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Super-simple group divisible designs with block size 4 and index 2
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Super-simple group divisible designs with block size 4 and index 2 H. Cao a,,1, F. Yan a, R. Wei b,2 a b
School of Mathematics Science, Nanjing Normal University, Nanjing 210046, China Department of Computer Science, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
a r t i c l e in fo
abstract
Article history: Received 27 April 2008 Received in revised form 22 June 2009 Accepted 24 February 2010 Available online 2 March 2010
Super-simple group divisible designs are useful for constructing other types of supersimple designs which can be applied to codes and designs. In this article, we investigate the existence of a super-simple (4,2)-GDD of type gu and show that such a design exists if and only if u 4, gðu2Þ 4, gðu1Þ 0 ðmod 3Þ and uðu1Þg 2 0 ðmod 6Þ. & 2010 Elsevier B.V. All rights reserved.
Keywords: Group divisible design Pairwise balanced design Super-simple
1. Introduction Let K be a set of positive integers and l a positive integer. A group divisible design ðK,lÞGDD is a triple ðX ,G,BÞ which satisfies the following properties: 1. X is a finite set of points. 2. G is a partition of X into subsets called groups. 3. B is a collection of subsets of X with sizes from K, called blocks, such that every pair of points from distinct groups occurs in exactly l blocks. 4. No pair of points belonging to a group occurs in any block. The type of a GDD ðX ,G,BÞ is the multiset fjGj : G 2 Gg. We usually use the ‘‘exponential’’ notation to describe types: g1u1 gkuk denotes ui occurrences of gi, 1 ri rk, in the multiset. If K ={k}, we write ðk,lÞGDD instead of ðfkg,lÞGDD. When l ¼ 1, it is omitted. Thus a k-GDD is a(k,1)-GDD. A transversal design TD ðk,l; nÞ is a ðk,lÞGDD of type nk. When l ¼ 1, we simply write TD(k,n). A ðK,lÞGDD with group type 1v is called a pairwise balanced design, denoted by ðv,K,lÞPBD. A ðk,lÞGDD with group type 1v is called a balanced incomplete block design, denoted by ðv,k,lÞBIBD. T A design is called simple if it contains no repeated blocks. A design ðX ,BÞ is said to be super-simple if jB1 B2 j r 2 for any two blocks B1 ,B2 2 B and B1 aB2 . When jBj ¼ 3 for any B 2 B, a super-simple design is just a simple design. When l ¼ 1, the designs are necessarily super-simple. The concept of super-simple designs was introduced by Gronau and Mullin (1992). The existence of super-simple designs is an interesting extremal problem by itself, but there are also some useful applications. For example, such super-simple designs Corresponding author.
E-mail addresses: [email protected] (H. Cao), [email protected] (R. Wei). Research supported by the National Natural Science Foundation of China under Grant no. 10971101. Research supported by NSERC discovery Grant 239135-06.
1 2
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.02.020
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are used in perfect hash families (Stinson et al., 2000) and covering Bluskov and H¨am¨al¨ainen (1998), in the construction of new designs (Bluskov, 1997) and in the construction of superimposed codes (Kim and Lebedev, 2004). There are many research papers for constructions of super-simple BIBDs and TDs. Therefore, the existence of super-simple GDDs, the more general designs, is an interesting design problem. On the other hand, super-simple group divisible designs are useful for the construction of other types of super-simple designs. The main purpose of this paper is to start the research on super-simple GDDs. The first interesting case is for GDDs with block size 4 and index 2, because a super-simple GDD of block size 3 is just a simple GDD. It is easy to see that the followings are the necessary conditions for the existence of a super-simple ðk,lÞGDD of type gu. Theorem 1.1. If there exists a super-simple ðk,lÞGDD of type gu, then u Z k, l r gðu2Þ=ðk2Þ, lðu1Þg 0 ðmod k1Þ and
luðu1Þg 2 0 ðmod kðk1ÞÞ. There are known results for the existence of super-simple designs, especially for super-simple ðv,k,lÞBIBDs. When k=4 or 5 the necessary conditions for super-simple ðv,k,lÞBIBDs are known to be sufficient for 2 r l r 6 or 2 r l r5 with few possible exceptions. We summarize these known results in the following theorems. Theorem 1.2 (Gronau and Mullin, 1992; Khodkar, 1994; Chen, 1995; Adams et al., 1996; Chen, 1996; Gronau, 2007; Cao et al., 2009; Chen et al., 2005). There exists a super-simple ðv,4,lÞBIBD with 2 r l r6 if and only if 1. 2. 3. 4. 5.
l ¼ 2, l ¼ 3, l ¼ 4, l ¼ 5, l ¼ 6,
v 1 ðmod 3Þ and v Z 7; v 0,1 ðmod 4Þ and v Z 8; v 1 ðmod 3Þ and v Z 10; v 1,4 ðmod 12Þ and v Z13; v Z14.
Theorem 1.3 (Gronau et al., 2004; Chen and Wei, 2007, 2006; Gronau, 2007; Abel and Bennett, 2008; Abel et al., 2008). A super-simple ðv,5,lÞBIBD exists with 2 r l r 5 if and only if 1. 2. 3. 4.
l ¼ 2, l ¼ 3, l ¼ 4, l ¼ 5,
v v v v
1,5 ðmod 10Þ, va5,15; 1 ðmod 20Þ and v Z21, and except possibly when v 5 ðmod 20Þ and v Z 25; 0,1 ðmod 5Þ and v Z 15; 1 ðmod 4Þ and v Z 17, except possibly when v =21.
In this paper, we investigate the existence of a super-simple (4,2)-GDD of type gu. We shall prove that there exists a super-simple (v,{4,5},2)-PBD for any v Z7 with four exceptions in Section 2. Then we shall apply this result to show that the necessary conditions for the existence of a super-simple (4,2)-GDD of type gu are also sufficient. The main results of this paper are summarized as follows. Theorem 1.4. A super-simple (v,{4,5},2)-PBD exists if and only if v Z 7 except when v 2 f8,9,12,14g. Theorem 1.5. A super-simple (4,2)-GDD of type gu exists if and only if u Z 4, gðu2Þ Z 4, gðu1Þ 0 ðmod 3Þ and uðu1Þg 2 0 ðmod 6Þ. 2. Proof of Theorem 1.4 In this section, we shall prove Theorem 1.4. First, we shall use direct constructions to obtain super-simple (v,{4,5}, 2)-PBD for some small values of v. These designs have been obtained after computer-assisted searches. The way to check the supersimplicity is essentially the same as what was used in Bluskov (1997). Suppose that a design is obtained by developing m base blocks modulo v. Let S ¼ fb1 ,b2 ,b3 g, b1 o b2 o b3 , be a 3-subset contained in a base block. Instead of developing S modulo v we form the following three representatives of the orbit corresponding to S: fb1 bi ,b2 bi ,b3 bi g,
i ¼ 1,2,3:
It is mentioned in Bluskov (1997) that if these 3-subsets are pairwise distinct, then the design is super-simple. Lemma 2.1. There exists a super-simple (15,{4,5}, 2)-PBD. Proof. Let the point set be Z15. Below are the required base blocks. The required design is obtained by developing the base blocks modulo 15. Here, the first base block {0,3,6,9,12} has a short orbit of order 3. f0,3,6,9,12g
f0,1,4,5g
f0,2,7,9g:
&
Lemma 2.2. There exists a super-simple (v,{4,5}, 2)-PBD for any v 2 M, where M = {17,23,27, 33,39,47}.
