 ). Here, we need a 3 × 4 grid-block design of type 1013 (Lemma 2.23). For convenience in the constructions of a D3×4 (K60n+36 ) for larger n’s, we split the proof into three cases. Case 1. n ≡ 3, 7 (mod 10), n ≥ 17. By Theorem 2.28, there exists a 5-GDD of type 4t 81 for each t ≡ 0, 2 (mod 5) and t ≥ 7, t = 10. Now, for the group of size 8, we assign weights of (15, 15, 15, 30, 30, 30, 30, 30) to these points; all the points in the other groups are assigned weight 30. Therefore, by adding 21 points, we conclude that a D3×4 (K60n+36 ) exists. This is by the fact that there exists a D3×4 (K60k+21 ) for each k ≥ 0 which has a subdesign D3×4 (K21 ) and, also, there exist a D3×4 (K5 (30)), a D3×4 (K216 ), and a 3 × 4 grid-block design of type 304 151 (Lemma 2.24). Since t = 10, the case n = 23 has to be considered separately. By a similar technique with a 5-GDD of type 86 , this can be done. Case 2. n ≡ 1, 7, 9 (mod 10), n ≥ 39. Apply a 5-GDD of type 4k 201 (n = 2k + 7 or n = 2k + 9); then we have the construction by using a similar technique as in Case 1. Case 3. n ≡ 5, 7, 9 (mod 10), n ≥ 45. Apply a 5-GDD of type 4k 241 (n = 2k + 7 or n = 2k + 9) and a similar technique. Now, we have to handle the small cases. Since the technique is similar, we simply list their corresponding 5-GDD, as constructed in Theorem 2.28: n. Type. 19 23 25 29. 85 86 83 47 135. Types of 5-GDD used in construction. We are left with n = 31, 35. For n = 31 and 35, we start from a 5-GDD of type 75 and give weight 0 to 3 or 4 points in one group, and weight 60 to other points. Then, we add 36 points and fill in the holes to obtain the desired designs, since we have a D3×4 (K60k+36 ) containing a D3×4 (K36 ) as a subdesign for k = 3, 7. Here, the input designs, a D3×4 (K4 (60)), and a D3×4 (K5 (60)) come from Lemmas 2.2 and 2.2, respectively. Lemma 2.30. There exists a D3×4 (K60n+36 ) for each even positive integer, where n ∈ {2, 4, 10, 20, 22, 26}. Proof. First, if n = 6, then a D3×4 (K396 ) can be obtained by a D3×4 (K36 ) (11 copies) and a D3×4 (K11 (36)) (Lemma 2.25). For n = 8, we first use a 7-GDD of type 87 to obtain a 3 × 4 grid-block design of type 806 351 by assigning appropriate weights (0 or 5 or 10). Then, we add one point to obtain the desired grid-block design. Similarly, if n = 14, 28, take a 7-GDD of type 6n+1 (Theorem 2.28) and assign weights. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(11) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. 1055. 0 or 5 or 10 to points in one group and weight 10 to all the other points; then we obtain a 3 × 4 grid-block design of type 60n 351 . Therefore, the construction follows by adding one more point properly. The case n = 16 can be obtained by taking a 7-GDD of type 167 , assigning weights as in the above case, and adding 21 points. Note that there exists a D3×4 (K181 ) which contains a subdesign D3×4 (K21 ); see Lemma 2.26. Finally, for n = 34, take a 5-GDD of type 75 and give weights 0 to 60 to points in one group and weight 60 to all the other points to obtain a 3 × 4 grid-block design of type 42043601 . Then, the design will be obtained by adding 36 points and filling in the holes with a D3×4 (K396 ) (see Lemma 2.29) and a D3×4 (K456 ) containing a D3×4 (K36 ) as a subdesign (see Lemma 2.29). When n = 12, 18, 24, 30, 36, we first obtain a D3×4 (K10t+1 (36)) for t = 2, 3, 4, 5, 6, and then fill in the holes with a D3×4 (K36 ) to obtain the desired design. For t = 2, 6, we start from a D3×4 (K10t+1 (1)) and inflate by 36. For t = 3, take a D3×4 (K6 (12)) from Lemma 