GDD(n,2,4;λ1,λ2) with equal number of even and odd blocks

Discrete Mathematics 339 (2016) 1344–1354

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Discrete Mathematics journal homepage: www.elsevier.com/locate/disc

GDD(n, 2, 4; λ1 , λ2 ) with equal number of even and odd blocks Issa Ndungo a , Dinesh G. Sarvate b,∗ a

Mbarara University of Science and Technology, Uganda

b

College of Charleston, Charleston, SC 29424, USA

article

info

Article history: Received 16 April 2015 Received in revised form 3 November 2015 Accepted 7 November 2015 Available online 17 December 2015 Keywords: Group divisible designs Latin squares Even and odd GDDs IMOLS SOLS

abstract We prove that the necessary condition, n ≡ 0(mod 3), is sufficient for the existence of GDD(n, 2, 4; 3, 4) except possibly for n = 18. We prove that necessary conditions for the existence of group divisible designs GDD(n, 2, 4; λ1 , λ2 ) with equal number of even and odd blocks are sufficient for GDD(n, 2, 4; 5n, 7(n−1)) for all n ≥ 2, GDD(7s, 2, 4; 5s, 7s−1) for all s, GDD(5t + 1, 2, 4; 5t + 1, 7t ) for t ≡ 0(mod 2) and GDD(5t + 1, 2, 4; 2(5t + 1), 14t ) for all t. To complete the existence of such GDDs, one needs to construct two more families: GDD(5t + 1, 2, 4; 5t + 1, 7t ) for all odd t, and GDD(35s + 21, 2, 4; 5s + 3, 7s + 4) for all positive integers s. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Definition 1.1. A group divisible design GDD(n, m, k; λ1 , λ2 ), is a collection of k-element subsets, called blocks, of an nm-set X where the elements of X are partitioned into m subsets (called groups) of size n each; pairs of distinct elements within the same group are called first associates of each other and appear together in λ1 blocks while any two elements not in the same group are called second associates of each other and appear together in λ2 blocks [11]. A GDD(n, 2, 4; λ1 , λ2 ) with two groups and block size four in which every block intersects each group in exactly two points is called an even GDD while GDD(n, 2, 4; λ1 , λ2 ) in which each block intersects each group either in one or three points is called an odd GDD. These were first introduced by Hurd and Sarvate [11]. They proved that necessary conditions for the existence of even and odd GDD(n, 2, 4; λ1 , λ2 ) are sufficient. A GDD(n, k, k; 0, 1) is also called a transversal design TD(n, k) and it has n2 blocks. Example 1.2. A GDD(3, 2, 4; 3, 2) with two groups {1, 2, 3} and {4, 5, 6} is {{1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 3, 6}, {4, 5, 6, 1}, {4, 5, 6, 2}, {4, 5, 6, 3}}. Observe that every element in the example above occurs in exactly 4 blocks. In fact, in any GDD every element occurs a fixed number of times, this replication number is usually denoted by r. Also note that in the example there are only two groups of the same size, each block intersects each group in exactly three points or in exactly one point. GDDs are building blocks for constructions of many other designs including balanced incomplete block designs which will be defined soon. Several mathematical studies have been carried out pertaining group divisible designs. Historically,

∗

Corresponding author. E-mail address: [email protected] (D.G. Sarvate).

http://dx.doi.org/10.1016/j.disc.2015.11.004 0012-365X/© 2015 Elsevier B.V. All rights reserved.

