Incomplete Group Divisible Designs with Block Size Four J. Wang Department of Mathematics, Suzhou University, Suzhou 215006, P. R. China Received October 2, 2002; revised March 24, 2003
Abstract: The object of this paper is the construction of incomplete group divisible designs (IGDDs) with block size four, group-type ðg; hÞu and index unity. It is shown that the necessary conditions for the existence of such an IGDD are also sufficient with three exceptions and six possible exceptions. # 2003 Wiley Periodicals, Inc. J Combin Designs 11: 442–455, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10055
Keywords: GDD; IGDD; HGDD
1.
INTRODUCTION
A group divisible design (GDD) of index unity is a triple (X; G; B) where X is a finite set (of points), G is a partition of X into subsets (called groups) and B is a family of subsets (called blocks) of X such that every pair of distinct points of X occurs in exactly one block or one group, but not both. The group-type (or type) of the GDD is the multiset fjGj : G 2 Gg which is usually denoted by an ‘‘exponential’’ notation: a group-type 1i 2j 3k . . . means i occurrences of 1; j occurrences of 2, etc. The notation K-GDD stands for a GDD having block-sizes from a set of positive integers K. Roughly speaking, an incomplete group divisible design (IGDD) is a GDD from which a sub-GDD is missing. In fact, the missing sub-GDD need not exist. More formally, an IGDD of index unity is a quadruple ðX; Y; G; BÞ where X is a set (of points), Y is a subset (called hole) of X; G is a partition of X into subsets (called groups), and B is a collection of subsets (called blocks) of X such that (1) each block intersects each group in at most one point, (2) no pair of distinct points of Y occurs in any block, and (3) every pair of points fx; yg from distinct groups with the exception of those in which both x and y lie in Y occurs in exactly one block of B. Correspondence to: J. Wang, E-mail:
[email protected] # 2003 Wiley Periodicals, Inc.
442
IGDDs WITH BLOCK SIZE FOUR
443
The multiset fðjGj; jG \ YjÞ : G 2 Gg is defined to be the type of the IGDD. As with GDDs, we use an ‘‘exponential’’ notation for its description. We say that an IGDD is a K-IGDD if jBj 2 K for every block B 2 B. When K ¼ fkg, we simply write k for K. Note that if Y ¼ ;, then the IGDD is a GDD. IGDDs play an important role in combinatorial design theory, being of interest in their own right, as well as having many applications in the construction of other type of designs (see, for example, [13,14,16]). Here we are interested in the existence of k-IGDDs of type ðg; hÞu . It is well known that the existence of a k-IGDD of type ðg; hÞk is equivalent to the existence of k 2 incomplete mutually orthogonal Latin squares (IMOLS) of type (g; h) (see [4,11]). So IGDDs can be thought of as a natural generalization of IMOLS. By simple calculation, we can obtain the following result. Theorem 1.1 [12]. The necessary conditions for the existence of a k-IGDD of type ðg; hÞu are that u k; g ðk 1Þh; gðu 1Þ 0 ðmod k 1Þ; ðg hÞðu 1Þ 0 ðmod k 1Þ, and uðu 1Þðg2 h2 Þ 0 ðmod kðk 1ÞÞ. In [12] it is proved that the above necessary conditions are also sufficient when k ¼ 3. However, these conditions are not sufficient when k ¼ 4. For example, there does not exist a 4-IGDD of type ð6; 1Þ4 because there do not exist two mutually orthogonal Latin squares (MOLS) of order 6. The main purpose of this paper is to investigate the existence spectrum for 4-IGDDs of type ðg; hÞu . By Theorem 1.1, we have the following lemma. Lemma 1.1. Let g; h and u be nonnegative integers. The necessary conditions for the existence of a 4-IGDD of type ðg; hÞu are that u 4; g 3h; gðu 1Þ 0 ðmod 3Þ, ðg hÞðu 1Þ 0 ðmod 3Þ and uðu 1Þðg2 h2 Þ 0 ðmod 12Þ. Now we must deal with the real issue: sufficiency. Some work has been done for the existence of 4-IGDDs of type ðg; hÞu . In [6,17] (or see [7]), the following result is proved. Theorem 1.2. There exists a 4-IGDD of type ðg; hÞu where u 4; u 1 ðmod 3Þ, u 62 f10; 19g, and ðg; hÞ 6¼ ð6; 1Þ if and only if ð1Þ g 3h for u 1 or 4 ðmod 12Þ, or ð2Þ g 3h and g h is even for u 7 or 10 ðmod 12Þ. When g ¼ ðk 1Þh, a k-IGDD of type ðg; hÞu is equivalent to a ðk 1Þ-frame of type ðg hÞu (see [14]). Since the existence of a 3-frame of type gu was determined in [15], we have the following theorem. Theorem 1.3. If g ¼ 3h, then a 4-IGDD of type ðg; hÞu exists if and only if u 4 and hðu 1Þ 0 ðmod 3Þ. The spectra for 4-GDDs of type gu , which can be considered as 4-IGDDs of type ðg; 0Þu , has been determined in [5]. Theorem 1.4. A 4-GDD of type gu (or a 4-IGDD of type ðg; 0Þu ) exists if and only if gðu 1Þ 0 ðmod 3Þ and g2 uðu 1Þ 0 ðmod 12Þ except for ðu; gÞ ¼ ð4; 2Þ or ð4; 6Þ.
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In what follows, we employ both direct and recursive methods of construction to handle the remaining cases. We will show that the conditions in Lemma 1.1 are also sufficient for the existence of a 4-IGDD of type ðg; hÞu , except for ðu; g; hÞ 2 fð4; 2; 0Þ; ð4; 6; 0Þ; ð4; 6; 1Þg and except possibly for ðu; g; hÞ 2 fð14; 15; 3Þ; ð14; 21; 3Þ; ð14; 93; 27Þ; ð18; 15; 3Þ; ð18; 21; 3Þ; ð18; 93; 27Þg. Since the cases g ¼ 3h and h ¼ 0 were handled in Theorems 1.3 and 1.4, we can restrict ourselves to the case 0 < 3h < g throughout the paper. 2.