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Proof. For each v 2 M, let the point set be Zv. Instead of listing all the required blocks, we only list the base blocks and all the required blocks can be generated from them by ( + 1 mod v). v= v= v= v= v= v=
17: 23: 27: 33: 39: 47:
{0,1,2,5,8} {0,1,2,5,11} {0,1,2,5,11} {0,1,2,4,11} {0,1,2,4,9} {0,1,3,9,27} {0,8,18,34}
{0,2,7,11} {0,2,6,16} {0,2,7,15,19} {0,3,8,15,21} {0,3,13,19,28} {0,1,4,6} {0,9,20,34}
{0,3,8,15} {0,3,14,21} {0,4,14,20} {0,4,17,25} {0,4,11,23}
{0,5,14,22} {0,5,17,24} {0,5,17,33}
{0,6,16,27} {0,7,17,32}
& Lemma 2.3. There exists a super-simple (v,{4,5}, 2)-PBD for any v 2 M ¼ f18,26,32,36,38,42,48,56,66g. Proof. For each v 2 M, let the point set be Zv. All the required blocks can be generated from the following base blocks by ( +2 mod v). v = 18: v = 26: v = 32: v = 36:
v = 38:
v = 42:
v = 48:
v = 56:
v = 66:
{0,1,2,4,8} {0,1,2,3,6} {0,8,16,25} {0,1,3,9,27} {0,6,17,18} {0,1,3,9,27} {0,1,13,17} {0,4,12,29} {0,1,3,9,27} {0,10,21,31} {0,19,23,33} {0,1,3,9,27} {0,6,14,26} {0,2,23,28} {0,1,3,9,27} {0,6,13,18} {0,2,21,35} {0,1,2,3,6} {0,12,25,28} {0,4,35,43} {1,7,25,35} {0,1,3,9,27} {0,4,43,65} {0,16,33,47} {0,2,40,47} {0,15,23,59}
{0,1,3,9} {0,2,5,9,15} {1,5,13,19} {0,1,2,4,8} {0,7,19,23} {0,6,15,19} {0,2,18,29}
{0,3,6,13} {0,4,14,19}
{0,5,7,11} {0,6,17,19}
{0,5,9,10} {0,7,12,23}
{0,3,7,18} {0,9,19,21} {0,7,21,22} {0,2,25,35}
{0,5,12,22} {0,11,16,29} {0,5,7,10} {0,3,8,23}
{0,5,15,24} {0,5,11,19} {0,4,10,24}
{0,1,4,12,36} {0,10,25,26}
{0,2,7,8} {0,11,18,33}
{0,4,9,18} {0,13,17,25}
{0,7,13,22} {0,17,19,35}
{0,11,21,33} {0,7,18,33} {0,4,19,22} {0,15,25,37} {0,8,16,34} {0,2,25,26} {0,2,5,7,11} {0,14,29,34} {0,6,32,45} {1,13,27,39} {0,20,45,55} {0,6,13,63} {0,18,41,51} {0,9,14,36}
{0,3,4,10} {0,8,25,35} {0,11,12,25} {0,4,19,35} {0,11,20,36} {0,3,10,39} {0,20,43,47} {0,14,32,47} {0,8,25,45}
{0,5,7,10} {0,1,23,29}
{0,5,9,13} {0,2,19,31}
{0,5,7,10} {0,11,28,42} {0,17,37,47} {0,7,21,41} {0,18,41,49} {0,9,30,46}
{0,5,9,13} {0,1,4,33} {0,21,27,41} {0,10,21,22} {0,19,29,48} {0,17,33,49}
{0,4,29,41} {0,5,43,58} {0,19,31,49} {0,11,27,42}
{0,3,29,49} {0,11,17,21} {0,1,33,65} {0,12,26,42}
{0,6,38,59} {0,5,8,18} {0,2,22,56} {0,15,19,57}
& For our recursive constructions, we shall use the following standard recursive constructions, the proofs of which can be found in Chen (1996). S Construction 2.4 (Weighting). Let ðX ,G,BÞ be a super-simple GDD with index l1 , and let o : X-Z þ f0g be a weighting + function on X, where Z is the set of positive integers. Suppose that for each block B 2 B, there exists a super-simple P ðk,l2 ÞGDD of type foðxÞ : x 2 Bg. Then there exists a super-simple ðk,l1 l2 ÞGDD of type f x2Gi oðxÞ : Gi 2 Gg. Construction 2.5 (Breaking up groups). If there exists a super-simple ðK,lÞGDD of type hu11 hut t and a super-simple P ðhi þ Z,K,lÞPBD for each i, 1 r ir t, then there exists a super-simple ð ti ¼ 1 hi ui þ Z,K,lÞPBD, where Z ¼ 0 or 1. We also need the following known results about super-simple ðv,f4,5g,lÞGDDs. Lemma 2.6 (Abel et al., 2008). Necessary conditions for a super-simple (5,2)-GDD of type hn to exist are: (1) n Z5, (2) nðn1Þh2 0 ðmod 5Þ, and (3) if h is odd, then n is odd. These conditions are also sufficient, except for (h,n)= (1, 5), (1,15), (2,5) and possibly for h 2 f13,17,19g, n = 15. Lemma 2.7 (Hartmann, 2000). A super-simple TD ð4,l; vÞ exists if and only if l rv and ðl,vÞ is neither (1,2) nor (1,6). It is obvious that we can obtain a super-simple ðv,fk1,kg,lÞPBD by deleting one point from the point set of a supersimple ðv,k,lÞBIBD. So, we have the following lemma. Lemma 2.8. If there exists a super-simple ðv,k,lÞBIBD, then there exists a super-simple ðv1,fk1,kg,lÞ- PBD. More generally, we can obtain a super-simple ðvs,fk1,kg,lÞGDD by deleting s points from the last group of a supersimple ðv,k,lÞGDD.
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Lemma 2.9. If there exists a super-simple ðv,k,lÞGDD with type h1 h2 ht , then there exists a super-simple ðvs,fk1,kg,lÞGDD with type h1 h2 ht1 ðht sÞ. Lemma 2.10. There exists a super-simple (v,{4,5},2)-PBD for any v 0,4 ðmod 10Þ and va4,14. Proof. By Theorem 1.3, there exists a super-simple ðv,5,2ÞBIBD for any v 1,5 ðmod 10Þ and va5,15. Applying Lemma 2.8 with k= 5 and l ¼ 2, we can get a super-simple (v,{4,5},2)-PBD for any v 0,4 ðmod 10Þ and va4,14. & Lemma 2.11. There exists a super-simple (v,{4,5},2)-PBD for any v 2 M ¼ f29,53,59,78, 83,102,107,119,132,135,138, 143,156,162g [ f12 t þ d : t 2 ½5,13\f6,9,11g,d ¼ 2,3g. Proof. For each v 2 M, let v =4g+ x. The two parameters g and x are listed in the following table. We start from a supersimple (5,2)-GDD of type g5 which exists by Lemma 2.6. Removing g x points from the last group of the super-simple (5,2)-GDD of type g5, we get a super-simple ({4,5},2)-GDD of type g4 x1 by Lemma 2.9. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (4g + x,{4,5},2)-PBD. Here, all these input super-simple (g,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.1–2.3, 2.10. v= 4g + x
g
x
v =4g+ x
g
x
v= 4g + x
g
x
29 62 83 98 107 123 138 147 159
7 13 17 20 23 25 28 31 33
1 10 15 18 15 23 26 23 27
53 63 86 99 119 132 143 156 162
13 13 19 20 25 28 31 33 33
1 11 10 19 19 20 19 24 30
59 78 87 102 122 135 146 158
13 17 19 21 25 28 31 33
7 10 11 18 22 23 22 26
& Lemma 2.12. There exists a super-simple (v,{4,5},2)-PBD for any v 2 f72,77,89,96,108, 113,137,149g [ f12 t þ d : t 2 f5, 7,9,10,12g,d ¼ 8,9g. Proof. By Lemma 2.7, there exists a super-simple (4,2)-GDD of type g4. Starting form this GDD and applying Construction 2.5, we obtain a super-simple ð4g þ Z,f4,5g,2ÞPBD, where Z ¼ 0 or 1. Here, the input designs ðg þ Z,f4,5g,2ÞPBDs come from Theorems 1.2, 1.3 and 2.2, 2.3, 2.10, 2.11. We list all the parameters in the following table. v ¼ 4g þ Z
g
Z
v ¼ 4g þ Z
g
Z
v ¼ 4g þ Z
g
Z
68 77 93 113 128 149
17 19 23 28 32 37
0 1 1 1 0 1
69 89 96 116 129 152
17 22 24 29 32 38
1 1 0 0 1 0
72 92 108 117 137 153
18 23 27 29 34 38
0 0 0 1 1 1
& Lemma 2.13. There exists a super-simple (126,{4,5},2)-PBD. Proof. Start from a super-simple (5,2)-GDD of type 255 which exists by Lemma 2.6. Applying Construction 2.5 with Z ¼ 1, we get a super-simple (126, {4,5},2)-PBD. Here, the input super-simple ( 26,{4,5},2)-PBD comes from Lemma 2.3. & Now, we shall prove Theorem 1.4. By Theorem 1.2, there exists a super-simple (v,{4,5}, 2)-PBD for any v 1 ðmod 3Þ and v Z7. Then we only need to prove that there exists a super-simple (v,{4,5}, 2)-PBD for any v 0,2 ðmod 3Þ and v Z8. For convenience, 0,2 we use [a,b]1,2 3 or [a,b]3 to denote a set of positive integers v such that a rv r b and v 1,2 ðmod 3Þ or v 0,2 ðmod 3Þ. Lemma 2.14. There exists a super-simple (v,{4,5},2)-PBD for any v Z 11, va12,14 and v= 48t + m, where m 2 ½0,470,2 3 , t Z0. 0,2 0,2 Proof. We distinguish m into three cases: m 2 ½0,180,2 3 , m 2 ½20,413 and m 2 ½42,473 .