I. Ndungo, D.G. Sarvate / Discrete Mathematics 339 (2016) 1344–1354

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GDDs were first defined with nonzero λ1 . In a long and important paper, Hanani [8] treated construction methods for block designs. In Section 6 of that paper it is shown that the obvious necessary conditions for the existence of a group divisible design with block size k = 3 are also sufficient but the definition of GDD excluded nonzero values of λ1 . Brouwer, Schrijver and Hanani [2] proved the existence of GDDs with block size 4 with λ1 = 0 only. Fu, Rodgers and Sarvate [5,6] proved the existence of GDD with block size 3 with nonzero λ1 . Another shorter proof of the same result is also given by Colbourn and Rosa (Section 6.6 [3]). The proofs given in [15] and in other subsequent papers suggest that the existence of such GDDs with nonzero λ1 and λ2 is much harder when the number of groups is less than the block size. Hurd and Sarvate [11] introduced special GDDs known as odd and even GDDs and proved that the necessary conditions are sufficient for the existence of odd GDD(n, 2, 4; λ1 , λ2 ) and even GDD(n, 2, 4; λ1 , λ2 ). Hurd and Sarvate [12] gave some constructions of infinite families of GDD(n, 2, 4; λ1 , λ2 ), including one which uses the existences of Bharsker Rao designs. They also showed that the necessary conditions are sufficient for 3 ≤ n ≤ 8. Henson, Sarvate, and Hurd [9,10] also presented constructions on GDDs with three groups and block size four, they gave a construction for a new family of group divisible designs GDD(6s + 2, 3, 4; 2, 1) using mutually orthogonal Latin squares for all positive integers s and proved that the necessary conditions are sufficient for the existence of GDDs of block size four with three groups, λ1 = 2 and λ2 = 1. Keranen and Laffin [13] studied GDDs with two groups and block size six in which each block has exactly s points from one group and t points from the other group. For s = 3 = t, they showed that the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1 , λ2 ) with fixed block configuration (s, t ) = (3, 3) and for s = 1, t = 5 they also gave minimal or near minimal index examples for all group size n ≥ 5 except n = 10, 15, 160 or 190. Clatworthy’s table list only 11 designs with two associate classes that have block size four, three groups and replication number at most 10. Rodger and Rodgers [14,15] provided neat constructions for families of group divisible designs that generalize some of these designs. In each case (namely, λ1 = 4 and λ2 = 5, λ1 = 4 and λ2 = 2, and λ1 = 8 and λ2 = 4), they proved that the necessary conditions found are also sufficient for the existence of such GDDs with block size four and three groups, with one possible exception. Together with [9,14,15], Gao and Ge [7] gave a complete generalization of all 11 designs. Present work continues the study of existence of GDDs with block size 4 and 2 groups with nonzero λ1 and λ2 including those which cannot be multiple of smaller GDDs as illustrated in Example 1.12. Definition 1.3 ([18]). A Balanced Incomplete Block Design, BIBD(v, b, r , k, λ), (V , B ), is a finite non-empty collection B of bk-subsets (called blocks) of a v -set V , such that each element appears in exactly r of the b blocks, every pair of distinct elements of V occurs in λ blocks and k < v . A BIBD(v, b, r , k, λ) can also be denoted by a BIBD(v, k, λ) since the rest of the parameters can be obtained using the given parameters. For convenience, k = v is allowed and BIBD(v, v, λ) is used to denote λ copies of set V . Definition 1.4. Suppose (X , A) is a BIBD(v, k, λ). A parallel class in (X , A) is a subset of disjoint blocks from A whose union is X . A partition of A into r parallel classes is called a resolution, and (X , A) is said to be a resolvable BIBD if A has at least one resolution. In other words, a resolvable balanced incomplete block design is a BIBD(v, k, λ) whose blocks can be partitioned into r parallel classes. Observe that a parallel class contains vk blocks, and therefore a BIBD can have a parallel class only if v ≡ 0 (mod k). Similarly one can define resolvability for any block design including GDDs. A design is called α -resolvable if its block can be partitioned into classes in which each point occurs α times. A design is near α -resolvable if its blocks can be partitioned into classes so that each class fails to contain exactly one point but contains each other point α times. For results on the existence of such designs, see [1]. It is known that a resolvable BIBD(v, 2, 1) exists if and only if v is an even integer and v ≥ 4. Following theorems including the results on Latin squares play an important role in this paper. For most of these results and for more information we refer to [3,18,19]. Theorem 1.5 ([11]). There exists a near 3-resolvable triple system TS (n, 3, 6) for every n ≥ 4. Theorem 1.6 ([16]). The necessary conditions for the existence of a (k, λ) − RGDD of type nm i.e., a RGDD(n, m, k; 0, λ) are (1) m ≥ k, (2) nm ≡ 0(mod k), and (3) λn(m − 1) ≡ 0 (mod (k − 1)). Theorem 1.7 ([16]). A (3, λ) − RGDD of type nm exists if and only if m ≥ 3, λn(m − 1) is even, nm ≡ 0(mod 3), and (λ, n, m) ̸∈ {(1, 2, 6), (1, 6, 3)} ∪ {(2j + 1, 2, 3), (4j + 2, 1, 6) : j ≥ 0}. Shen and Shen [17] studied the existence of resolvable group divisible designs with block size four and gave the necessary conditions for the existence of such GDDs. They proved that for m ̸≡ 0, 2, 6, 10(mod 12), there exists a resolvable group divisible design of order v , block size 4 and group size m if and only if v ≡ 0(mod 4), v ≡ 0(mod m), v − m ≡ 0(mod 3), except when (n, m) = (3, 12) and possibly when (n, m) = (3, 264), (3, 372), (8, 80), (8, 104), (9, 396), (40, 400) or (40, 520). Definition 1.8. A Latin square of order n is an n × n array with entries from n distinct symbols, say {1, . . . , n}, such that each element occurs exactly once in each row and column.