RECURSIVE CONSTRUCTIONS
Before describing our recursive constructions, we need to introduce some terminologies and notations. We use [3] as our standard design theory reference. A K-GDD of type 1v is referred to as a pairwise balanced design (PBD) of index unity, denoted by BðK; 1; vÞ. A transversal design, TD(k; n), is defined as a k-GDD of type nk . It is well known that a TD(k; n) is equivalent to k 2 MOLS of order n. Construction 2.1 to Construction 2.6 are commonly used techniques (see [6,12,13]). Construction 2.1. If there exist a K-GDD of type mt11 mtnn and a k-IGDD of type ðg; hÞu for every u 2 K, then there exists a k-IGDD of type ðgm1 ; hm1 Þt1 ðgmn ; hmn Þtn . Construction 2.2. If there exist a BðK; 1; vÞ and a k-IGDD of type ðg; hÞu for every u 2 K, then there exists a k-IGDD of type ðg; hÞv . Construction 2.3. Suppose that there exist a k-IGDD of type ðg; hÞu and a TDðk; mÞ. Then there exists a k-IGDD of type ðgm; hmÞu . Construction 2.4. Suppose that there exist a TDðu þ 1; qÞ and a k-IGDD of type ðm þ ti ; ti Þu for 1 i q. Then there exists a k-IGDD of type ðmq þ r; rÞu , where r ¼ t1 þ þ tq . Construction 2.5. Suppose that there exist k-IGDDs of type ðg; mÞu and ðm; hÞu . Then there exists a k-IGDD of type ðg; hÞu . Construction 2.6. ð1Þ Suppose that there exist k-IGDDs of type ðtg; thÞu and ðg; hÞtþ1 . Then there exists a k-IGDD of type ðg; hÞtuþ1 . ð2Þ Suppose that there exist k-IGDDs of type ðg; hÞu ðgt; htÞ1 and ðg; hÞt . Then there exists a k-IGDD of type ðg; hÞuþt . Construction 2.7. Suppose that there exist a TDðu þ s; q þ 1Þ, a k-IGDD of type ðm þ ti ; ti Þu for 1 i s; and k-GDDs of type mu and mu1 w1 . Then there exists a k-IGDD of type ðmq þ w þ r; w þ rÞu , where r ¼ t1 þ þ ts . Proof. Suppose (X; G; B) is a TDðu þ s; q þ 1Þ, where G ¼ fG1 ; G2 ; . . . ; Guþs g. Take a block S B0 ¼ fx1 ; x2 ; . . . ; xuþs g from B such that xi 2 Gi for 1 i u þ s, and let Y ¼ uþ1iuþs ðGinfxi gÞ. Let fLx : x 2 Xg be pairwise disjoint sets with jLx j ¼ lx , where lx ¼ m if x 2 Ginfxi g for 1 i u; lxS¼ 0 if x 2 Y; lxi ¼ w for 1 i u, and lxi ¼ ti for u þ 1 i u þ s. Write LZS¼ x2Z Lx forSZ X. Now we construct the required IGDD on point set X ¼ ð 1iu LGi Þ [ ð uþ1iuþs ðLxi Iu ÞÞ, where Iu ¼ f1; 2; . . . ; ug.
IGDDs WITH BLOCK SIZE FOUR
445
Let B0 ¼ BnfB0 g. For each B 2 B0 , construct a k-GDD of type mu 0s or mu1 w1 0s with the group set fLx : x 2 Bg if B \ B0 ¼ ; or B \ B0 ¼ fxi g ð1 i uÞ, and a k-IGDD of type ðm þ ti ; ti Þu with group set fLx [ ðLxi fjgÞ : x 2 B \ Gj ; 1 j ug if S B \ B0 ¼ fxi g ðu þ 1 i u þSsÞ. Write AB for its block set. Let Y ¼ G ¼ fLGj [ ð uþ1iuþs ðLxi fjgÞÞ : 1 j ug, and B ¼ Suþ1iuþs ðLxi Iu Þ; & B2B0 AB . Then ðX ; Y ; G ; B Þ is the desired IGDD. We also require the notion of a holey group divisible design (HGDD). A K-HGDD is a quadruple (X; H; G; B) which satisfies the following properties: (1) X is a uðg1 þ þ gt Þ-set (of points), P (2) G ¼ fG1 ; . . . ; Gu g is a partition of X into u groups of ti¼1 gi points each, (3) H ¼ fH1 ; . . . ; Ht g is another partition of X into t holes; each hole Hi ð1 i t) is a set of ugi points such that jHi \Gj j ¼ gi for 1 j u, (4) B is a collection of subsets (called blocks) of X such that jBj 2 K for every B 2 B, where K is a set of positive integers, (5) no block contains two distinct points of any group or any hole, but any other pairset of X occurs in exactly one block of B. If H contains ti holes of size ugi ; 1 i r, we then use an exponential notation gt11 gtrr to denote the multiset T ¼ fgj : j ¼ 1; 2; . . . ; tg and call (u; T) the type of the HGDD. The following construction is essentially the same as Theorem 2.8 in [12]. Construction 2.8. Suppose that there exist a k-IGDD of type ðm þ r; rÞt , a kHGDD of type ðu; 1k Þ and a k-IGDD of type ðm þ r þ h; r þ hÞu . Then there exists a k-IGDD of type ðtm þ tr þ h; tr þ hÞu . The technique of ‘‘filling in holes’’ is simple but useful. Construction 2.9. Suppose that there exist a K-HGDD of type ðu; gt11 gtrr Þ and a K-IGDD of type ðgi þ h; hÞu for 1 i r. Then there exist K-IGDDs of type ðg þ h; hÞu and ðg þ h; gi þ hÞu for 1 i r, where g ¼ g1 t1 þ þ gr tr .
3.
PRELIMINARIES
In this section we give some preliminary results which will be used in the subsequent sections. Lemma 3.1 [5]. There exists a 4-GDD of type 2u 51 for every u 0 (mod 3) with u 9. Lemma 3.2 [7]. There exists a 4-GDD of type 3u m1 if and only if either u 0 ðmod 4Þ and m 0 ðmod 3Þ; 0 m ð3u 6Þ=2; or u 1 ðmod 4Þ and m 0 ðmod 6Þ; 0 m ð3u 3Þ=2; or u 3 ðmod 4Þ and m 3 ðmod 6Þ; 0 m ð3u 3Þ=2. Lemma 3.3 [9]. There exists a 4-GDD of type 6t m1 for every t 4 and m 0 (mod 3) with 0 m 3t 3 except for ðt; mÞ ¼ ð4; 0Þ and except possibly for ðt; mÞ 2 fð7; 15Þ; ð11; 21Þ; ð11; 24Þ; ð11; 27Þ; ð13; 27Þ; ð13; 33Þ; ð17; 39Þ; ð17; 42Þ; ð19; 45Þ; ð19; 48Þ; ð19; 51Þ; ð23; 60Þ; ð23; 63Þg.
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Lemma 3.4 [8]. (1) Let g 2 ðmod 6Þ and u 0 ðmod 3Þ, u 6. Then there exist 4-GDDs of type gu 21 and gu ðgðu 1Þ=2Þ1 , except that there is no 4-GDD of type 26 51 . (2) Let g 4 ðmod 6Þ and u 0 ðmod 3Þ, u 6. Then there exist 4-GDDs of type gu 11 and gu ðgðu 1Þ=2Þ1 , except possibly for type g21 11 when g 10 ðmod 12Þ. Lemma 3.5 [7]. Let BðKÞ ¼ fv :a BðK; 1; vÞ existsg. Then v 2 Bðf4; 5; 6; 9gÞ when v 2; 3; 6 or 11 (mod 12); v 6 and v 62 f11; 14; 15; 18; 23g. Lemma 3.6 [1, 7]. k q þ 1.