(1) m 2 ½0,180,2 3 : For t r3, the existence of a super-simple (v,{4,5},2)-PBD has been solved by Theorem 1.3 and Lemmas 2.1–2.3, 2.10–2.12. For t Z 4, we start with a super-simple (5,2)-GDD of type (12t 5)5. Removing 12t 5 x, 4 1 x 2 ½20,381,2 3 , points from the last group, we get a super-simple ({4,5},2)-GDD of type (12t 5) x by Lemma 2.9. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (48t + m,{4,5},2)-PBD. Here, all these input super-simple (12t 5,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.2, 2.3, 2.10.
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(2) m 2 ½20,410,2 3 : For t r 2, there exists a super-simple (v,{4,5},2)-PBD by Theorem 1.3 and Lemmas 2.2, 2.3, 2.10–2.13. For t Z 3, we start from a super-simple (5,2)-GDD of type (12t+ 1)5. Remove 12t + 1 x, x 2 ½16,371,2 3 , points from the last group to get a super-simple ({4,5},2)-GDD of type (12t + 1)4 x1. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (v,{4,5},2)-PBD. Here, all these input super-simple (12t +1,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.2, 2.3, 2.10, and 2.11. (3) m 2 ½42,470,2 3 : For t r2, there exists a super-simple (v,{4,5},2)-PBD by Theorem 1.3 and Lemmas 2.2, 2.3, 2.10–2.12. For t Z 3, we start with a super-simple (5,2)-GDD of type (12t + 4)5. Removing 12t +4 x, x 2 ½26,311,2 3 , points from the last group, we get a super-simple ({4,5},4)-GDD of type (12t + 4)4 x1. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (v,{4,5},2)-PBD. Here, all these input super-simple (12t +4,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.3, 2.11. & By some simple computations and a computer exhaustive search, there does not exist a super-simple (v,{4,5},2)-PBD for any v 2 f8,9,12,14g. So, combining Theorem 1.2 and Lemma 2.14, we have proved Theorem 1.4. 3. Proof of Theorem 1.5 It is easy to see that the necessary conditions can be distinguished into two cases, namely, g 0 ðmod 3Þ, u Z4 and g 1,2 ðmod 3Þ, u 1 ðmod 3Þ, gðu2Þ Z 4. Firstly, we settle sufficiency when g 0 ðmod 3Þ and u Z 4. We start with some direct constructions. Lemma 3.1. There exists a super-simple (4,2)-GDD of type 3u for any u 2 f5,6,8,9,12,14g. Proof. For each u 2 f5,6,8,9,12,14g, let the point set be Z3u and the group set be ffi,u þi,2u þ ig : 0 ri r u1g. Below are the required base blocks. u = 5: u = 6: u = 8: u = 9: u = 12: u = 14:
{0,1,3,7} {0,1,2,3} {0,1,2,3} {0,9,11,21} {0,1,2,4} {0,1,2,3} {0,9,17,22} {0,1,2,24} {0,8,16,25} {0,23,27,35}
{0,1,9,13} {0,2,5,10} {0,2,5,9} {0,13,17,23} {0,3,10,15} {0,2,5,6} {0,11,19,22} {0,1,20,37} {0,10,31,39}
{0,4,9,13} {0,4,14,19}
{0,4,11,15} {0,4,15,21}
{0,4,11,17} {0,4,9,11} {0,13,16,31} {0,2,5,6} {0,11,15,33}
{0,5,11,19} {0,6,23,27} {0,13,19,29} {0,3,12,24} {0,13,29,31}
{0,7,15,17} {0,6,12,19}
{0,7,21,27} {0,15,25,29} {0,4,9,11} {0,15,27,37}
+1 mod 15 +2 mod 18 +2 mod 24 +1 mod 27 {0,8,16,26} +2 mod 36 {0,6,13,16} {0,19,25,35} +2 mod 42
& Lemma 3.2. There exists a super-simple (4,2)-GDD of type 6u for any u 2 f6,11,14,15,18,23g. Proof. Let the point set be Z6u and the group set be ffju þi : 0 r j r 5g : 0 ri r u1g. All the required blocks can be generated from the following base blocks by ( +1 mod 6u). u = 6: u = 11: u = 14:
u = 15:
u = 18:
u = 23:
{0,1,2,4} {0,5,15,35} {0,17,41,63} {0,13,36,57} {0,5,31,43} {0,19,48,70} {0,14,28,61} {0,4,41,50} {0,22,60,85} {0,14,28,43} {0,1,4,42} {0,31,68,104} {0,14,28,43} {0,29,62,97} {0,3,12,60}
{0,3,7,22} {0,9,25,43} {0,2,34,59} {0,15,35,66}
{0,5,13,28} 13i{0,1,3,7} {0,5,11,17} {0,1,4,66}
{0,5,14,25} 13i {0,2,15,32} {0,7,15,23} {0,1,34,37}
{0,7,16,26} ð0 r ir 3Þ {0,9,19,29} {0,2,32,39}
{0,11,24,40} {0,4,30,39}
{0,3,35,58} {0,16,50,72} {0,5,12,59} {0,3,47,77} {0,16,33,53} {0,2,52,59} {0,2,61,64} {0,16,32,50} {0,30,75,105} {0,5,49,91}
{0,5,11,17} {0,19,63,88}
{0,7,16,24} {0,1,4,42}
{0,8,18,28} {0,1,27,52}
{0,11,24,37} {0,2,23,59}
{0,5,11,17} {0,19,40,65} {0,4,39,67} {0,7,61,97} {0,19,38,58} {0,31,66,98} {0,6,17,73}
{0,7,15,23} {0,21,55,82} {0,5,32,44} {0,7,15,24} {0,21,42,64} {0,1,40,52} {0,6,44,93}
{0,9,19,29} {0,22,46,78} {0,9,40,75} {0,8,18,28} {0,22,47,72} {0,1,5,83}
{0,11,24,37} {0,1,58,60} {0,11,24,37} {0,26,53,80} {0,2,55,59}
& We also need the following known results which can be found in Abel et al. (2007). Lemma 3.3 (Abel et al., 2007). 1. A 4-GDD of type mu exists if and only if u Z 4, ðu1Þm 0 ðmod 3Þ and uðu1Þm2 0 ðmod 12Þ except ðm,uÞ 2 fð2,4Þ,ð6,4Þg. 2. A(v,{4,5,6,8},1)-PBD exists for all v Z8 and va9,10,11,12,14,15,18,19,23. 3. A (v,{4,7},1)-PBD exists for all v 1 ðmod 3Þ and va10,19. Lemma 3.4. There exists a super-simple(4,2)-GDD of type 6u for any u 2 f5,8,9,12g.