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Two Latin squares L1 and L2 are said to be orthogonal if for each (x, y) ∈ {1, 2, . . . , n} x {1, 2, . . . , n} there is exactly one ordered pair (i, j) such that cell (i, j) of L1 contains the symbol x and cell (i, j) of L2 contains the symbol y. The Latin squares L1 , L2 , . . . , Lt are said to be mutually orthogonal(MOLS) if for all a, b, 1 ≤ a ̸= b ≤ t, La and Lb are orthogonal. A Latin square L = lxy is self orthogonal if it is orthogonal to its transpose, it is symmetric if lxy = lyx and it is idempotent if lxx = x [18]. Theorem 1.9 ([18]). An idempotent self orthogonal Latin square of order n exists if and only if n > 3. Theorem 1.10 ([19]). Let N (n) denote the maximum number of MOLS of order n. There exists a set of N (n) − 1 idempotent mutually orthogonal Latin squares (IMOLS ) of order n. It is known that N (n) ≤ n − 1 and for n a prime power the upper bound is achieved. So for a prime or a prime power order n, there exists a set of n − 2 IMOLS of order n. Theorem 1.11 ([19]). There are at least three MOLS and hence 2 IMOLS of order n except for n = 1, 2, 3, 6 and possibly 10. Example 1.12. A GDD(4, 2, 4; λ1 = 20, λ2 = 21) on groups {1, 2, 3, 4} and {a, b, c , d} with 96 blocks having the required property is shown below. 1 2 3 a a b c 4 1 2 a b 2 3 a c