(1) Let q be a prime power. Then there exists a TDðk; qÞ for 3
(2) There exists a TDð7; tÞ for every integer t 7 except possibly for t 2 f10; 14; 15; 18; 20; 22; 26; 30; 34; 38; 46; 60; 62g. Lemma 3.7. Let v be a positive integer and v 62 f2x1 3x2 5x3 7x4 : 0 x1 3; 0 x2 2; 0 x3 ; x4 1gnf210; 315; 2520g. Then there exists a divisor p of v such that a TDð11; pÞ exists. Proof. For every v 2 f210; 315; 2520g, there exists a TDð11; vÞ (see [7]), then take p ¼ v to get the result. For the other stated values of v, there must exist a prime power p satisfying pjv and p 11. By Lemma 3.6 (1), there exists a TDð11; pÞ. & Lemma 3.8 [10]. There exists a 4-HGDD of type ðu; gt Þ if and only if u; t 4 and ðu 1Þðt 1Þg 0 ðmod 3Þ except for ðu; g; tÞ 6¼ ð4; 1; 6Þ and except possibly for g ¼ 3 and ðu; tÞ 2 fð6; 14Þ; ð6; 15Þ; ð6; 18Þ; ð6; 23Þg. In order to obtain the required 4-IGDDs, we need to construct more HGDDs. We will make use of the following constructions for HGDDs (see [10,18]). Construction 3.1. If there exist a K-GDD of type mt11 mt22 mtrr and a k-HGDD of type ðu; gn Þ for each n 2 K, then there exists a k-HGDD of type ðu; ðgm1 Þt1 ðgm2 Þt2 ðgmr Þtr Þ. Construction 3.2. If there exist a K-HGDD of type ðu; mt11 mt22 mtrr Þ and a k-GDD of type gn for each n 2 K, then there exists a k-HGDD of type ðu; ðgm1 Þt1 ðgm2 Þt2 ðgmr Þtr Þ. Lemma 3.9. Let v 2 f20; 60g, e be odd satisfying 1 e v 1. Then there exists a 4-HGDD of type ð6; 6v ð3e 3Þ1 Þ. 1 Proof. Apply Construction 3.1 with a 4-GDD of type 3v ð3e3 2 Þ from Lemma 3.2 4 and a 4-HGDD of type ð6; 2 Þ from Lemma 3.8 to get a 4-HGDD of type ð6; 6v ð3e 3Þ1 Þ. &
Lemma 3.10. There exists a 4-HGDD of type ð10; g7 w1 Þ for ðg; wÞ ¼ ð1; 3Þ; ð2; 5Þ, or ð4; 10Þ. Proof. For each stated pair (g; w), we take the point set X ¼ Z10 ðZ7g [ f1i : i 2 Zw gÞ, the group set G ¼ ffjg ðZ7g [ f1i : i 2 Zw gÞ : j 2 Z10 g and the hole set H ¼ fZ10 f0 þ i; 7 þ i; . . . ; 7ðg 1Þ þ ig : i 2 Z7 g [ fZ10 f1i : i 2 Zw gg. The required base blocks listed below are developed over Z10 Z7g, where the notation ‘‘þd mod g’’ denotes that all elements of the base blocks should be taken cyclically
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IGDDs WITH BLOCK SIZE FOUR
by adding d (mod g) to them, while the infinite point, if it occurs in the base block, is always fixed. ðg; wÞ ¼(1, 3): developed by ðþ2 mod 10; þ1 mod 7Þ ((1,10 ),(0,0),(4,2),(9,3)), ((1,11 ),(0,5),(3,1),(7,6)), ((1,12 ),(0,3),(2,2),(5,6)), ((2,10 ),(0,2),(6,1),(7,0)), ((2,11 ),(0,3),(3,1),(7,5)), ((2,12 ),(0,5),(4,1),(6,6)),
((1,10 ),(2,1),(5,2),(6,5)), ((1,11 ),(2,2),(4,6),(6,0)), ((1,12 ),(3,2),(4,4),(6,6)), ((2,10 ),(1,1),(3,6),(8,2)), ((2,11 ),(1,1),(8,6),(9,0)), ((2,12 ),(1,2),(7,1),(9,0)),
((1,10 ),(3,2),(7,5),(8,4)), ((1,11 ),(5,1),(8,4),(9,0)), ((1,12 ),(7,2),(8,3),(9,5)), ((2,10 ),(4,1),(5,5),(9,0)), ((2,11 ),(4,2),(5,0),(6,5)), ((2,12 ),(3,2),(5,6),(8,0)).
ðg; wÞ ¼(2, 5): developed by ðþ2 mod 10; þ1 mod 14Þ ((1,10 ),(0,3),(6,1),(8,5)), ((1,11 ),(0,2),(6,1),(9,6)), ((1,12 ),(0,0),(2,1),(3,4)), ((1,13 ),(0,0),(6,1),(8,4)), ((1,14 ),(0,7),(5,1),(6,3)), ((2,10 ),(0,10),(7,1),(9,9)), ((2,11 ),(0,6),(4,1),(8,12)), ((2,12 ),(0,13),(3,1),(6,5)), ((2,13 ),(0,5),(5,1),(6,9)), ((2,14 ),(0,3),(3,1),(6,12)), ((1,1),(2,12),(7,2),(0,3)),
((1,10 ),(2,1),(4,7),(5,12)), ((1,11 ),(2,1),(3,11),(4,3)), ((1,12 ),(4,1),(5,5),(7,11)), ((1,13 ),(2,1),(5,4),(7,3)), ((1,14 ),(2,1),(4,6),(9,7)), ((2,10 ),(1,1),(4,7),(6,4)), ((2,11 ),(1,1),(5,10),(9,6)), ((2,12 ),(1,1),(4,0),(5,9)), ((2,13 ),(1,1),(4,11),(7,3)), ((2,14 ),(1,1),(4,3),(8,6)), ((1,1),(2,13),(3,5),(8,7)),
((1,10 ),(3,1),(7,2),(9,4)), ((1,11 ),(5,1),(7,4),(8,13)), ((1,12 ),(6,1),(8,0),(9,2)), ((1,13 ),(3,1),(4,5),(9,10)), ((1,14 ),(3,1),(7,4),(8,9)), ((2,10 ),(3,1),(5,13),(8,2)), ((2,11 ),(3,1),(6,6),(7,5)), ((2,12 ),(7,1),(8,4),(9,12)), ((2,13 ),(3,1),(8,12),(9,13)), ((2,14 ),(5,1),(7,6),(9,7)), ((1,1),(2,0),(3,11),(8,2)).