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Proof. We start from a (4,1)-GDD of type 3u which exists by Lemma 3.3. Applying Construction 2.4 with weight 2, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDD of type 24 comes from Lemma 2.7. & Lemma 3.5. There exists a super-simple (4,2)-GDD of type 6u for any u 2 f7,10,19g. Proof. We start from a (4,1)-GDD of type 2u which exists by Lemma 3.3. Applying Construction 2.4 with weight 3, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDD of type 34 comes from Lemma 2.7. & Lemma 3.6. There exists a super-simple (4,2)-GDD of type 6u for any u Z4. Proof. If u 2 ½4,12 [ f14,15,18,19,23g, then a super-simple (4,2)-GDD of type 6u exists by Lemma 2.7 and Lemmas 3.2–3.5. For other values u, start from a ðu,f4,5,6,8g,1ÞPBD which exists by Lemma 3.3. Applying Construction 2.4 with weight 6, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDDs of type 6m ,m 2 f4,5,6,8g come from Lemmas 2.7, 3.2 and 3.4. & Lemma 3.7. There exists a super-simple (4,2)-GDD of type 18u for any u 2 f6,14g. Proof. Start form a super-simple (4,2)-GDD of type 6u which exists by Lemma 3.2. Applying Construction 2.4 with a (4,1)GDD of type 34 form Lemma 3.3, we can obtain a super-simple (4,2)-GDD of type 18u. & Lemma 3.8. There exists a super-simple(4,2)-GDD of type gu for any g 0 ðmod 3Þ and u 2 f6,14g. Proof. Let g= 3m, m Z 1. If m 2 f1,2,6g, then a super-simple (4,2)-GDD of type gu exists by Lemmas 3.1, 3.2 and 3.7. If ma1,2,6, we start from a super-simple (4,2)-GDD of type 3u which exists by Lemma 3.1. Applying Construction 2.4 with weight m, we obtain a super-simple (4,2)-GDD of type (3m)u, where the input (4,1)-GDD of type m4 comes from Lemma 3.3. & Lemma 3.9. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ and u 2 f5,8,9,12g. Proof. Let g = 3m, m Z1. If m =1, then there exists a super-simple (4,2)-GDD by Lemma 3.1. Suppose m 4 1. Start from a (4,1)-GDD of type 3u which exists by Lemma 3.3. Applying Construction 2.4 with weight m, we obtain a super-simple (4,2)-GDD of type (3m)u , where the input super-simple (4,2)-GDD of type m4 comes from Lemma 2.7. & Lemma 3.10. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ, ga6 and u Z7, u= 2f8,9,12,14g. Proof. By Theorem 1.4, there exists a super-simple ðu,f4,5g,2ÞPBD. Applying Construction 2.4 with weight g, we obtain a super-simple (4,2)-GDD of type gu, where the input designs of (4,1)-GDD of type g4 and the (4,1)-GDD of type g5 come from Lemma 3.3. & Combining Lemmas 2.7, 3.6 and Lemmas 3.8–3.10, we have the following theorem. Theorem 3.11. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ and u Z 4. Now, we study the case when g 1,2 ðmod 3Þ and u 1 ðmod 3Þ, gðu2Þ Z 4. First, we shall use direct constructions to obtain some super-simple (4,2)-GDDs. Lemma 3.12. There exists a super-simple (4,2)-GDD of type 2u for any u 2 f7,10,19g. Proof. Let the point set be Z2u and the group set be ffi,u þig : 0 r ir u1g. All the required blocks can be generated from the following base blocks by ( +1 mod 2u). u= 7: u= 10: u= 19:
{0,1,4,6} {0,1,2,6} 7i {0,1,2,4}
{0,1,9,11} {0,2,9,16} 7i {0,3,9,29}
{0,3,8,11} ð0 r i r 2Þ
& Lemma 3.13. There exists a super-simple (4,2)-GDD of type 2u for any u 1 ðmod 3Þ. Proof. There exists a super-simple (4,2)-GDD of type 2u for any u 2 f10,19g by Lemma 3.12. Now, let u 1 ðmod 3Þ and u= 2f10,19g. By Lemma 3.3, there exists a (u,{4,7},1)- PBD. Applying Construction 2.4 with weight 2, we obtain a super-simple (4,2)-GDD of type 2u, where the input super-simple (4,2)-GDDs of type 24 and 27 come from Lemmas 2.7 and 3.12. & Lemma 3.14. There exists a super-simple (4,2)-GDD of type gu for any g 1,2 ðmod 3Þ, ga2 and u 1 ðmod 3Þ, u Z7. Proof. We start from a super-simple (u,4,2)-BIBD, u 1 ðmod 3Þ and u Z 7, which exists by Theorem 1.2. Applying Construction 2.4 with weight g, g 1,2 ðmod 3Þ and ga2, we obtain a super-simple (4,2)-GDD of type gu, where the input design of a (4,1)-GDD of type g4 comes from Lemma 3.3. &
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Combining Lemmas 2.7, 3.13 and 3.14, we have the following theorem. Theorem 3.15. There exists a super-simple (4,2)-GDD of type gu for any g 1,2 ðmod 3Þ, u 1 ðmod 3Þ and gðu2Þ Z 4. Combining Theorems 3.11 and 3.15, we have proved Theorem 1.5.