1 2 3 c a b c 2 1 2 c d 2 3 b d

1 2 3 b a b c 3 1 2 a c 2 3 a d

1 2 3 d a b c 1 1 2 b d 2 3 b c

1 2 3 a a b c 4 1 2 a b 2 3 a c

1 2 3 c a b c 2 1 2 c d 2 3 b d

1 2 4 b a b d 3 1 2 a c 2 3 a d

1 2 4 d a b d 1 1 2 b d 2 3 b c

1 2 4 a a b d 4 1 3 a b 2 4 a b

1 2 4 c a b d 2 1 3 c d 2 4 c d

1 2 4 b a b d 3 1 3 a d 2 4 a d

1 2 4 d a b d 1 1 3 b c 2 4 b c

1 3 4 a a c d 4 1 3 a b 2 4 a b

1 3 4 c a c d 2 1 3 c d 2 4 c d

1 3 4 b a c d 3 1 3 a d 2 4 a d

1 3 4 d a c d 1 1 3 b c 2 4 b c

1 3 4 a a c d 4 1 4 a c 3 4 a b

1 3 4 c a c d 2 1 4 b d 3 4 c d

2 3 4 b b c d 3 1 4 a d 3 4 a c

2 3 4 d b c d 1 1 4 b c 3 4 b d

2 3 4 a b c d 4 1 4 a c 3 4 a b

2 3 4 c b c d 2 1 4 b d 3 4 c d

2 3 4 b b c d 3 1 4 a d 3 4 a c

2 3 4 d b c d 1 1 4 b c 3 4 b d

In the next section, the necessary conditions for the existence of the GDDs with block size four and two groups with equal number of even and odd blocks are given. In the subsequent sections, we give some new constructions to prove that the necessary conditions are sufficient for the existence of certain families of GDDs with equal number of even and odd blocks. We have also constructed two families where there is no restriction on the equality of the number of even and odd blocks. 1.1. Necessary conditions A natural question one may ask before deciding on the existence of GDD(n, 2, 4; λ1 , λ2 ) with equal number of even and odd blocks is that: Is it true that such a GDD is always a union of an odd design and an even design? (Does such GDD(n, 2, 4; λ1 , λ2 ) always split into an even GDD(n, 2, 4; λ′1 , λ′2 ) and an odd GDD(n, 2, 4; λ′′1 , λ′′2 ) such that λ′1 + λ′′1 = λ1 and λ′2 + λ′′2 = λ2 ?) The answer is not always, as demonstrated by Example 3.3, but we will see in Section 5 a family which is constructed using known even and odd families of GDDs. If b is the number of blocks of the GDD(n, 2, 4; λ1 , λ2 ), then we want the number of even blocks and odd blocks to be b . Counting the number of first associate pairs in the GDD if it exists, we have 2 · 2b + 3 · 2b = 5b = λ1 n(n − 1). Similarly, 2 2 there are 3 ·

λ2 =

b 2

7λ1 (n−1) 5n

+ 4 · 2b = 7b second associate pairs in the GDD if it exists and hence λ2 n2 = 2 5nλ2 or λ1 = 7(n−1) . Hence 7 divides nλ2 , i.e., 7|n or 7|λ2 . 7λ (n−1)

7b . 2

As b =

2λ1 n(n−1) 5

=

2n2 λ2 , 7

5nλ

Therefore we are interested in constructing GDD(n, 2, 4; λ1 , 1 5n ) = GDD(n, 2, 4; 7(n−21) , λ2 ) with equal number of even and odd blocks. As 5nλ2 = 7λ1 (n − 1) for any given n we would like to know the minimum values of indices of λ1 and λ2 such that other values of indices are multiples of the minimum values. Hence there are four cases to consider. 1. 2. 3. 4.

gcd(n − 1, 5) gcd(n − 1, 5) gcd(n − 1, 5) gcd(n − 1, 5)

= 1 and 7 ̸ |n then λ1 = 5n and λ2 = 7(n − 1). = 1 and 7|n, if n = 7t, λ1 = 5t and λ2 = 7t − 1. = 5 and 7 ̸ |n, if n = 5t + 1, then λ1 = 5t + 1, λ2 = 7t. = 5 and 7|n. Here n ≡ 21(mod 35), so if n = 35s + 21, λ1 =

n 7

= 5s + 3 and λ2 = 7s + 4.

In this paper we complete Cases 1, 2, partially solve Case 3 and in Section 2 provide the first example for Case 4, but first we prove in Section 2 that the necessary conditions are sufficient for the existence of GDD(n, 2, 4; 3, 4) except possibly at n = 18. Note that the number of even blocks need not be equal to the number of odd blocks in the constructions given in Section 2 but both constructions provide examples where the number of even blocks is equal to the number of odd blocks.

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2. General construction 2.1. GDD(n, 2, 4; 3, 4) for n not equal to 18 A necessary condition for the existence of GDD(n, 2, 4; 3, 4) is n ≡ 0(mod 3). We need the following three examples as the first general construction given below cannot be applied for these parameters. Example 2.1. A GDD(3, 2, 4; 3, 4) with groups {1, 2, 3} and {4, 5, 6} exists and the blocks are given below in the columns. 1 2 4 5

1 2 4 6

1 2 5 6

1 3 4 5

1 3 4 6

1 3 5 6

2 3 4 5

2 3 4 6

2 3 5 6

Example 2.2. A GDD(6, 2, 4; 3, 4) exists and is constructed on the groups {1, 2, 3, 4, 5, 6} and {7, 8, 9, 10, 11, 12} as shown below. 1 2 3 7