2.25 and inflate by 15 to obtain a D3×4 (K6 (180)). Then, add 36 points and fill in the holes with a D3×4 (K6 (36)) (Lemma 2.25) to obtain a D3×4 (K31 (36)). For t = 4, take a D3×4 (K8 (15)) and inflate by 12. Then, fill in the holes with a D3×4 (K6 (36)) to obtain a D3×4 (K41 (36)). For t = 5, take a D3×4 (K10 (20)) (Lemma 2.14), inflate by nine, and fill in the holes to obtain a D3×4 (K51 (36)). Now, consider larger n’s. If n ≡ 2, 6 (mod 10), n ≥ 32, the construction of a D3×4 (K60n+36 ) can be obtained by taking a 5-GDD of type 4k 161 to obtain a 3 × 4 grid-block design of type 120k 3751 (by assigning appropriate weights and adding 21 points). For n ≡ 0, 4, 8 (mod 10), we use a 5-GDD of type 4k y 1 , where y = 20 or 24 for n ≥ 38 by a similar construction. Therefore, we have the proof. 2.4. v ≡ 16 (mod 60). First, we show the nonexistence of order 16. Lemma 2.31. There does not exist a D3×4 (K16 ). Proof. Suppose not. Since a G(3, 4) is a 5-regular graph, each vertex of K16 is incident to exactly three grid-blocks. Moreover, since a G(3, 4) has 30 edges, K16 can be decomposed into four grid-blocks. This implies that every grid-block Bi corresponds to a set Si of four vertices which is vertex-disjoint from Bi , i = 1, 2, 3, 4, and also that {S1 , S2 , S3 , S4 } forms a partition of V (K16 ). Let S1 = {a, b, c, d}. Then, the edges in S1 (the subgraph of K16 induced by S1 ) must be in B2 , B3 , and (or) B4 . Thus, S1 ⊆ V (Bi ), i = 2, 3, 4. Now, if the vertices of S1 occur in a Bi (say B2 ) which are in at most two rows, then let {e, f, g, h} be the set of vertices which occur in a row of B2 which contains no vertices of {a, b, c, d}. Clearly, the vertices in {e, f, g, h} are in the grid-block B1 . Since e, f, g, and h are in the same row of B2 , {e, f, g, h} ∼ = K4 and, therefore, every edge of this graph cannot be an edge in B1 , i.e., they must be in distinct rows and columns. But, this is impossible, since we have only three rows. Thus, the vertices of S1 occur in three rows of Bi , i = 2, 3, 4. Hence, B2 , B3 , and B4 contain at least one edge of S1 , respectively. Without loss of generality, let B2 contain the maximum number of edges in S1 and let B4 contain the minimum number of edges in S1 . Therefore, we have the following cases to consider: (i) 4, 1, 1 (see Figure 1), (ii) 3, 2, 1 (see Figure 2), and (iii) 2, 2, 2 (see Figure 3), where the first coordinate represents the number of edges in S1 which are in B2 and so on. (i) 4, 1, 1 . Here, consider ∗ in B3 ; let it be x. Since x must be incident with every vertex in V (K16 ) \ {x}, x has to be in B2 or B4 such that x is incident to a and no other vertices of {b, c, d}. But, clearly this is not possible. (ii) 3, 2, 1 . Similarly, consider ∗ in B3 and the vertices in B2 or B4 which are incident to c and also incident to one of the vertices in {a, b, d}. This is not possible either.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(12) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1056. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. a b c. d. b. ∗. d. c. d. a. a c. B2. b. B3. B4. Fig. 1. Obtained by permuting rows, columns, and/or vertices in {a, b, c, d}.. a b. a c. c d. d. ∗. b. d a. b B3. B2. c B4. Fig. 2. Obtained by permuting rows, columns, and/or vertices in {a, b, c, d}.. a. b c. a d B2. c d ∗. a b B3. d b c B4. Fig. 3. Obtained by permuting rows, columns, and/or vertices in {a, b, c, d}.. (iii) 2, 2, 2 . Similarly, consider ∗ in B3 and the vertices in B2 or B4 which are incident to a and also incident to a vertex in {b, c, d}. This case is not possible either. Since (after all cases are considered) there are no suitable decompositions obtained, we conclude that no D3×4 (K16 ) exists. Lemma 2.32. There exists a D3×4 (K76 ). Proof. Let the point set be V = Z19 × (Z3 ∪ {x}). We view x as an infinite point, that is, we consider x + 1 = x. It is easy to see that 7 is the cube root of unity in Z19 . Now, we can define the automorphism group to be a non-Abelian group generated by the following two automorphisms: σ : (a, b) −→ (a + 1, b), τ : (a, b) −→ (7a, b + 1). Here, the order of σ is 19 and that of τ is 3. Hence, the automorphism group is of order 57 and the design has three base grid-blocks; the third one is of full orbit and the other two are of short orbits. (4,0) (9,1) (6,2). (2,0) (14,1) (3,2). (5,0) (16,1) (17,2). (16,0) (17,1) (5,2). (0,x) (7,0) (13,1). (3,x) (6,2) (18,0). (0,0) (13,1) (6,2). (4,0) (0,1) (15,2). (18,1) (14,x) (13,x). (14,0) (4,1) (18,2). (10,0) (9,1) (0,2). (1,x) (7,x) (11,x). Lemma 2.33. There exists a D3×4 (K13 (15)). Proof. The design is constructed on the Abelian group V = Z195 generated by the group V acting on the points. Since 16 is the cube root of unity in Z195 , we can first find one base grid-block and get all three base grid-blocks by multiplying 1, 16, and 61, respectively. Here is the first one:. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(13) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. 0 5 59. 1 15 166. 3 23 50. 1057. 7 147 185. Lemma 2.34. There exists a D3×4 (K60n+76 ) for each even positive integer n, where n ∈ {2, 6, 22}. Proof. Clearly, if there exists a 5-GDD with at least one group of size 4, then we give weight 15 to three points in the group and weight 30 to all the other points in the 5-GDD. Now, we have a 3 × 4 grid-block GDD with a group of size 75 and all the other groups of size a multiple of 120, provided that each group in the 5-GDD is of size a multiple of 4. Now, by adding one point to this inflated GDD and filling in the holes with proper 3 × 4 grid-block designs, we are able to obtain the desired designs. So, it is not difficult to see that by applying a 5-GDD of type 4k+1 , where n = 2k and n ≥ 8, we have the cases n ≡ 0, 8 (mod 10). Then, by applying a 5-GDD of type 4k 81 , where n = 2(k + 1) and n ≥ 16, we have the cases n ≡ 2, 6 (mod 10) except when n = 22. Finally, we apply a 5-GDD of type 4k 121 , where n = 2(k + 2) and n ≥ 24, and we have the cases n ≡ 4 (mod 10). Combining the results obtained above, we have the cases n = 4, 12, 14 remaining. For n = 4, we first obtain a D3×4 (K9 (35)) by assigning weight 7 to each point of a D3×4 (K9 (5)) (Lemma 2.26). Then, the case n = 4 follows by adding a point and using a D3×4 (K36 ) to fill in the holes. For n = 12, take a 7-GDD of type 127 , give weight 10 to six points in one specific group, weight 5 to one point in the same group, and weight 0 to the remaining five points of the group. Moreover, we give weight 10 to all the points in the other six groups. Now, add 10 points and fill in the six groups mentioned above with a D3×4 (K13 (10)) from Lemma 2.23. Now, by using one point with weight 0 to redefine the groups, we have a 3 × 4 grid-block design of type 6012 751 , and the case n = 12 follows by adding one more point and filling in the holes with a D3×4 (K61 ) and a D3×4 (K76 ), respectively. Finally, for n = 14, take a 7-GDD of type 615 and give weight 5 to one point and 10 to all the other points. Then, we have a 3 × 4 grid-block design of type 6014 551 . Therefore, we have the construction