ðg; wÞ ¼ (4, 10): developed by ðþ1 mod 10; þ1 mod 28Þ ((0,10 ),(1,1),(5,19),(7,25)), ((0,11 ),(1,1),(4,26),(7,21)), ((0,12 ),(1,1),(2,27),(6,25)), ((0,13 ),(1,1),(2,2),(4,3)), ((0,14 ),(1,1),(7,5),(8,13)), ((0,15 ),(1,1),(2,20),(4,18)), ((0,16 ),(1,1),(4,4),(9,27)), ((0,17 ),(1,1),(2,19),(5,3)), ((0,18 ),(1,1),(2,25),(5,21)), ((0,19 ),(1,1),(3,20),(8,9)), ((1,1),(2,21),(4,9),(6,20)),
((0,10 ),(2,1),(3,16),(4,0)), ((0,11 ),(2,1),(6,6),(8,18)), ((0,12 ),(3,1),(7,24),(8,19)), ((0,13 ),(5,1),(8,27),(9,2)), ((0,14 ),(2,1),(4,18),(5,12)), ((0,15 ),(3,1),(5,5),(6,14)), ((0,16 ),(2,1),(6,4),(7,9)), ((0,17 ),(3,1),(4,18),(6,5)), ((0,18 ),(3,1),(7,17),(8,14)), ((0,19 ),(2,1),(5,10),(6,23)), ((1,1),(3,6),(5,26),(7,16)),
((0,10 ),(6,1),(8,24),(9,2)), ((0,11 ),(3,1),(5,25),(9,14)), ((0,12 ),(4,1),(5,17),(9,23)), ((0,13 ),(3,1),(6,0),(7,10)), ((0,14 ),(3,1),(6,11),(9,17)), ((0,15 ),(7,1),(8,3),(9,14)), ((0,16 ),(3,1),(5,23),(8,0)), ((0,17 ),(7,1),(8,0),(9,4)), ((0,18 ),(4,1),(6,26),(9,17)), ((0,19 ),(4,1),(7,19),(9,27)), ((1,1),(3,10),(5,20),(8,7)). &
Lemma 3.11. There exists a 4-HGDD of type ð10; gt w1 Þ for ðg; t; wÞ 2 fð2; 9; 5Þ; ð2; 21; 5Þ; ð4; 4; 2Þ; ð4; 4; 5Þ; ð4; 7; 12Þ; ð8; 4; 4Þ; ð10; 7; 25Þ; ð10; 7; 30Þ, ð20; 4; 10Þg [fð6; 7; wÞ : w ¼ 3; 6; 9; 12g. Proof. The result for ðg; t; wÞ 2 fð2; 9; 5Þ; ð2; 21; 5Þg [ fð6; 7; wÞ : w ¼ 3; 6; 9; 12g can be obtained by applying Construction 3.1 with a 4-GDD of type gt w1 from Lemmas 3.1 and 3.3 and a 4-HGDD of type ð10; 14 Þ. Delete two points from one group of a TD(5, 4) to get a f4; 5g-GDD of type 44 21 . Delete four points from one block of a TD(5, 5) to get a f4; 5g-GDD of type 44 51 . Delete four points from one group of a TD(5, 8) to get a f4; 5g-GDD of type 84 41 . Since there exists a 4-IGDD of type ð10; mr Þ for m 2 f1; 5g and r 2 f4; 5g by
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Lemma 3.8, we can apply Construction 3.1 to each of the above f4; 5g-GDDs to handle the case ðg; t; wÞ 2 fð4; 4; 2Þ; ð20; 4; 10Þ; ð4; 4; 5Þ; ð8; 4; 4Þg. For ðg; t; wÞ 2 fð4; 7; 12Þ; ð10; 7; 30Þg, give weight 4 or 10 to each point of a 4HGDD of type ð10; 17 31 Þ from Lemma 3.10. Then apply Construction 3.2, using a 4GDD of type 44 or 104, to yield the required HGDD. For ðg; t; wÞ ¼ ð10; 7; 25Þ, apply Construction 3.2 giving weight 5 to each point of a 4-HGDD of type ð10; 27 51 Þ from Lemma 3.10. This completes the proof. & Lemma 3.12. (1) Let u 2 f11; 14; 18g; v 4, e be odd satisfying 0 < e < v. There exists a 4-HGDD of type ðu; 6v ð3e 3Þ1 Þ except for ðv; eÞ 2 fð11; 9Þ; ð17; 15Þ; ð19; 17Þ; ð23; 21Þg. (2) There exists a 4-HGDD of type ðu; 617 451 Þ for u 2 f11; 14; 18g. Proof. Each of the required HGDDs can be obtained by applying Construction 3.1 with a suitable 4-GDD of type 6t m1 and a 4-HGDD of type ðu; 14 Þ. All auxiliary designs exist by Lemmas 3.3 and 3.8. & 4.
EXISTENCE OF 4-IGDDs OF TYPE ðg; hÞ6
By Lemma 1.1, there exists a 4-IGDD of type ðg; hÞ6 only if g h 0 (mod 3), g h is even and 0 3h g. Let g h ¼ 6v and h ¼ 3e, where 0 e v. Since the cases g ¼ 3h and h ¼ 0 have been handled in Theorems 1.3 and 1.4, respectively, we need only consider the case 0 < e < v. Lemma 4.1. There exist 4-IGDDs of type ð15; 3Þ6 and ð21; 3Þ6 . Proof. For ðg; hÞ ¼ ð15; 3Þ or ð21; 3Þ, let n ¼ ðg hÞ=3. We take the point set X ¼ Z6 Z3 ðZn [ f1gÞ, the group set G ¼ ffjg Z3 ðZn [ f1gÞ : j 2 Z6 g and the hole Y ¼ fZ6 Z3 f1gg. The required base blocks listed below are developed over Z6 Z3 Zn. ðg; hÞ ¼(15, 3): developed by ðþ2 mod 6; þ1 mod 3; þ1 mod 4Þ ((1,1,1), (4,0,1), (5,2,3), (0,1,1)), ((1,1,1), (3,1,1), (5,0,3), (0,2,3)), ((1,1,1), (2,0,1), (3,0,0), (5,1,0)), ((2,1,1), (1,1,1), (3,2,2), (5,1,3)), ((2,1,1), (3,0,1), (4,2,2), (0,0,3)), ((1,1,1), (2,1,1), (5,2,1), (0,0,3)), ((1,1,1), (2,0,0), (3,0,0), (0,0,0)), ((1,1,1), (2,2,0), (4,1,2), (5,0,2)).
((1,1,1), (2,2,1), (3,2,3), (4,2,0)), ((1,1,1), (2,1,1), (4,1,2), (0,0,2)), ((2,1,1), (1,2,1), (4,0,1), (0,2,0)), ((2,1,1), (3,1,1), (4,1,0), (5,2,3)), ((2,1,1), (1,0,1), (5,0,2), (0,1,2)), ((1,1,1), (2,2,2), (4,2,0), (5,1,1)), ((1,1,1), (2,1,3), (4,0,0), (5,1,0)),
ðg; hÞ ¼(21, 3): developed by ðþ2 mod 6; þ1 mod 3; þ1 mod 6Þ ((1,1,1), (2,1,1), (3,0,0), (0,0,2)), ((1,1,1), (2,0,1), (4,0,4), (5,1,3)), ((1,1,1), (3,2,1), (5,2,4), (0,1,0)), ((2,1,1), (1,0,1), (5,0,1), (0,0,2)), ((2,1,1), (3,2,1), (4,1,4), (5,1,5)), ((1,1,1), (2,1,1), (3,0,1), (4,1,0)), ((1,1,1), (2,1,4), (3,2,2), (4,2,1)),
((1,1,1), (2,2,1), (4,1,3), (5,0,5)), ((1,1,1), (3,1,1), (4,2,5), (0,2,5)), ((2,1,1), (1,1,1), (3,1,3), (0,1,4)), ((2,1,1), (1,2,1), (4,0,4), (5,2,2)), ((2,1,1), (3,0,1), (4,2,5), (0,2,1)), ((1,1,1), (2,1,3), (3,2,3), (5,0,1)), ((1,1,1), (2,2,1), (3,0,2), (0,1,1)),
449
IGDDs WITH BLOCK SIZE FOUR
((1,1,1), (2,2,2), (3,2,0), (5,2,4)), ((1,1,1), (2,2,4), (4,1,1), (0,0,5)), ((1,1,1), (2,0,2), (4,2,2), (0,2,0)),
((1,1,1), (2,2,3), (3,0,0), (4,0,4)), ((1,1,1), (2,0,1), (3,2,4), (5,1,0)), ((1,1,1), (2,0,0), (4,1,4), (0,1,5)).