1 2 3 8

2 4 7 11

1 2 3 9 2 4 8 12

4 5 6 10 2 4 9 10

4 5 6 11 2 5 7 12

4 5 6 12 2 5 8 10

7 8 9 4 2 5 9 11

7 8 9 5 2 6 7 10

7 8 9 6

10 11 12 1

2 6 8 11

10 11 12 2

2 6 9 12

10 11 12 3

3 4 7 12

1 4 7 10

3 4 8 10

1 4 8 11

3 4 9 11

1 4 9 12

3 5 7 10

1 5 7 11

3 5 8 11

1 5 8 12

3 5 9 12

1 5 9 10

3 6 7 11

1 6 7 12

3 6 8 12

1 6 8 10

1 6 9 11

3 6 9 10

Example 2.3. A GDD(9, 2, 4; 3, 4) with groups {1, 2 . . . 9} and {10, 11, . . . , 18} exists. The blocks are given below. 1 4 7 10

2 5 8 11

3 6 9 12

1 5 9 13

2 6 7 14

3 4 8 15

1 6 8 16

2 4 9 17

3 5 7 18

10 13 16 1

11 14 17 2

12 15 18 3

10 14 18 4

11 15 16 5

12 13 17 6

10 15 17 7

11 13 18 8

12 14 16 9

Each of these odd blocks is taken twice. 1 4 11 12

1 7 11 12

4 7 11 12

2 5 10 12

2 8 10 12

5 8 10 12

3 6 10 11

3 9 10 11

6 9 10 11

1 5 14 15

1 9 14 15

5 9 14 15

2 6 13 15

2 7 13 15

6 7 13 15

3 4 13 14

3 8 13 14

4 8 13 14

1 6 17 18

1 8 17 18

6 8 17 18

2 4 16 18

2 9 16 18

4 9 16 18

3 5 16 17

3 7 16 17

5 7 16 17

10 13 2 3

10 16 2 3

13 16 2 3

11 14 1 3

11 17 1 3

14 17 1 3

12 15 1 2

12 18 1 2

15 18 1 2

10 14 5 6

10 18 5 6

14 18 5 6

11 15 4 6

11 16 4 6

15 16 4 6

12 13 4 5

12 17 4 5

13 17 4 5

10 15 8 9

10 17 8 9

15 17 8 9

11 13 7 9

11 18 7 9

13 18 7 9

12 14 7 8

12 16 7 8

14 16 7 8

Theorem 2.4. For n ≡ 0(mod 3) and n ̸= 18, a GDD(n, 2, 4; 3, 4) exists. Hence necessary conditions are sufficient for the existence of GDD (n, 2, 4; 3, 4) except possibly for n = 18. Proof. For n = 3, 6 and 9 refer to Examples 2.1, 2.2 and 2.3 respectively. For n = 3s, s ≥ 4, let G1 = {a1 , a2 , . . . , as , 3s(7s−1) b1 , b2 , . . . , bs , c1 , c2 , . . . , cs } and G2 = {d1 , d2 , . . . , ds , e1 , e2 , . . . , es , f1 , f2 , . . . , fs }. Then the number of blocks, b = . 2 For convenience, we will assume that the order of the elements is fixed and we call ai , di ; bi , ei and ci , fi as corresponding elements for i = 1, 2, . . . , s. Construct a pair of IMOLSs of order s which exists according to [18] except for s = 6, say (L1 , ◦1 ) and (L2 , ◦2 ). Label the rows and the columns of L1 and L2 by A = {a1 , a2 , . . . , as }, for entries of L1 use elements of B = {b1 , b2 , . . . , bs }, while for the entries of L2 use elements of E = {e1 , e2 , . . . , es }. Repeat the same procedure where the rows and columns of L1 and L2 are labeled by B, entries of L1 are from C = {c1 , c2 , . . . , cs } and entries of L2 are from F = {f1 , f2 , . . . , fs }. Similarly, label the rows and columns of L1 and L2 by C and the entries of L1 and L2 by A and D = {d1 , d2 , . . . , ds } respectively. Form blocks (i, j, i ◦1 j, i ◦2 j) for each case. We have 3s(s − 1) odd blocks where 3 elements come from G1 . Repeat the same procedures