Acknowledgement The authors would like to thank the referee for his careful reading of the manuscript and helpful comments which have improved the manuscript of this paper. References Abel, R.J.R., Bennett, F.E., 2008. Super-simple Steiner pentagon systems. Discrete Appl. Math. 156, 780–793. Abel, R.J.R., Bennett, F.E., Ge, G., 2008. Super-simple holey Steiner pentagon systems and related designs. J. Combin. Des. 16, 301–328. Abel, R.J.R., Colbourn, C.J., Dinitz, J.H., 2007. Mutually orthogonal latin squares. In: Colbourn, C.J., Dinitz, J.H. (Eds.), Handbook of Combinatorial Designs, second ed. Chapman & Hall/CRC, pp. 160–193. Adams, P., Bryant, D., Khodkar, A., 1996. On the existence of super-simple designs with block size 4. Aequationes Math. 52, 230–246. Bluskov, I., 1997. New designs. J. Combin. Math. Combin. Comput. 23, 212–220. ¨ al ¨ ainen, ¨ Bluskov, I., Ham H., 1998. New upper bounds on the minimum size of covering designs. J. Combin. Des. 6, 21–41. Chen, K., 1995. On the existence of super-simple (v,4,3)-BIBDs. J. Combin. Math. Combin. Comput. 17, 149–159. Chen, K., 1996. On the existence of super-simple (v,4,4)-BIBDs. J. Statist. Plann. Inference 51, 339–350. Cao, H., Chen, K., Wei, R., 2009. Super-simple balanced incomplete block designs with block size 4 and index 5. Discrete Math. 309, 2808–2814. Chen, K., Cao, Z., Wei, R., 2005. Super-simple balanced incomplete block designs with block size 4 and index 6. J. Statist. Plann. Inference 133, 537–554. Chen, K., Wei, R., 2007. Super-simple (v,5,4) designs. Discrete Appl. Math. 155, 904–913. Chen, K., Wei, R., 2006. Super-simple (v,5,5) designs. Des. Codes Cryptogr. 39, 173–187. Gronau, H.-D.O.F., 2007. Super-simple designs. In: Colbourn, C.J., Dinitz, J.H. (Eds.), Handbook of Combinatorial Designs, second ed. Chapman & Hall/ CRC, pp. 633–635. Gronau, H.-D.O.F., Mullin, R.C., 1992. On super-simple 2-(v,4,l) designs. J. Combin. Math. Combin. Comput. 11, 113–121. Gronau, H.-D.O.F., Kreher, D.L., Ling, A.C.H., 2004. Super-simple (v,5,2)-designs. Discrete Appl. Math. 138, 65–77. Hartmann, S., 2000. On simple and super-simple transversal. J. Combin. Des. 8, 311–320. Khodkar, A., 1994. Various super-simple designs with block size four. Australas. J. Combin. 9, 201–210. Kim, H.K., Lebedev, V., 2004. Cover-free families, superimposed codes and key distribution patterns. J. Combin. Des. 12, 79–91. Stinson, D.R., Wei, R., Zhu, L., 2000. New constructions for perfect hash families and related structures using related combinatorial designs and codes. J. Combin. Des. 8, 189–200.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi
Super-simple group divisible designs with block size 4 and index 2 H. Cao a,,1, F. Yan a, R. Wei b,2 a b
School of Mathematics Science, Nanjing Normal University, Nanjing 210046, China Department of Computer Science, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
a r t i c l e in fo
abstract
Article history: Received 27 April 2008 Received in revised form 22 June 2009 Accepted 24 February 2010 Available online 2 March 2010
Super-simple group divisible designs are useful for constructing other types of supersimple designs which can be applied to codes and designs. In this article, we investigate the existence of a super-simple (4,2)-GDD of type gu and show that such a design exists if and only if u 4, gðu2Þ 4, gðu1Þ 0 ðmod 3Þ and uðu1Þg 2 0 ðmod 6Þ. & 2010 Elsevier B.V. All rights reserved.
Keywords: Group divisible design Pairwise balanced design Super-simple
1. Introduction Let K be a set of positive integers and l a positive integer. A group divisible design ðK,lÞGDD is a triple ðX ,G,BÞ which satisfies the following properties: 1. X is a finite set of points. 2. G is a partition of X into subsets called groups. 3. B is a collection of subsets of X with sizes from K, called blocks, such that every pair of points from distinct groups occurs in exactly l blocks. 4. No pair of points belonging to a group occurs in any block. The type of a GDD ðX ,G,BÞ is the multiset fjGj : G 2 Gg. We usually use the ‘‘exponential’’ notation to describe types: g1u1 gkuk denotes ui occurrences of gi, 1 ri rk, in the multiset. If K ={k}, we write ðk,lÞGDD instead of ðfkg,lÞGDD. When l ¼ 1, it is omitted. Thus a k-GDD is a(k,1)-GDD. A transversal design TD ðk,l; nÞ is a ðk,lÞGDD of type nk. When l ¼ 1, we simply write TD(k,n). A ðK,lÞGDD with group type 1v is called a pairwise balanced design, denoted by ðv,K,lÞPBD. A ðk,lÞGDD with group type 1v is called a balanced incomplete block design, denoted by ðv,k,lÞBIBD. T A design is called simple if it contains no repeated blocks. A design ðX ,BÞ is said to be super-simple if jB1 B2 j r 2 for any two blocks B1 ,B2 2 B and B1 aB2 . When jBj ¼ 3 for any B 2 B, a super-simple design is just a simple design. When l ¼ 1, the designs are necessarily super-simple. The concept of super-simple designs was introduced by Gronau and Mullin (1992). The existence of super-simple designs is an interesting extremal problem by itself, but there are also some useful applications. For example, such super-simple designs Corresponding author.
E-mail addresses: [email protected] (H. Cao), [email protected] (R. Wei). Research supported by the National Natural Science Foundation of China under Grant no. 10971101. Research supported by NSERC discovery Grant 239135-06.
1 2
0378-3758/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2010.02.020
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are used in perfect hash families (Stinson et al., 2000) and covering Bluskov and H¨am¨al¨ainen (1998), in the construction of new designs (Bluskov, 1997) and in the construction of superimposed codes (Kim and Lebedev, 2004). There are many research papers for constructions of super-simple BIBDs and TDs. Therefore, the existence of super-simple GDDs, the more general designs, is an interesting design problem. On the other hand, super-simple group divisible designs are useful for the construction of other types of super-simple designs. The main purpose of this paper is to start the research on super-simple GDDs. The first interesting case is for GDDs with block size 4 and index 2, because a super-simple GDD of block size 3 is just a simple GDD. It is easy to see that the followings are the necessary conditions for the existence of a super-simple ðk,lÞGDD of type gu. Theorem 1.1. If there exists a super-simple ðk,lÞGDD of type gu, then u Z k, l r gðu2Þ=ðk2Þ, lðu1Þg 0 ðmod k1Þ and
luðu1Þg 2 0 ðmod kðk1ÞÞ. There are known results for the existence of super-simple designs, especially for super-simple ðv,k,lÞBIBDs. When k=4 or 5 the necessary conditions for super-simple ðv,k,lÞBIBDs are known to be sufficient for 2 r l r 6 or 2 r l r5 with few possible exceptions. We summarize these known results in the following theorems. Theorem 1.2 (Gronau and Mullin, 1992; Khodkar, 1994; Chen, 1995; Adams et al., 1996; Chen, 1996; Gronau, 2007; Cao et al., 2009; Chen et al., 2005). There exists a super-simple ðv,4,lÞBIBD with 2 r l r6 if and only if 1. 2. 3. 4. 5.
l ¼ 2, l ¼ 3, l ¼ 4, l ¼ 5, l ¼ 6,
v 1 ðmod 3Þ and v Z 7; v 0,1 ðmod 4Þ and v Z 8; v 1 ðmod 3Þ and v Z 10; v 1,4 ðmod 12Þ and v Z13; v Z14.
Theorem 1.3 (Gronau et al., 2004; Chen and Wei, 2007, 2006; Gronau, 2007; Abel and Bennett, 2008; Abel et al., 2008). A super-simple ðv,5,lÞBIBD exists with 2 r l r 5 if and only if 1. 2. 3. 4.
l ¼ 2, l ¼ 3, l ¼ 4, l ¼ 5,
v v v v
1,5 ðmod 10Þ, va5,15; 1 ðmod 20Þ and v Z21, and except possibly when v 5 ðmod 20Þ and v Z 25; 0,1 ðmod 5Þ and v Z 15; 1 ðmod 4Þ and v Z 17, except possibly when v =21.