by adding 21 points and filling in the holes with a D3×4 (K81 ) containing a D3×4 (K21 ) as a subdesign (see Theorem 2.17) and a D3×4 (K76 ), respectively. Lemma 2.35. There exists a D3×4 (K60n+76 ) for each odd positive integer n, where n ∈ {1, 3, 9, 17, 21}. Proof. Since a 5-GDD of type 5q , where q ≡ 1 (mod 4), exists [5] and a D3×4 (K5 (15)) exists, we have a D3×4 (Kq (75)) by giving weight 15. Now, by adding one point and filling in the holes with a D3×4 (K76 ), we obtain a D3×4 (K60n+76 ) for each n ≡ 0 (mod 5) and n ≥ 5. Note that 60n + 76 = 1 + (300k + 75) = 1 + 75q, where q = 4k + 1. By a similar idea to that used in Lemma 2.34, we take a 5-GDD of type 4k 161 (Theorem 2.28), and obtain a 3 × 4 grid-block design of type 120k 3751 by giving weight 30 to points in the groups of size 4 and weight 15 or 30 to points in the group of size 16. This implies that we have the cases n ≡ 1 (mod 10), n ≥ 31, by adding one more point. Also, take a 5-GDD of type 4k 201 to obtain a 3 × 4 grid-block design of type 120k x1 with x = 60 × 5 + 75 or x = 60 × 7 + 75 and a 5-GDD of type 4k 241 to obtain a 3 × 4 grid-block design of type 120k 3751 by assigning appropriate weights. Add one point and fill in the holes to settle the cases n ≡ 7, 9 (mod 10), n ≥ 37 and n ≡ 3 (mod 10), n ≥ 43. Here, we need a D3×4 (K60×7+75 ) as the ingredient design. Take a 7-GDD of type 68 to obtain a 3 × 4 grid-block design of type 607 551 and add. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(14) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1058. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. 21 points to finish off the construction. Now, it remains to consider the small cases. First, for n = 27, take a 7-GDD of type 628 to obtain a 3 × 4 grid-block design of type 6027 551 and add 21 points. For n = 11, we give weight 35 to each point of a D3×4 (K21 ) to obtain a D3×4 (K21 (35)) and add one point. For convenience, we list the rest of the cases. • n = 13. Take a 7-GDD of type 137 to obtain a 3 × 4 grid-block design of type 6013 751 first, and then add one point. • n = 19. By the existence of D3×4 (K16 (4)) (Lemma 2.25), we have a D3×4 (K16 (76)). Then, fill in the holes with a D3×4 (K76 ). • n = 23, 29. First, we construct a D3×4 (Kn+1 (10)). For n = 23, we use the existence of a D3×4 (K4 (60)) and for n = 29, we use a D3×4 (K5 (60)) and then fill in the holes of size 60 with a D3×4 (K6 (10)). Then, take a 7-GDD of type n7 , give weight 10 to six points in one specific group, weight 5 to one point in the same group, and weight 0 to the remaining five points of the group. Moreover, we give weight 10 to all the points in the other six groups. Now, add 10 points and fill in the six groups mentioned above with a D3×4 (Kn+1 (10)). Now, by using one point with weight 0 to redefine the groups, we have a 3 × 4 grid-block design of type 60n 751 , and the existence of n = 23, 29 follows by adding one more point and filling in the holes with a D3×4 (K61 ) and a D3×4 (K76 ), respectively. • n = 33. Take a 5-GDD of type 75 (Theorem 2.28) and give weight 60 to all but two points (weight 0) in one group. Then, we obtain a 3 × 4 grid-block design of type 4204 3001 . The proof follows by adding 76 points. To conclude this section, we would like to point out that the existence of a D3×4 (K60×21+76 ) can be obtained by using transversal designs with holes. Let S be a set of size s, and let H={S1 , S2 , . . . , Sn } be a set of subsets of S. A holey Latin square having hole set H is an s × s array L, whose rows and columns are indexed by elements of S and possess the following further properties: 1. Each cell in L is either empty or contains an element of S. 2. Every element of S appears at most once in any row or column of L. 3. The subarrays indexed by Si × Si are empty for 1 ≤ i ≤ n (these subarrays are referred to as holes). 