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Lemma 4.2. Let v 2 f2; 3; 4; 5; 20; 60g and 1 e v 1. Then there exists a 4IGDD of type ð6v þ 3e; 3eÞ6 . Proof. When e is even, take a 4-GDD of type 66 from Theorem 1.4 and a 4-IGDD of type ðv þ e=2; e=2Þ4 from Theorem 1.2, then apply Construction 2.1 to obtain the desired IGDD. This leaves the case ðv; eÞ ¼ ð5; 2Þ to be handled. We can fill five holes of a 4-HGDD of type ð6; 66 Þ from Lemma 3.8 with 4-GDDs of type 66 to get the result. When e is odd, the result for v 2 f2; 3g was provided in Lemma 4.1. For v 2 f4; 5g, the result follows from applying Construction 2.9 with a 4-IGDD of type ð9; 3Þ6 and a 4-HGDD of type ð6; 6v Þ or ð6; 6vþ1 Þ from Lemma 3.8. For v 2 f20; 60g, take a 4-HGDD of type ð6; 6v ð3e 3Þ1 Þ from Lemma 3.11, then fill v holes of the HGDD using 4-IGDDs of type ð9; 3Þ6 to get the desired result. & Lemma 4.3. Let v 2 f6; 10; 15; 30g and 1 e v 1. Then there exists a 4-IGDD of type ð6v þ 3e; 3eÞ6 . Proof. Let 3e ¼ t1 þ t2 þ þ tv4 þ w, where ti 2 f0; 3g for 1 i v 4 and w 2 f3; 12g. Since there exist a TD(v þ 2; v þ 1) by Lemma 3.6 (1), a 4-IGDD of type ð6 þ ti ; ti Þ6 for 1 i v 4, and 4-GDDs of type 66 and 65 w1 from Lemma 3.3, the conclusion follows from Construction 2.7. & Lemma 4.4. Let v 2 f14; 18; 22; 26; 34; 38; 46; 62g and 1 e v 1. Then there exists a 4-IGDD of type ð6v þ 3e; 3eÞ6 . Proof. There exists a TD(7, v=2) by Lemma 3.6 (2). Let 3e ¼ t1 þ t2 þ þ tv=2 , where ti 2 f0; 3; 6g for 1 i v=2. Since there exists a 4-IGDD of type ð12 þ ti ; ti Þ6 for any ti, the result is obtained by applying Construction 2.4. & Lemma 4.5. Let v 7; v 62 f10; 14; 15; 18; 20; 22; 26; 30; 34; 38; 46; 60; 62g and 1 e v 1. Then there exists a 4-IGDD of type ð6v þ 3e; 3eÞ6 . Proof. Choose suitable ti 2 f0; 3g such that t1 þ þ tv ¼ 3e. Since there exist a TD(7, v) by Lemma 3.6 (2) and a 4-IGDD of type ð6 þ ti ; ti Þ6 for 1 i v, we can apply Construction 2.4 to get the result. & Combining the above results, we get the following theorem. Theorem 4.1. Let g h 0 (mod 3), g h be even and 0 < 3h < g. Then there exists a 4-IGDD of type ðg; hÞ6 .
5.
EXISTENCE OF 4-IGDDs OF TYPE ðg; hÞ10
By Lemma 1.1, there exists a 4-IGDD of type ðg; hÞ10 only if 0 3h g and g h is even. We need only consider 0 < 3h < g. Let g h ¼ 2v, where 0 < h < v.
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Lemma 5.1. Let h be even and 2 h v 1. Then there exists a 4-IGDD of type ð2v þ h; hÞ10 . Proof. For all stated values of v and h but ðv; hÞ ¼ ð5; 2Þ, take a 4-GDD of type 210 from Theorem 1.4 and a 4-IGDD of type ðv þ h=2; h=2Þ4 from Theorem 1.2, then apply Construction 2.1 to obtain the required IGDD. For the remaining case, fill five holes of a 4-HGDD of type ð10; 26 Þ from Lemma 3.8 using 4-GDDs of type 210 . &
Lemma 5.2. There exist 4-IGDDs of type ð2v þ 1; 1Þ10 for v > 1 and ð2v þ 3; 3Þ10 for v > 3. Proof. The result for v 4 follows from applying Construction 2.9 with a 4-HGDD of type ð10; 2n Þ, n 2 fv; v þ 1g, and a 4-IGDD of type ð3; 1Þ10 . For v ¼ 2 or 3, we directly construct a 4-IGDD of type ð2v þ 1; 1Þ10 . Take the point set X ¼ Z10 ðZ2v [ f1gÞ, the group set G ¼ ffjg ðZ2v [ f1gÞ : j 2 Z10 g and the hole Y ¼ fZ10 f1gg. The required base blocks listed below are developed over Z10 Z2v . v ¼ 2: developed by ðþ2 mod 10; þ1 mod 4Þ ((1,1), (0,0), (5,1), (9,3)), ((2,1), (0,1), (7,1), (9,3)), ((0,0), (1,1), (2,1), (8,1)),
((1,1), (2,1), (3,1), (7,0)), ((2,1), (1,1), (3,0), (6,3)), ((1,1), (2,0), (5,1), (8,3)),
((1,1), (4,1), (6,1), (8,3)), ((2,1), (4,1), (5,3), (8,0)), ((1,1), (3,1), (5,2), (0,2)).