In this paper, we investigate the existence of a super-simple (4,2)-GDD of type gu. We shall prove that there exists a super-simple (v,{4,5},2)-PBD for any v Z7 with four exceptions in Section 2. Then we shall apply this result to show that the necessary conditions for the existence of a super-simple (4,2)-GDD of type gu are also sufficient. The main results of this paper are summarized as follows. Theorem 1.4. A super-simple (v,{4,5},2)-PBD exists if and only if v Z 7 except when v 2 f8,9,12,14g. Theorem 1.5. A super-simple (4,2)-GDD of type gu exists if and only if u Z 4, gðu2Þ Z 4, gðu1Þ 0 ðmod 3Þ and uðu1Þg 2 0 ðmod 6Þ. 2. Proof of Theorem 1.4 In this section, we shall prove Theorem 1.4. First, we shall use direct constructions to obtain super-simple (v,{4,5}, 2)-PBD for some small values of v. These designs have been obtained after computer-assisted searches. The way to check the supersimplicity is essentially the same as what was used in Bluskov (1997). Suppose that a design is obtained by developing m base blocks modulo v. Let S ¼ fb1 ,b2 ,b3 g, b1 o b2 o b3 , be a 3-subset contained in a base block. Instead of developing S modulo v we form the following three representatives of the orbit corresponding to S: fb1 bi ,b2 bi ,b3 bi g,
i ¼ 1,2,3:
It is mentioned in Bluskov (1997) that if these 3-subsets are pairwise distinct, then the design is super-simple. Lemma 2.1. There exists a super-simple (15,{4,5}, 2)-PBD. Proof. Let the point set be Z15. Below are the required base blocks. The required design is obtained by developing the base blocks modulo 15. Here, the first base block {0,3,6,9,12} has a short orbit of order 3. f0,3,6,9,12g
f0,1,4,5g
f0,2,7,9g:
&
Lemma 2.2. There exists a super-simple (v,{4,5}, 2)-PBD for any v 2 M, where M = {17,23,27, 33,39,47}.
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Proof. For each v 2 M, let the point set be Zv. Instead of listing all the required blocks, we only list the base blocks and all the required blocks can be generated from them by ( + 1 mod v). v= v= v= v= v= v=
17: 23: 27: 33: 39: 47:
{0,1,2,5,8} {0,1,2,5,11} {0,1,2,5,11} {0,1,2,4,11} {0,1,2,4,9} {0,1,3,9,27} {0,8,18,34}
{0,2,7,11} {0,2,6,16} {0,2,7,15,19} {0,3,8,15,21} {0,3,13,19,28} {0,1,4,6} {0,9,20,34}
{0,3,8,15} {0,3,14,21} {0,4,14,20} {0,4,17,25} {0,4,11,23}
{0,5,14,22} {0,5,17,24} {0,5,17,33}
{0,6,16,27} {0,7,17,32}
& Lemma 2.3. There exists a super-simple (v,{4,5}, 2)-PBD for any v 2 M ¼ f18,26,32,36,38,42,48,56,66g. Proof. For each v 2 M, let the point set be Zv. All the required blocks can be generated from the following base blocks by ( +2 mod v). v = 18: v = 26: v = 32: v = 36:
v = 38:
v = 42:
v = 48:
v = 56:
v = 66:
{0,1,2,4,8} {0,1,2,3,6} {0,8,16,25} {0,1,3,9,27} {0,6,17,18} {0,1,3,9,27} {0,1,13,17} {0,4,12,29} {0,1,3,9,27} {0,10,21,31} {0,19,23,33} {0,1,3,9,27} {0,6,14,26} {0,2,23,28} {0,1,3,9,27} {0,6,13,18} {0,2,21,35} {0,1,2,3,6} {0,12,25,28} {0,4,35,43} {1,7,25,35} {0,1,3,9,27} {0,4,43,65} {0,16,33,47} {0,2,40,47} {0,15,23,59}
{0,1,3,9} {0,2,5,9,15} {1,5,13,19} {0,1,2,4,8} {0,7,19,23} {0,6,15,19} {0,2,18,29}
{0,3,6,13} {0,4,14,19}
{0,5,7,11} {0,6,17,19}
{0,5,9,10} {0,7,12,23}
{0,3,7,18} {0,9,19,21} {0,7,21,22} {0,2,25,35}
{0,5,12,22} {0,11,16,29} {0,5,7,10} {0,3,8,23}
{0,5,15,24} {0,5,11,19} {0,4,10,24}
{0,1,4,12,36} {0,10,25,26}
{0,2,7,8} {0,11,18,33}
{0,4,9,18} {0,13,17,25}
{0,7,13,22} {0,17,19,35}
{0,11,21,33} {0,7,18,33} {0,4,19,22} {0,15,25,37} {0,8,16,34} {0,2,25,26} {0,2,5,7,11} {0,14,29,34} {0,6,32,45} {1,13,27,39} {0,20,45,55} {0,6,13,63} {0,18,41,51} {0,9,14,36}
{0,3,4,10} {0,8,25,35} {0,11,12,25} {0,4,19,35} {0,11,20,36} {0,3,10,39} {0,20,43,47} {0,14,32,47} {0,8,25,45}
{0,5,7,10} {0,1,23,29}
{0,5,9,13} {0,2,19,31}
{0,5,7,10} {0,11,28,42} {0,17,37,47} {0,7,21,41} {0,18,41,49} {0,9,30,46}
{0,5,9,13} {0,1,4,33} {0,21,27,41} {0,10,21,22} {0,19,29,48} {0,17,33,49}
{0,4,29,41} {0,5,43,58} {0,19,31,49} {0,11,27,42}
{0,3,29,49} {0,11,17,21} {0,1,33,65} {0,12,26,42}
{0,6,38,59} {0,5,8,18} {0,2,22,56} {0,15,19,57}
& For our recursive constructions, we shall use the following standard recursive constructions, the proofs of which can be found in Chen (1996). S Construction 2.4 (Weighting). Let ðX ,G,BÞ be a super-simple GDD with index l1 , and let o : X-Z þ f0g be a weighting + function on X, where Z is the set of positive integers. Suppose that for each block B 2 B, there exists a super-simple P ðk,l2 ÞGDD of type foðxÞ : x 2 Bg. Then there exists a super-simple ðk,l1 l2 ÞGDD of type f x2Gi oðxÞ : Gi 2 Gg. Construction 2.5 (Breaking up groups). If there exists a super-simple ðK,lÞGDD of type hu11 hut t and a super-simple P ðhi þ Z,K,lÞPBD for each i, 1 r ir t, then there exists a super-simple ð ti ¼ 1 hi ui þ Z,K,lÞPBD, where Z ¼ 0 or 1. We also need the following known results about super-simple ðv,f4,5g,lÞGDDs. Lemma 2.6 (Abel et al., 2008). Necessary conditions for a super-simple (5,2)-GDD of type hn to exist are: (1) n Z5, (2) nðn1Þh2 0 ðmod 5Þ, and (3) if h is odd, then n is odd. These conditions are also sufficient, except for (h,n)= (1, 5), (1,15), (2,5) and possibly for h 2 f13,17,19g, n = 15. Lemma 2.7 (Hartmann, 2000). A super-simple TD ð4,l; vÞ exists if and only if l rv and ðl,vÞ is neither (1,2) nor (1,6). It is obvious that we can obtain a super-simple ðv,fk1,kg,lÞPBD by deleting one point from the point set of a supersimple ðv,k,lÞBIBD. So, we have the following lemma. Lemma 2.8. If there exists a super-simple ðv,k,lÞBIBD, then there exists a super-simple ðv1,fk1,kg,lÞ- PBD. More generally, we can obtain a super-simple ðvs,fk1,kg,lÞGDD by deleting s points from the last group of a supersimple ðv,k,lÞGDD.