4. Symbol s ∈ S occurs in row or column t if and only if (s, t) ∈ (S × S) \ ∪1≤i≤n (Si × Si ). L is said to have order s; if S1 , S2 , . . . , Sn are disjoint, it is also said to have type (|S1 |, |S2 |, . . . , |Sn |). Alternatively, if for 1 ≤ i ≤ m there are ui holes of size ti , then we can write the type of S as tu1 1 , tu2 2 , . . . , tumm . Two holey Latin squares of the same type on the same set S of size s and hole set H are said to be orthogonal if their superposition yields every ordered pair in (S × S) \ ∪1≤i≤n (Si × Si ). A set of k holey Latin squares is said to be orthogonal if every pair of them is orthogonal. It is not difficult to notice that the existence of k holey mutually orthogonal Latin squares of type g n is equivalent to the existence of a (k + 2)-HTD (holey transversal design) of type g n . Lemma 2.36. If there exists a resolvable 6-HTD of type g n , and there exists a 3 × 4 grid-block design of type (10g)n x1 , then there exists a 3 × 4 grid-block design of type (60g)n (x + y)1 , where 0 ≤ y ≤ 10g(n − 1) and x + y is admissible. Proof. Since a resolvable 6-HTD has g(n−1) parallel classes of blocks, we can add g(n − 1) extra points to extend the blocks of these parallel classes. Assign weights 0, 5, or 10 to these new g(n − 1) points and let the weight of each point in the 6-HTD be 10. Here, the input designs 3 × 4 grid-block designs of types 106 , 106 51 , and 107 from. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(15) 1059. Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. Lemmas 2.21, 2.20, 2.19, respectively. Then, by adjoining x additional points and filling in the groups from the HTD with a 3 × 4 grid-block design of type (10g)n x1 , we obtain the desired 3 × 4 grid-block design of type (60g)n (x + y)1 . Lemma 2.37. There exists a K3×4 (K60n+76 ) when n = 21. Proof. Apply Lemma 2.36 with a resolvable 6-HTD of type 37 together with a 3×4 grid-block design of type 307 to obtain a 3 × 4 grid-block design of type 1807 (0 + 75)1 . Then, adjoin an extra point and fill in the holes to obtain the desired design. Here, the resolvable 6-HTD of type 37 comes from [1, Theorem 3.4]. The existence of a D3×4 (K7 (10)) (Lemma 2.19) implies that of a D3×4 (K7 (30)). We summarize these results in this section with the following theorem. Theorem 2.38. A D3×4 (Kv ) exists if and only if v ≡ 1, 16, 21, 36 (mod 60) except for when v = 16 and possibly except when v ∈ {60n+36|n = 1, 2, 4, 5, 10, 20, 22, 26}∪ {60n + 16|n = 2, 3, 4, 7, 10, 18, 23}. 3. 4 × 4 grid-block designs. By Lemma 1.1, it is easy to find a necessary condition for the existence of a D4×4 (Kv ). Lemma 3.1. If a D4×4 (Kv ) exists, then v ≡ 1 (mod 96). In what follows, we prove that the above necessary condition is also sufficient by constructing a D4×4 (Kv ) for each v ≡ 1 (mod 96). We start with the smallest nontrivial case: v = 97. Lemma 3.2. There exists a D4×4 (K97 ). Proof. By a computer search, we obtain a base grid-block. (There are exactly 48 distinct differences, 1, 2, . . . , 48, which are obtained from the four rows and four columns of the array.) 0 5 16 46. 1 13 60 74. 3 81 26 61. 