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to create 3s(s − 1) odd blocks with 3 elements coming from G2 . Until now, except pairs of the corresponding elements and the pairs (ai , bi ), (ai , ci ), (ai , ei ), (ai , fi ), (bi , ci ), (bi , di ), (bi , fi ), (ci , di ), (ci , ei ), (di , ei ), (di , fi ), (ei , fi ) which do not occur, all first associate pairs and second associate pairs occur twice. Now use triple {ai , bi , ci } and {di , ei , fi } to create 6 even blocks {ai , bi , di , ei }, {ai , ci , di , fi }, and {bi , ci , ei , fi } each taken twice for i = 1, 2, . . . , s to form 6s blocks in which pairs of the corresponding elements occur four times and the other pairs occur twice. We know that there exists a self orthogonal Latin square of order 3s on {1, 2, 3, . . . , 3s} as s ≥ 4 except for s = 6. One can form sets {i, j, i ◦ j, j ◦ i}, 1 ≤ i < j ≤ 3s and construct the remaining blocks using {i, j, i ◦ j, j ◦ i} by relabeling i, j by elements from G1 and the entries i ◦ j, j ◦ i by elements from G2 . From these blocks, the first associate pairs occur once and second associate pairs occur twice except pairs of the corresponding elements which do not occur at all. Throughout the whole construction therefore, we have first associate pairs occurring three times and second associate 3s(3s−1) pairs occurring four times. The total number of blocks is + 6s + 6s(s − 1) = 3s(7s2−1) which is exactly the required 2 number of blocks.  Remark 2.5. 1. A slight variation of Bose’s construction has been used to construct odd blocks. 9s(s+1) 2. In the construction above the number of even blocks is 2 and number of odd blocks is 6s(s − 1). If we wish to have equal number of even and odd blocks, then we must have hence we have the next corollary.

9s(s+1) 2

= 6s(s − 1), implying that s = 7 and n = 3s = 21,

Corollary 2.6. A GDD (21, 2, 4; 3, 4) with equal number of even and odd blocks exists. Corollary 2.7. The necessary conditions are sufficient for the existence of GDD (n, 2, 4; 3t , 4t ) for all t ≡ 1, 2(mod 3), except possibly for n = 18. In the rest of the paper we will, by the nature of the problem under consideration, deal with much higher values of λ1 and λ2 . A general construction is given below for such high indices. 2.2. Gdds with higher values of λ1 and λ2 Theorem 2.8. If a BIBD (v, 3, λ) exists, then a GDD(v, 2, 4; λ1 = λv + number of even and odd blocks may not be equal.

v(v−1) 2

, λ2 = λ(v − 1) + (v − 1)2 ) exists, where the

Proof. Take v copies of a BIBD(v, 3, λ) on one group (say G1 ) and attach ith element of the second group (say G2 ) with the ith copy of the BIBD(v, 3, λ). Similarly take v copies of a BIBD(v, 3, λ) on G2 and attach ith element of G1 with the ith copy 2λ(v−1) = λ(v − 1) times of the BIBD. Note that the first associate pairs occur λv times and the second associate pairs occur 2 as the replication number r for a BIBD(v, 3, λ) is To construct even blocks, write each of the First associate pairs occur

v 2

λ(v−1)

v 2

2

.

pairs of G1 with all

v 2

pairs of G2 . The number of even blocks is (

times and the second associate pairs occur (v − 1) times. 2

Corollary 2.9. For all odd v , BIBD (v, 3, 3) exists. Therefore GDD(v, 2, 4; λ1 = 3v + (v − 1)(v + 2)) exists where the number of even and odd blocks may not be equal.

v(v−1) 2

v(v−1) 2 2

) .