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Lemma 2.9. If there exists a super-simple ðv,k,lÞGDD with type h1 h2 ht , then there exists a super-simple ðvs,fk1,kg,lÞGDD with type h1 h2 ht1 ðht sÞ. Lemma 2.10. There exists a super-simple (v,{4,5},2)-PBD for any v 0,4 ðmod 10Þ and va4,14. Proof. By Theorem 1.3, there exists a super-simple ðv,5,2ÞBIBD for any v 1,5 ðmod 10Þ and va5,15. Applying Lemma 2.8 with k= 5 and l ¼ 2, we can get a super-simple (v,{4,5},2)-PBD for any v 0,4 ðmod 10Þ and va4,14. & Lemma 2.11. There exists a super-simple (v,{4,5},2)-PBD for any v 2 M ¼ f29,53,59,78, 83,102,107,119,132,135,138, 143,156,162g [ f12 t þ d : t 2 ½5,13\f6,9,11g,d ¼ 2,3g. Proof. For each v 2 M, let v =4g+ x. The two parameters g and x are listed in the following table. We start from a supersimple (5,2)-GDD of type g5 which exists by Lemma 2.6. Removing g x points from the last group of the super-simple (5,2)-GDD of type g5, we get a super-simple ({4,5},2)-GDD of type g4 x1 by Lemma 2.9. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (4g + x,{4,5},2)-PBD. Here, all these input super-simple (g,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.1–2.3, 2.10. v= 4g + x
g
x
v =4g+ x
g
x
v= 4g + x
g
x
29 62 83 98 107 123 138 147 159
7 13 17 20 23 25 28 31 33
1 10 15 18 15 23 26 23 27
53 63 86 99 119 132 143 156 162
13 13 19 20 25 28 31 33 33
1 11 10 19 19 20 19 24 30
59 78 87 102 122 135 146 158
13 17 19 21 25 28 31 33
7 10 11 18 22 23 22 26
& Lemma 2.12. There exists a super-simple (v,{4,5},2)-PBD for any v 2 f72,77,89,96,108, 113,137,149g [ f12 t þ d : t 2 f5, 7,9,10,12g,d ¼ 8,9g. Proof. By Lemma 2.7, there exists a super-simple (4,2)-GDD of type g4. Starting form this GDD and applying Construction 2.5, we obtain a super-simple ð4g þ Z,f4,5g,2ÞPBD, where Z ¼ 0 or 1. Here, the input designs ðg þ Z,f4,5g,2ÞPBDs come from Theorems 1.2, 1.3 and 2.2, 2.3, 2.10, 2.11. We list all the parameters in the following table. v ¼ 4g þ Z
g
Z
v ¼ 4g þ Z
g
Z
v ¼ 4g þ Z
g
Z
68 77 93 113 128 149
17 19 23 28 32 37
0 1 1 1 0 1
69 89 96 116 129 152
17 22 24 29 32 38
1 1 0 0 1 0
72 92 108 117 137 153
18 23 27 29 34 38
0 0 0 1 1 1
& Lemma 2.13. There exists a super-simple (126,{4,5},2)-PBD. Proof. Start from a super-simple (5,2)-GDD of type 255 which exists by Lemma 2.6. Applying Construction 2.5 with Z ¼ 1, we get a super-simple (126, {4,5},2)-PBD. Here, the input super-simple ( 26,{4,5},2)-PBD comes from Lemma 2.3. & Now, we shall prove Theorem 1.4. By Theorem 1.2, there exists a super-simple (v,{4,5}, 2)-PBD for any v 1 ðmod 3Þ and v Z7. Then we only need to prove that there exists a super-simple (v,{4,5}, 2)-PBD for any v 0,2 ðmod 3Þ and v Z8. For convenience, 0,2 we use [a,b]1,2 3 or [a,b]3 to denote a set of positive integers v such that a rv r b and v 1,2 ðmod 3Þ or v 0,2 ðmod 3Þ. Lemma 2.14. There exists a super-simple (v,{4,5},2)-PBD for any v Z 11, va12,14 and v= 48t + m, where m 2 ½0,470,2 3 , t Z0. 0,2 0,2 Proof. We distinguish m into three cases: m 2 ½0,180,2 3 , m 2 ½20,413 and m 2 ½42,473 .
(1) m 2 ½0,180,2 3 : For t r3, the existence of a super-simple (v,{4,5},2)-PBD has been solved by Theorem 1.3 and Lemmas 2.1–2.3, 2.10–2.12. For t Z 4, we start with a super-simple (5,2)-GDD of type (12t 5)5. Removing 12t 5 x, 4 1 x 2 ½20,381,2 3 , points from the last group, we get a super-simple ({4,5},2)-GDD of type (12t 5) x by Lemma 2.9. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (48t + m,{4,5},2)-PBD. Here, all these input super-simple (12t 5,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.2, 2.3, 2.10.
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(2) m 2 ½20,410,2 3 : For t r 2, there exists a super-simple (v,{4,5},2)-PBD by Theorem 1.3 and Lemmas 2.2, 2.3, 2.10–2.13. For t Z 3, we start from a super-simple (5,2)-GDD of type (12t+ 1)5. Remove 12t + 1 x, x 2 ½16,371,2 3 , points from the last group to get a super-simple ({4,5},2)-GDD of type (12t + 1)4 x1. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (v,{4,5},2)-PBD. Here, all these input super-simple (12t +1,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.2, 2.3, 2.10, and 2.11. (3) m 2 ½42,470,2 3 : For t r2, there exists a super-simple (v,{4,5},2)-PBD by Theorem 1.3 and Lemmas 2.2, 2.3, 2.10–2.12. For t Z 3, we start with a super-simple (5,2)-GDD of type (12t + 4)5. Removing 12t +4 x, x 2 ½26,311,2 3 , points from the last group, we get a super-simple ({4,5},4)-GDD of type (12t + 4)4 x1. Applying Construction 2.5 with Z ¼ 0, we get a super-simple (v,{4,5},2)-PBD. Here, all these input super-simple (12t +4,{4,5},2)-PBD and (x,{4,5},2)-PBD come from Theorems 1.2, 1.3 and Lemmas 2.3, 2.11. & By some simple computations and a computer exhaustive search, there does not exist a super-simple (v,{4,5},2)-PBD for any v 2 f8,9,12,14g. So, combining Theorem 1.2 and Lemma 2.14, we have proved Theorem 1.4. 3. Proof of Theorem 1.5 It is easy to see that the necessary conditions can be distinguished into two cases, namely, g 0 ðmod 3Þ, u Z4 and g 1,2 ðmod 3Þ, u 1 ðmod 3Þ, gðu2Þ Z 4. Firstly, we settle sufficiency when g 0 ðmod 3Þ and u Z 4. We start with some direct constructions. Lemma 3.1. There exists a super-simple (4,2)-GDD of type 3u for any u 2 f5,6,8,9,12,14g. Proof. For each u 2 f5,6,8,9,12,14g, let the point set be Z3u and the group set be ffi,u þi,2u þ ig : 0 ri r u1g. Below are the required base blocks. u = 5: u = 6: u = 8: u = 9: u = 12: u = 14:
{0,1,3,7} {0,1,2,3} {0,1,2,3} {0,9,11,21} {0,1,2,4} {0,1,2,3} {0,9,17,22} {0,1,2,24} {0,8,16,25} {0,23,27,35}
{0,1,9,13} {0,2,5,10} {0,2,5,9} {0,13,17,23} {0,3,10,15} {0,2,5,6} {0,11,19,22} {0,1,20,37} {0,10,31,39}
{0,4,9,13} {0,4,14,19}
{0,4,11,15} {0,4,15,21}
{0,4,11,17} {0,4,9,11} {0,13,16,31} {0,2,5,6} {0,11,15,33}
{0,5,11,19} {0,6,23,27} {0,13,19,29} {0,3,12,24} {0,13,29,31}
{0,7,15,17} {0,6,12,19}
{0,7,21,27} {0,15,25,29} {0,4,9,11} {0,15,27,37}
+1 mod 15 +2 mod 18 +2 mod 24 +1 mod 27 {0,8,16,26} +2 mod 36 {0,6,13,16} {0,19,25,35} +2 mod 42