v ¼ 3: developed by ðþ2 mod 10; þ1 mod 6Þ ((1,1), (0,3), (5,1), (8,0)), ((2,1), (0,0), (5,1), (6,0)), ((1,1), (4,5), (6,1), (7,0)), ((1,1), (2,3), (4,1), (0,3)),
((1,1), (2,1), (4,0), (7,3)), ((2,1), (1,1), (3,0), (7,2)), ((1,1), (2,2), (3,4), (8,1)), ((1,1), (2,4), (3,5), (9,1)),
((1,1), (3,1), (6,3), (9,1)), ((2,1), (4,1), (8,0), (9,0)), ((1,1), (2,1), (4,2), (8,5)), ((1,1), (3,3), (9,0), (0,4)). &
Lemma 5.3. Let v 2 A ¼ f6; 7; 8; 9; 14; 21; 35g and h be odd satisfying 5 h v 1. Then there exists a 4-IGDD of type ð2v þ h; hÞ10 . Proof. When v ¼ 6, apply Construction 2.9 with a 4-HGDD of type ð10; 44 Þ and a 4-IGDD of type ð5; 1Þ10 from Lemma 5.2 to create a 4-IGDD of type ð17; 5Þ10 . When v ¼ 7, take a 4-HGDD of type ð10; 44 21 Þ from Lemma 3.11, then apply Construction 2.9 to obtain a 4-IGDD of type ð19; 5Þ10 , using 4-IGDDs of type ð3; 1Þ10 and ð5; 1Þ10 . When v ¼ 8, the result for h ¼ 5 follows from applying Construction 2.9 with a 4HGDD of ð10; 45 Þ and a 4-IGDD of type ð5; 1Þ10 . For h ¼ 7, take a 4-HGDD of type ð10; 44 51 Þ from Lemma 3.11, then fill four holes of the HGDD using 4-IGDDs of type ð6; 2Þ10 . When v ¼ 9, we have a 4-HGDD of type ð10; 29 51 Þ from Lemma 3.11. Filling nine holes of the HGDD with 4-GDDs of type 210 creates a 4-IGDD of type ð23; 5Þ10 . The result for h ¼ 7 follows from applying Construction 2.9 with a 4-HGDD of type ð10; 64 Þ and a 4-IGDD of type ð7; 1Þ10 . When v ¼ 14, the conclusion for h 2 f5; 9; 11; 13g holds by Construction 2.9, since there exist 4-IGDDs of type ð5; 1Þ10 and ð9; 1Þ10 , 4-HGDDs of type ð10; 48 Þ,
IGDDs WITH BLOCK SIZE FOUR
451
ð10; 84 41 Þ, ð10; 47 101 Þ and ð10; 47 121 Þ from Lemmas 3.8, 3.10, and 3.11. For h ¼ 7, apply Construction 2.3 with a 4-IGDD of type ð5; 1Þ10 and a TD(4, 7). When v ¼ 21, the case for 5 h 15 is handled by applying Construction 2.9 with a 4-HGDD of type ð10; 67 w1 Þ from Lemma 3.11 and a 4-IGDD of type ð6 þ r; rÞ10 , where w 2 f3, 6, 9, 12g, 0 r 3 and w þ r ¼ h. Apply Construction 2.9 again with a 4-HGDD of type ð10; 144 Þ and a 4-IGDD of type ð14 þ t; tÞ10 for t 2 f3; 5g to handle the case h 2 f17, 19g. When v ¼ 35, apply Construction 2.9 with a 4-HGDD of type ð10; 145 Þ and a 4-IGDD of type ð14 þ h; hÞ10 to handle the case h 2 f5; 7g; with a 4-HGDD of type ð10; 108 Þ and a 4-IGDD of type ð10 þ t; tÞ10 for t 2 f1; 3; 5g to handle the case h 2 f11; 13; 15g; with a 4-HGDD of type ð10; 146 Þ and a 4-IGDD of type ð14 þ t; tÞ10 for t 2 f3; 5; 7g to handle the case h 2 f17; 19; 21g; with a 4-HGDD of type ð10; 204 101 Þ from Lemma 3.11 and 4-IGDDs of type ð13; 3Þ10 and ð23; 3Þ10 to handle the case h ¼ 23; with a 4-HGDD of type ð10; 107 251 Þ and a 4-IGDD of type ð10 þ t; tÞ10 for t 2 f0; 2; 4g to handle the case h 2 f25; 27; 29g; with a 4-HGDD of type ð10; 107 301 Þ and a 4-IGDD of type ð10 þ t; tÞ10 for t 2 f1; 3g to handle the case h 2 f31; 33g. The result for the remaining case h ¼ 9 follows from applying Construction 2.8 with a 4-HGDD of type ð10; 14 Þ, 4-IGDDs of type ð13; 3Þ10 and ð11; 1Þ7 from Theorem 1.2. & Lemma 5.4. Let v 2 B ¼ f10; 12; 15; 18; 24; 28; 30; 36; 40; 42; 60; 63; 70; 72; 120; 126; 180; 280; 360; 420; 630; 840g and h be odd satisfying 5 h v 1. Then there exists a 4-IGDD of type ð2v þ h; hÞ10 . Proof. Let h ¼ t1 þ t2 þ þ tv8 þ w, where ti 2 f0; 1g for 1 i v 8 and w 2 f5; 8g. Since there exist a TD(v þ 2; v þ 1) by Lemma 3.6 (1), a 4-IGDD of type ð2 þ ti ; ti Þ10 for 1 i v 8, and 4-GDDs of type 210 and 29 w1 from Lemmas 3.1 and 3.4, the conclusion holds by Construction 2.7. & Lemma 5.5. Let v 2 C ¼ f20; 45; 56; 84; 90; 105; 140; 168; 252; 504; 1260g and h be odd satisfying 5 h v 1. Then there exists a 4-IGDD of type ð2v þ h; hÞ10 . Proof. When v 2 f20; 56; 84; 140; 252; 1260g, the desired result, except for v ¼ 20 and 9 h 15, is obtained by applying Construction 2.7 to a TD(v=2 þ 2; v=2 þ 1) with 4-GDDs of type 410 and 49 w1 ; w 2 f4; 16g (see Lemma 3.4 (2)), and 4IGDDs of type ð4 þ ti ; ti Þ10 , where 0 ti 2 for i ¼ 1; . . . ; v=2 8 and t1 þ þ tv=28 þ w ¼ h. For the remaining cases, apply Construction 2.9 with a 4-HGDD of type ð10; 86 Þ or ð10; 105 Þ and a 4-IGDD of type ð9; 1Þ10 or ð10 þ n; nÞ10, where n 2 f1; 3; 5g. When v ¼ 45, apply Construction 2.7 to a TD(17, 16) with 4-GDDs of type 610 and 69 w1 ; w 2 f3; 24g (see Lemma 3.3), and 4-IGDDs of type ð6 þ ti ; ti Þ10 , where 0 ti 3 for i ¼ 1; . . . ; 7 and t1 þ þ t7 þ w ¼ h. When v ¼ 90, apply Construction 2.7 to a TD(20, 19) with 4-GDDs of type 1010 and 109 w1 ; w 2 f1; 40g (see Lemma 3.4 (2)), and 4-IGDDs of type ð10 þ ti ; ti Þ10 , where 0 ti 5 for i ¼ 1; . . . ; 10 and t1 þ þ t10 þ w ¼ h. When v ¼ 105, apply Construction 2.7 to a TD(17, 16) with 4-GDDs of type 1410 and 149 w1 ; w 2 f2; 14; 56g (see Lemma 3.4 (1)), and 4-IGDDs of type ð14 þ ti ; ti Þ10 , where 0 ti 7 for i ¼ 1; . . . ; 7 and t1 þ þ t7 þ w ¼ h.