, λ2 = 3(v − 1) + (v − 1)2 =

Example 2.10. For v = 5, Corollary 2.9 gives a GDD(5, 2, 4; 25, 28) which is not a multiple of any smaller GDD and has equal number of odd and even blocks. Remark 2.11. Instead of writing each pair with pairs from the other group (so the number of even blocks is ( can sometimes increase the number of even blocks by taking x times λv 2 (v−1)

x(v(v−1))

v 2

v(v−1) 2 2

) ), we

pairs instead of only one as done in the construction

and force the number of odd blocks, equal to , number of even blocks. Implying that if x = 2λv , we will have 3 2 3 a GDD with equal number of even and odd blocks. For example, for BIBD(v = 8, 3, λ = 6), we have x = 32. Unless otherwise stated from now on a GDD refers to a group divisible design with equal number of even and odd blocks. 3. GDD(7t , 2, 4; 5t , 7t − 1) Theorem 3.1. The necessary conditions for the existence of GDD (7t , 2, 4; 5t , 7t − 1) are sufficient for all odd t. Proof. Let t = 2s + 1, G1 = {a1 , a2 , . . . , a7t } and G2 = {b1 , b2 , . . . , b7t }. We need to construct 2 · 7t 2 (7t − 1) = 2 · 7t (2s + 1)(7t − 1) blocks. Take a BIBD(7t , 3, 3) on G1 which exists constructed from an idempotent symmetric Latin square (Q , ◦) of order 7t. Construct odd blocks by replacing each block {ai , aj , ai ◦ aj = ak } of the BIBD(7t , 3, 3) on G1 with

{ai , aj , ak , bk } to get once.

7t (7t −1) 2

odd blocks in which pairs (ai , aj ) occur three times, (ak , bk ) occur

7t −1 2

and (ai , bk )i ̸= k occur

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Repeating this using a BIBD(7t , 3, 3) on G2 increases the number of blocks to 7t (7t − 1) and occurrences of the pairs (ak , bk ) to 7t − 1 time, pairs (ai , bk ) i ̸= k to two times, and pairs (bi , bj ) occur three times. We now take s copies of BIBD(7t , 3, 6) on G1 which exists to get 7st near 3-resolvable classes [11] and for each class missing ai , construct odd blocks by adding bi i.e. if {a1 , a2 , a3 } is a block of a near 3-resolvable class missing ai , then construct odd block {a1 , a2 , a3 , bi }. Repeat the same using BIBD(7t , 3, 6) on G2 to get 2 · 7ts(7t − 1)s odd blocks in which each of the pairs (ai , aj ) and (ai , bk ) occur 6s times. Until now we have constructed 7t (2s + 1)(7t − 1) odd blocks in which all the first associate pairs occur 6s + 3 = 3t times and second associate pairs occur 6s + 2 = 3t − 1 times except the pairs (ai , bk ) which have occurred 7t − 1 times. We construct even blocks using 2 idempotent mutually orthogonal Latin squares of order 7t, (Q , ◦1 ) and (Q , ◦2 ). Label the rows and columns by the elements from G1 and the entries from G2 and take t copies of {ai , aj , ai ◦1 aj , ai ◦2 aj } to get 7t (2s + 1)(7t − 1) even blocks in which pairs (ai , aj ), (bi , bj ) occur 2t times and (ai , bk ) occur 4t times. Altogether in the construction, 2 · 7t 2 (7t − 1) blocks have been constructed with first associate pairs occurring 5t times and second associate pairs occurring 7t − 1 times.  Theorem 3.2. The necessary conditions for the existence of GDD(7s, 2, 4; 5s, 7s − 1) are sufficient for all even s. Proof. Let s = 2t, G1 = {a1 , a2 , . . . , a14t } and G2 = {b1 , b2 , . . . , b14t }. We note that in this case b = 4t (14t )(14t − 1) and first associate pairs need to occur 10t times while second associate pairs need to occur 14t − 1 times. We take a near 3-resolvable BIBD(14t , 3, 6t ) on G1 to get 14t 2 near 3-resolvable classes. For t near 3-resolvable class with ai missing, we create an odd block by adding in each triple a corresponding element bi from G2 . We repeat the same using near 3-resolvable BIBD(14t , 3, 6t ) on G2 and obtain 2t (14t )(14t − 1) odd blocks with first and second associate pairs both occurring   6t times except for pairs (ai , bi ) which do not occur at all. We then use 4t copies of K14t on G1 and G2 which have 4t