& Lemma 3.2. There exists a super-simple (4,2)-GDD of type 6u for any u 2 f6,11,14,15,18,23g. Proof. Let the point set be Z6u and the group set be ffju þi : 0 r j r 5g : 0 ri r u1g. All the required blocks can be generated from the following base blocks by ( +1 mod 6u). u = 6: u = 11: u = 14:
u = 15:
u = 18:
u = 23:
{0,1,2,4} {0,5,15,35} {0,17,41,63} {0,13,36,57} {0,5,31,43} {0,19,48,70} {0,14,28,61} {0,4,41,50} {0,22,60,85} {0,14,28,43} {0,1,4,42} {0,31,68,104} {0,14,28,43} {0,29,62,97} {0,3,12,60}
{0,3,7,22} {0,9,25,43} {0,2,34,59} {0,15,35,66}
{0,5,13,28} 13i{0,1,3,7} {0,5,11,17} {0,1,4,66}
{0,5,14,25} 13i {0,2,15,32} {0,7,15,23} {0,1,34,37}
{0,7,16,26} ð0 r ir 3Þ {0,9,19,29} {0,2,32,39}
{0,11,24,40} {0,4,30,39}
{0,3,35,58} {0,16,50,72} {0,5,12,59} {0,3,47,77} {0,16,33,53} {0,2,52,59} {0,2,61,64} {0,16,32,50} {0,30,75,105} {0,5,49,91}
{0,5,11,17} {0,19,63,88}
{0,7,16,24} {0,1,4,42}
{0,8,18,28} {0,1,27,52}
{0,11,24,37} {0,2,23,59}
{0,5,11,17} {0,19,40,65} {0,4,39,67} {0,7,61,97} {0,19,38,58} {0,31,66,98} {0,6,17,73}
{0,7,15,23} {0,21,55,82} {0,5,32,44} {0,7,15,24} {0,21,42,64} {0,1,40,52} {0,6,44,93}
{0,9,19,29} {0,22,46,78} {0,9,40,75} {0,8,18,28} {0,22,47,72} {0,1,5,83}
{0,11,24,37} {0,1,58,60} {0,11,24,37} {0,26,53,80} {0,2,55,59}
& We also need the following known results which can be found in Abel et al. (2007). Lemma 3.3 (Abel et al., 2007). 1. A 4-GDD of type mu exists if and only if u Z 4, ðu1Þm 0 ðmod 3Þ and uðu1Þm2 0 ðmod 12Þ except ðm,uÞ 2 fð2,4Þ,ð6,4Þg. 2. A(v,{4,5,6,8},1)-PBD exists for all v Z8 and va9,10,11,12,14,15,18,19,23. 3. A (v,{4,7},1)-PBD exists for all v 1 ðmod 3Þ and va10,19. Lemma 3.4. There exists a super-simple(4,2)-GDD of type 6u for any u 2 f5,8,9,12g.
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Proof. We start from a (4,1)-GDD of type 3u which exists by Lemma 3.3. Applying Construction 2.4 with weight 2, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDD of type 24 comes from Lemma 2.7. & Lemma 3.5. There exists a super-simple (4,2)-GDD of type 6u for any u 2 f7,10,19g. Proof. We start from a (4,1)-GDD of type 2u which exists by Lemma 3.3. Applying Construction 2.4 with weight 3, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDD of type 34 comes from Lemma 2.7. & Lemma 3.6. There exists a super-simple (4,2)-GDD of type 6u for any u Z4. Proof. If u 2 ½4,12 [ f14,15,18,19,23g, then a super-simple (4,2)-GDD of type 6u exists by Lemma 2.7 and Lemmas 3.2–3.5. For other values u, start from a ðu,f4,5,6,8g,1ÞPBD which exists by Lemma 3.3. Applying Construction 2.4 with weight 6, we obtain a super-simple (4,2)-GDD of type 6u, where the input super-simple (4,2)-GDDs of type 6m ,m 2 f4,5,6,8g come from Lemmas 2.7, 3.2 and 3.4. & Lemma 3.7. There exists a super-simple (4,2)-GDD of type 18u for any u 2 f6,14g. Proof. Start form a super-simple (4,2)-GDD of type 6u which exists by Lemma 3.2. Applying Construction 2.4 with a (4,1)GDD of type 34 form Lemma 3.3, we can obtain a super-simple (4,2)-GDD of type 18u. & Lemma 3.8. There exists a super-simple(4,2)-GDD of type gu for any g 0 ðmod 3Þ and u 2 f6,14g. Proof. Let g= 3m, m Z 1. If m 2 f1,2,6g, then a super-simple (4,2)-GDD of type gu exists by Lemmas 3.1, 3.2 and 3.7. If ma1,2,6, we start from a super-simple (4,2)-GDD of type 3u which exists by Lemma 3.1. Applying Construction 2.4 with weight m, we obtain a super-simple (4,2)-GDD of type (3m)u, where the input (4,1)-GDD of type m4 comes from Lemma 3.3. & Lemma 3.9. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ and u 2 f5,8,9,12g. Proof. Let g = 3m, m Z1. If m =1, then there exists a super-simple (4,2)-GDD by Lemma 3.1. Suppose m 4 1. Start from a (4,1)-GDD of type 3u which exists by Lemma 3.3. Applying Construction 2.4 with weight m, we obtain a super-simple (4,2)-GDD of type (3m)u , where the input super-simple (4,2)-GDD of type m4 comes from Lemma 2.7. & Lemma 3.10. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ, ga6 and u Z7, u= 2f8,9,12,14g. Proof. By Theorem 1.4, there exists a super-simple ðu,f4,5g,2ÞPBD. Applying Construction 2.4 with weight g, we obtain a super-simple (4,2)-GDD of type gu, where the input designs of (4,1)-GDD of type g4 and the (4,1)-GDD of type g5 come from Lemma 3.3. & Combining Lemmas 2.7, 3.6 and Lemmas 3.8–3.10, we have the following theorem. Theorem 3.11. There exists a super-simple (4,2)-GDD of type gu for any g 0 ðmod 3Þ and u Z 4. Now, we study the case when g 1,2 ðmod 3Þ and u 1 ðmod 3Þ, gðu2Þ Z 4. First, we shall use direct constructions to obtain some super-simple (4,2)-GDDs. Lemma 3.12. There exists a super-simple (4,2)-GDD of type 2u for any u 2 f7,10,19g. Proof. Let the point set be Z2u and the group set be ffi,u þig : 0 r ir u1g. All the required blocks can be generated from the following base blocks by ( +1 mod 2u). u= 7: u= 10: u= 19:
{0,1,4,6} {0,1,2,6} 7i {0,1,2,4}
{0,1,9,11} {0,2,9,16} 7i {0,3,9,29}
{0,3,8,11} ð0 r i r 2Þ
& Lemma 3.13. There exists a super-simple (4,2)-GDD of type 2u for any u 1 ðmod 3Þ. Proof. There exists a super-simple (4,2)-GDD of type 2u for any u 2 f10,19g by Lemma 3.12. Now, let u 1 ðmod 3Þ and u= 2f10,19g. By Lemma 3.3, there exists a (u,{4,7},1)- PBD. Applying Construction 2.4 with weight 2, we obtain a super-simple (4,2)-GDD of type 2u, where the input super-simple (4,2)-GDDs of type 24 and 27 come from Lemmas 2.7 and 3.12. & Lemma 3.14. There exists a super-simple (4,2)-GDD of type gu for any g 1,2 ðmod 3Þ, ga2 and u 1 ðmod 3Þ, u Z7. Proof. We start from a super-simple (u,4,2)-BIBD, u 1 ðmod 3Þ and u Z 7, which exists by Theorem 1.2. Applying Construction 2.4 with weight g, g 1,2 ðmod 3Þ and ga2, we obtain a super-simple (4,2)-GDD of type gu, where the input design of a (4,1)-GDD of type g4 comes from Lemma 3.3. &
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Combining Lemmas 2.7, 3.13 and 3.14, we have the following theorem. Theorem 3.15. There exists a super-simple (4,2)-GDD of type gu for any g 1,2 ðmod 3Þ, u 1 ðmod 3Þ and gðu2Þ Z 4. Combining Theorems 3.11 and 3.15, we have proved Theorem 1.5.
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