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When v 2 f168; 504g, the desired result is obtained by applying Construction 2.7 to a TD(v=4 þ 2; v=4 þ 1) with 4-GDDs of type 810 and 89 w1 ; w 2 f2; 32g (see Lemma 3.4 (1)), and 4-IGDDs of type ð8 þ ti ; ti Þ10 , where 0 ti 4 for i ¼ 1; . . . ; v=4 8 and t1 þ þ tv=48 þ w ¼ h. & Lemma 5.6. Let v > 5; v 62 D ¼ A [ B [ C and h be odd satisfying 5 h v 1, where A; B, and C are given in the statement of Lemmas 5:3, 5:4, and 5.5, respectively. Then there exists a 4-IGDD of type ð2v þ h; hÞ10 . Proof. Let set T ¼ fv : v > 5; v 62 D and there does not exist a 4-IGDD of type ð2v þ h; hÞ10 for some odd h satisfying 5 h v 1g. It is obvious that the conclusion holds if and only if T ¼ ;. Suppose T 6¼ ; and v 0 is the smallest value in T. Since v 0 62 D, we can write v 0 ¼ pq such that there exists a TD(11, p) by Lemma 3.7. Since 0 < q < v 0 , we have a 4-IGDD of type ð2q þ t; tÞ10 for 0 t q. Now apply Construction 2.4 to a TD(11, p) with 4-IGDDs of type ð2q þ ti ; ti Þ10 for i ¼ 1; . . . ; p, where t1 þ þ tp ¼ h, to obtain a 4-IGDD of type ð2v 0 þ h; hÞ10 . Thus v 0 62 T, a contradiction. & The foregoing can be summarized into the following theorem. Theorem 5.1. Let 0 < 3h < g and g h is even. Then there exists a 4-IGDD of type ðg; hÞ10 . 6.
EXISTENCE OF 4-IGDDs OF TYPE ðg; hÞu FOR u 2 f11; 14; 18g
By Lemma 1.2, there exists a 4-IGDD of type ðg; hÞu for u 2 f11; 14; 18g only if g h 0 (mod 3), g h is even and 0 3h g. Let g h ¼ 6v and h ¼ 3e, where 0 e v. We need only consider the case 0 < e < v. Lemma 6.1. Let u 2 f11; 14; 18g; g h 0 (mod 6) and 0 < 3h < g. Then there exists a 4-IGDD of type ðg; hÞu . Proof. A 4-IGDD of type ð36; 6Þu can be obtained by filling five holes of a 4-HGDD of type ðu; 66 Þ from Lemma 3.8 with 4-GDDs of type 6u . For the other cases, the result follows from applying Construction 2.1 with a 4-GDD of type 6u and a 4-IGDD of type ðg=6; h=6Þ4 . & Lemma 6.2. There exist 4-IGDDs of type ð15; 3Þ11 and ð21; 3Þ11 . Proof. For ðg; hÞ ¼(15, 3) or (21, 3), let n ¼ ðg hÞ=3. We take the point set X ¼ Z11 Z3 ðZn [ f1gÞ, the group set G ¼ ffjg Z3 ðZn [ f1gÞ : j 2 Z11 g and the hole Y ¼ fZ11 Z3 f1gg. The required base blocks listed below are developed over Z11 Z3 Zn. ðg; hÞ ¼(15, 3): developed by ðþ1 mod 11; þ1 mod 3; þ1 mod 4Þ ((1,1,1), ((1,1,1), ((1,1,1), ((1,1,1), ((1,1,1),
(2,0,1), (6,0,1), (4,1,1), (4,2,1), (2,1,1),
(3,0,0), (8,2,2), (9,2,1), (7,2,2), (3,1,2),
(10,0,2)), (9,1,3)), (10,1,1)), (0,0,2)), (8,0,2)),
((1,1,1), ((1,1,1), ((1,1,1), ((1,1,1), ((1,1,1),
(5,2,1), (2,2,1), (5,1,1), (5,0,1), (4,0,1),
(6,1,3), (3,2,3), (6,2,1), (7,0,1), (9,0,3),
(0,1,0)), (10,2,3)), (7,1,0)), (8,1,2)), (0,2,2)),
453
IGDDs WITH BLOCK SIZE FOUR
((1,1,1), (2,1,1), (5,0,2), (8,1,2)), ((1,1,1), (3,0,1), (5,2,3), (8,2,3)), ((1,1,1), (3,1,3), (5,2,2), (7,0,3)).
((1,1,1), (2,2,0), (4,0,0), (9,2,3)), ((1,1,1), (2,2,3), (4,2,0), (10,0,3)),
ðg; hÞ ¼ ð21; 3Þ: developed by ðþ1 mod 11; þ1 mod 3; þ1 mod 6Þ ((1,1,1), (8,0,1), (9,2,3), (10,1,3)), ((1,1,1), (2,0,1), (3,0,4), (0,2,3)), ((1,1,1), (4,2,1), (5,0,2), (7,2,1)), ((1,1,1), (2,2,1), (3,2,3), (10,0,2)), ((1,1,1), (6,0,1), (8,2,5), (9,1,3)), ((1,1,1), (3,1,1), (7,1,3), (9,1,4)), ((1,1,1), (3,1,4), (5,2,3), (9,1,2)), ((1,1,1), (2,2,3), (4,2,1), (8,1,3)), ((1,1,1), (2,1,5), (6,1,2), (8,0,2)), ((1,1,1), (2,1,2), (6,0,2), (10,2,0)),
((1,1,1), (4,1,1), (10,2,2), (0,1,1)), ((1,1,1), (5,2,1), (6,1,4), (7,1,3)), ((1,1,1), (2,1,1), (3,1,1), (9,0,3)), ((1,1,1), (0,0,0), (5,1,1), (6,2,1),), ((1,1,1), (4,0,1), (7,0,2), (8,1,1)), ((1,1,1), (2,2,5), (7,2,4), (10,0,1)), ((1,1,1), (3,1,3), (5,2,4), (10,1,2)), ((1,1,1), (2,2,4), (6,2,3), (9,1,3)), ((1,1,1), (2,0,2), (5,0,4), (10,2,4)), ((1,1,1), (3,2,4), (6,1,1), (9,0,2)).