7 38 86 29. Lemma 3.3. There exists a D4×4 (K4 (4)). Proof. Let the four partite sets of K4 (4) be A0 = {0, 4, 8, 12}, A1 = {1, 5, 9, 13}, A2 = {2, 6, 10, 14}, and A3 = {3, 7, 11, 15}. Then a D4×4 (K4 (4)) contains exactly two 4 × 4 grid-blocks as follows: 0 5 10 15. 1 4 11 14. 2 7 8 13. 3 6 9 12. 0 13 6 11. 9 4 15 2. 14 3 8 5. 17 10 1 12. We note here that if Km (n) has a decomposition into subgraphs K4 , then we have a GDD, GD[4, 1, n; mn]; see [3] for a reference. The following result has been known for some time. Theorem 3.4. A GD[4, 1, n; mn] exists if and only if 1. 3|(m − 1)n and 2. 12|n2 m(m − 1). By using Lemma 3.3 and Theorem 3.4, we can prove the following lemma. Lemma 3.5. For each m ≥ 4, a D4×4 (Km (96)) exists. Proof. First, by partitioning each partite set of Km (96) into 24 subsets (each of size 4), we have a new graph G ∼ = Km (24) in which each point represents a subset of size 4. Denote the m partite sets of G by A0 , A1 , . . . , Am−1 , where Ai = {Ai,0 , Ai,1 , . . . , Ai,23 }, i ∈ Zm . Therefore, V (G) = {Ai,j |i ∈ Zm and j ∈ Z24 }.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(16) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1060. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. By Theorem 3.4, a GD[4, 1, 24; 24m] exists for each m ≥ 4. Therefore, we conclude the proof by using Lemma 3.3 to obtain two grid-blocks for each K4 in a K4 -design of G. Lemma 3.6. There exists a D4×4 (K193 ). Proof. First, we construct a base grid-block of a D4×4 (K193 ) using a computer; see the following array:. A:. 0 5 35 82. 1 14 72 150. 3 25 131 110. 7 39 62 183. Now we check whether the array A satisfies the condition of Lemma 1.7. First, we have q = 193, e = 48, and f = (q − 1)/e = 4. We let F ∗ = GF (q) \ {0}, ω be the primitive element of F ∗ (here ω = 5), and ε = ω e . Second, we compute the cosets of C0e : C0e = {1 112 192 81} C2e = {25 98 168 95} C4e = {46 134 147 59} C6e = {185 69 8 124} C8e = {186 181 7 12} e C10 = {18 86 175 107} e C12 = {64 27 129 166} e C14 = {56 96 137 97} e C16 = {49 84 144 109} e C18 = {67 170 126 23} e C20 = {131 4 62 189} e C22 = {187 100 6 93} e C24 = {43 184 150 9} e C26 = {110 161 83 32} e C28 = {48 165 145 28} e C30 = {42 72 151 121} e C32 = {85 63 108 130} e C34 = {2 31 191 162} e C36 = {50 3 143 190} e C38 = {92 75 101 118} e C40 = {177 138 16 55} e C42 = {179 169 14 24} e C44 = {36 172 157 21} e C46 = {128 54 65 139}. C1e = {5 174 188 19} C3e = {125 104 68 89} C5e = {37 91 156 102} C7e = {153 152 40 41} C9e = {158 133 35 60} e C11 = {90 44 103 149} e C13 = {127 135 66 58} e C15 = {87 94 106 99} e C17 = {52 34 141 159} e C19 = {142 78 51 115} e C21 = {76 20 117 173} e C23 = {163 114 30 79} e C25 = {22 148 171 45} e C27 = {164 33 29 160} e C29 = {47 53 146 140} e C31 = {17 167 176 26} e C33 = {39 122 154 71} e C35 = {10 155 183 38} e C37 = {57 15 136 178} e C39 = {74 182 119 11} e C41 = {113 111 80 82} e C43 = {123 73 70 120} e C45 = {180 88 13 105} e C47 = {61 77 132 116}. = {5, 30, 35, 47, 77, 82, 13, Third, by using the differences less than 97, we have ∂A 58, 71, 78, 57, 44, 22, 87, 65, 21, 85, 86, 32, 23, 55, 72, 49, 17, 1, 2, 3, 4, 6, 7, 9, 11, 20, 14, 25, 34, 37, 59, 96, 69, 10, 27, 68, 40, 28, 73, 33, 92}. Therefore, by Lemma 1.7, a D4×4 (K193 ) exists. Note that all grid-blocks are obtained by calculating cA+x for c ∈ C0e /{1, −1} =. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(17) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. GRID-BLOCK DESIGNS. 1061. {1, 112} and x ∈ GF (q). Lemma 3.7. There exists a D4×4 (K289 ). Proof. By a similar idea, we find A first. Let F ∗ = GF (172 ) \ {0} and ω be a primitive element of F ∗ ; here we let ω be the root of the primitive polynomial ω 2 + ω + 3. Then, we have the following:. A:. ω∞ ω3 ω 20 ω 46. ω0 ω4 ω 17 ω 70. ω1 ω5 ω 155 ω 221. ω2 ω6 ω 83 ω7. Now, the cosets of C0e are as follows: C0e = {1, ω 48 , ω 96 , ω 144 , ω 192 , ω 240 } C1e = {ω, ω 49 , ω 97 , ω 145 , ω 193 , ω 241 } .. . e C46 = {ω 46 , ω 94 , ω 142 , ω 190 , ω 238 , ω 286 } e C47 = {ω 47 , ω 95 , ω 143 , ω 191 , ω 239 , ω 287 } = {ω 3 , ω 8 , ω 20 , ω 30 , ω 21 , ω 46 , Therefore, we can obtain the set of differences ∂A 36 5 7 15 35 12 39 27 22 23 24 ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω , ω 13 , ω 16 , ω 14 , ω 31 , ω 43 , ω 19 , 1, ω 37 , ω, ω 38 , ω 6 , ω 2 , ω 40 , ω 41 , ω 9 , ω 42 , ω 10 , ω 4 , ω 18 , ω 45 , ω 33 , ω 29 , ω 28 , ω 47 , ω 26 , ω 25 , ω 32 , ω 44 , ω 34 , ω 17 }. Hence, we have a D4×4 (K289 ). Here, all grid-blocks are obtained by Lemma 1.7, calculating cA + x for c ∈ C0e /{1, −1} and x ∈ GF (q). Theorem 3.8. A D4×4 (Kv ) exists if and only if v ≡ 1 (mod 96). Proof. By Lemmas 3.2 and 3.5, we know that a D4×4 (K96t+1 ) exists for each positive integer t, t = 1, and t ≥ 4. So, by using Lemmas 3.6 and 3.7, we conclude that a D4×4 (Kv ) exists for each v ≡ 1 (mod 96). Hence, by Lemma 3.1 the theorem is proved. 11. Acknowledgments. The authors thank Dr. Jyhmin Kuo for constructing a D4×4 (K97 ) by a computer search. Also, we thank the referees for their helpful comments in revising the paper. REFERENCES [1] R. J. R. Abel, F. E. Bennett, and G. Ge, The existence of four HMOLS with equal sized holes, Des. Codes Cryptogr., 26 (2002), pp. 7–31. [2] R. J. R. Abel, G. Ge, M. Greig, and A. C. H. Ling, Further result on (v, {5, w ∗ }, 1)-PBDs, Discrete Math., to appear. [3] A. E. Brouwer, A. Shrijver, and H. Hanani, Group divisible designs with block size 4, Discrete Math., 20 (1997), pp. 1–10. [4] J. E. Carter, Designs on Cubic Multigraphs, Ph.D. thesis, McMaster University, Hamilton, ON, Canada, 1989. [5] C. J. Colbourn and J. H. Dinitz, eds., Handbook of Combinatorial Designs, 2nd ed., Chapman and Hall/CRC, Boca Raton FL, 2007. [6] H.-L. Fu, F. K. Hwang, M. Jimbo, Y. Mutoh, and C.-L. Shiue, Decomposing complete graphs into Kr × Kc ’s, J. Statist. Plann. Inference, 119 (2004), pp. 225–236. [7] G. Ge and A. C. H. Ling, Asymptotic results on the existence of 4-RGDDs and uniform 5GDDs, J. Combin. Des., 13 (2005), pp. 222–237. [8] F. K. Hwang, An isomorphic factorization of the complete graph, J. Graph Theory, 19 (1995), pp. 333–337. [9] Y. Li, J. Yin, R. Zhang, and G. Ge, The decomposition of Kv into K2 × K5 ’s, Sci. China Ser. A, 50 (2007), pp. 1382–1388.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(18) Downloaded 04/25/14 to 140.113.38.11. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php. 1062. R. ZHANG, G. GE, A. C. H. LING, H.-L. FU, AND Y. MUTOH. [10] Y. Mutoh, M. Jimbo, and H.-L. Fu, A resolvable r × c grid-block packing and its application to DNA library screening, Taiwanese J. Math., 8 (2004), pp. 713–737. [11] Y. Mutoh, T. Morihara, M. Jimbo, and H.-L. Fu, The existence of 2 × 4 grid-block designs and their applications, SIAM J. Discrete Math., 16 (2003), pp. 173–178. [12] D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York, 1971. [13] D. B. West, Introduction to Graph Theory, Prentice–Hall, Upper Saddle River, NJ, 1996. [14] R. M. Wilson, Cyclotomy and difference families in elementary abelian groups, J. Number Theory, 4 (1972), pp. 17–47. [15] F. Yates, Lattice squares, J. Agri. Sci., 30 (1940), pp. 672–687.. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited..

(19)