&
Lemma 6.3. Let u 2 f11; 14; 18g; v 4, e be odd satisfying 0 < e < v, and ðu; v; eÞ 62 fð14; 11; 9Þ; ð18; 11; 9Þg. Then there exists a 4-IGDD of type ð6v þ 3e; 3eÞu . Proof. Take a 4-HGDD of type ðu; 6v ð3e 3Þ1 Þ from Lemma 3.12 (1), then apply Construction 2.9 with a 4-IGDD of type ð9; 3Þu from Theorem 1.3 to obtain a 4-IGDD of type ð6v þ 3e; 3eÞu . This handles all cases but ðu; v; eÞ ¼ ð11; 11; 9Þ, and u 2 f11; 14; 18g and ðv; eÞ 2 fð17; 15Þ; ð19; 17Þ; ð23; 21Þg. For u 2 f11; 14; 18g and ðv; eÞ ¼ ð17; 15Þ, there exists a 4-HGDD of type ðu; 617 451 Þ by Lemma 3.12 (2), then fill 17 holes using 4-GDDs of type 6u to obtain the desired result. For the remaining cases, the result follows from applying Construction 2.4 to a TD(u þ 1; v) with 4-IGDDs of type ð6 þ ti ; ti Þu for i ¼ 1; . . . ; v, where t1 þ þ tv ¼ 3e and ti 2 f0; 3g for 1 i v. & Combining Lemmas 6.1, 6.2, and 6.3 gives the following result. Theorem 6.1. Let u 2 f11; 14; 18g; g h 0 (mod 3), g h is even, 0 < 3h < g and ðu; g; hÞ 62 fð14; 15; 3Þ; ð14; 21; 3Þ; ð14; 93; 27Þ ð18; 15; 3Þ; ð18; 21; 3Þ; ð18; 93; 27Þg. Then there exists a 4-IGDD of type ðg; hÞu . 7.
MAIN RESULTS
In this section, we shall establish our main results concerning the existence of 4IGDDs of type ðg; hÞu . Lemma 7.1. Let u 1 or 4 (mod 12), u 4; 0 < 3h < g and ðu; g; hÞ 6¼ ð4; 6; 1Þ. Then there exists a 4-IGDD of type ðg; hÞu . Proof. By Theorem 1.2 the case when u > 4 and ðg; hÞ ¼ ð6; 1Þ remains to be handled. Since there exists a 4-HGDD of type ðu; 16 Þ by Lemma 3.8, we can fill five holes of the HGDD with 4-GDDs of type 1u to get the desired result. & Lemma 7.2. Let u 7 or 10 (mod 12), u 7; 0 < 3h < g and g h be even. Then there exists a 4-IGDD of type ðg; hÞu .
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WANG
Proof. By Theorems 1.2 and 5.1 we need only consider the case u ¼ 19. Take 4IGDDs of type ð3g; 3hÞ6 from Theorem 4.1 and ðg; hÞ4 from Lemma 7.1, then apply Construction 2.6 (1) to obtain the required IGDD. & Lemma 7.3. Let u 0; 5; 8 or 9 (mod 12), u 5, g h 0 (mod 3) and 0 < 3h < g. Then there exists a 4-IGDD of type ðg; hÞu . Proof. By filling five holes of a 4-HGDD of type ðu; 36 Þ from Lemma 3.8 with 4GDDs of type 3u , we have a 4-IGDD of type ð18; 3Þu . For the other cases, the result follows from applying Construction 2.1 with a 4-GDD of type 3u and a 4-IGDD of type ðg=3; h=3Þ4 . & Lemma 7.4. Let u 2; 3; 6 or 11 (mod 12), u 6; g h 0 (mod 3), g h is even, 0 < 3h < g and ðu; g; hÞ 62 fð14; 15; 3Þ; ð14; 21; 3Þ; ð14; 93; 27Þ; ð18; 15; 3Þ, ð18; 21; 3Þ; ð18; 93; 27Þg. Then there exists a 4-IGDD of type ðg; hÞu . Proof. The result for u 2 f6; 11; 14; 18g was given in Theorem 4.1 or 6.1. For u ¼ 15, take a 7-GDD of type 315 (see [2]) and a 4-IGDD of type ðg=3; h=3Þ7 from Lemma 7.2, then apply Construction 2.1 to create the desired IGDD. For u ¼ 23, since there exist a 4-GDD of type 317 181 from Lemma 3.2 and a 4-IGDD of type ðg=3; h=3Þ4 , we can apply Construction 2.1 to obtain a 4-IGDD of type ðg; hÞ17 ð6g; 6hÞ1 , then fill the long group of the IGDD with a 4-IGDD of type ðg; hÞ6 . For the other values of u, since u 2 Bðf4; 5; 6; 9gÞ by Lemma 3.5 and there exists a 4IGDD of type ðg; hÞn for n 2 f4; 5; 6; 9g, we can apply Construction 2.2 to obtain the result. & Combining the above results with Lemma 1.1, Theorems 1.3 and 1.4, we have proved the following theorem. Theorem 7.1. The necessary conditions for the existence of a 4-IGDD of type ðg; hÞu shown in Lemma 1.1 are also sufficient, except for ðu; g; hÞ 2 fð4; 2; 0Þ, ð4; 6; 0Þ; ð4; 6; 1Þg and except possibly for ðu; g; hÞ 2 fð14; 15; 3Þ; ð14; 21; 3Þ; ð14; 93; 27Þ, ð18; 15; 3Þ; ð18; 21; 3Þ; ð18; 93; 27Þg.
REFERENCES [1] R. J. R. Abel, Five MOLS of side 39 and 54, http://www.emba.uvm.edu/ dinitz/ newresults.part2.html [2] R. D. Baker, An elliptic semiplane, J Combin Theory, Ser A 25 (1978), 193–195. [3] T. Beth, D. Jungnickel, and H. Lenz, Design theory, Cambridge University Press, 1986. [4] A. E. Brouwer and G. H. J. Van Rees, More mutually orthogonal Latin squares, Discrete Math 39 (1982), 263–281. [5] A. E. Brouwer, A. Schrijver, and H. Hanani, Group divisible designs with block size four, Discrete Math 20 (1977), 1–10. [6] D. Chen and R. G. Stanton, On incomplete group divisible designs, JCMCC 6 (1989), 213–223. [7] C. J. Colbourn and J. H. Dinitz (eds.), CRC handbook of combinatorial designs, CRC Press, Boca Raton, FL (1996).
IGDDs WITH BLOCK SIZE FOUR
455
[8] G. Ge, R. S. Rees, and L. Zhu, Group-divisible designs with block size four and grouptype gu m1 with m as large or as small as possible, J Combin Theory, Ser A 98 (2002), 357– 376. [9] G. Ge and R. S. Rees, On group-divisible designs with block size four and group-type 6u m1 , Discrete Math (in press). [10] G. Ge and R. Wei, HGDDs with block size four, Discrete Math (in press). [11] K. Heinrich and L. Zhu, Existence of orthogonal Latin squares with aligned subsquares, Discrete Math 59 (1986), 69–78. [12] Y. Miao and L. Zhu, Existence of incomplete group divisible designs, JCMCC 6 (1989), 33–49. [13] R. C. Mullin, P. J. Schellenberg, S. A. Vanstone, and W. D. Wallis, On the existence of frames, Discrete Math 37 (1981), 79–104. [14] R. S. Rees and D. R. Stinson, On combinatorial designs with subdesigns, Discrete Math 77 (1989), 259–279. [15] D. R. Stinson, Frames for Kirkman triple systems, Discrete Math 65 (1987), 289–300. [16] D. R. Stinson, A general construction of group divisible designs, Discrete Math 33 (1981), 89–94. [17] Wu Dianhua and Wei Ruizhong, A note on IGDDs, J Suzhou Univ 8 (1992), 142–145. [18] L. Zhu, Some recent developments on BIBDs and related designs, Discrete Math 123 (1993), 189–214.