14t 2

= 2t (14t )(14t − 1) pairs. Take a copy of K14t on G1 and G2 and for each edges (ai , aj ) of K14t on G1 create 14t (14t −1)

even blocks with first and second associate pairs occurring once except for the pairs a block {ai , aj , bi , bj } to get 2 (ai , bi ) which occur 14t − 1 time. Now we have 4t − 1 copies of K14t on G1 left. We take a self orthogonal Latin square of size 14t which exists for all t ≥ 1. Label its rows and columns by a1 , a2 , . . . , a14t and the entries as b1 , b2 , . . . , b14t . Construct blocks {ai , aj , ai ◦ aj , aj ◦ ai } for 14t (4t −1)(14t −1)

blocks in which first associate pairs occur 4t − 1 i < j and take 4t − 1 copies of the resulting blocks to get 2 times and second associate pairs occur 2(4t − 1) times. 14t (14t −1) All together in the construction, we have 2t (14t )(14t − 1)+ + 14t (14t −21)(4t −1) = 4t (14t )(14t − 1) blocks in which 2 first associate pairs occur 6t + 1 + 4t − 1 = 10t = 5s and second associate pairs occur 6t + 1 + 2(4t − 1) = 14t − 1 = 7s − 1 times.  Hence for the case where gcd(n − 1, 5) = 1 and 7|n, the necessary conditions are sufficient for the existence of such GDDs. A different method is used in the following example. Example 3.3. Construction for GDD(7, 2, 4; 5, 6). Here number of blocks is 84. Let G1 = {a1 , a2 , a3 , a4 , a5 , a6 , a7 } and G2 = {b1 , b2 , b3 , b4 , b5 , b6 , b7 }. We construct blocks of the BIBD(7, 3, 1) on G1 and for each block {ai , aj , ak }, use the corresponding elements {bi , bj , bk } from G2 to construct blocks: {ai , aj , ak , bi }, {ai , aj , ak , bj }, {ai , aj , ak , bk }. Repeat this using BIBD(7, 3, 1) on G2 to create blocks {bi , bj , bk , ai }, {bi , bj , bk , aj }, {bi , bj , bk , ak }. This gives 21 + 21 = 42 odd blocks in which the first associate pairs occur three times and second associate pairs occur twice except the pairs of the corresponding elements such as (ai , bi ) which occur 6 times. To construct even blocks, take BIBD(7, 4, 2) on G1 ; each of the blocks contains 6 pairs, each pair from each block is written with the remaining elements relabeled into the corresponding elements from G2 . For example, block {ai , aj , ak , al } is replaced by {ai , aj , bk , bl }, {ai , ak , bj , bl }, {ai , al , bj , bk }, {aj , ak , bi , bl }, {aj , al , bi , bk }, {ak , al , bi , bj } This yields 6 × 7 = 42 even blocks. The theorems in this section, taken together prove the following: Theorem 3.4. The necessary conditions are sufficient for the existence of GDD (7t , 2, 4; 5t , 7t − 1) in which number of even blocks and odd blocks is equal for all t ≥ 1. 4. GDD(5t + 1, 2, 4; 5t + 1, 7t ) 5nλ

′

′

For the case of gcd(5, n − 1) = 5, and 7 does not divide n; let n = 5t + 1 and λ2 = 7s′ , λ1 = 7(n−21) = n5ns = nst . So −1 s′ = t · s. Hence we take λ2 = 7ts and λ1 = (5t + 1)s for all t except for t ≡ 4(mod 7) (here 7|n, hence not applicable for this case) and we need to construct GDD(5t + 1, 2, 4; (5t + 1)s, 7ts). As the smallest case, we are required to construct GDD(6, 2, 4, 6, 7). We start with a GDD(3, 4, 4; 0, 1). This gives the following blocks. 1 4 7 10

2 5 8 10

3 6 9 10

1 5 9 11

2 6 7 11

3 4 8 11

1 6 8 12

2 4 9 12

3 5 7 12

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Four copies of this design give 36 even blocks of GDD(6, 2, 4, 6, 7). The remaining 36 odd blocks are obtained as follows: 1 5 6 7

1 4 6 7

7 9 10 2

1 4 5 7 7 9 11 2

2 4 5 8

2 4 6 8

7 9 12 2

2 5 6 8

7 8 10 3

3 4 5 9

7 8 11 3

3 4 6 9

3 5 6 9

7 8 12 3

1 2 3 10

9 8 10 1

1 2 3 10

9 8 11 1

4 5 6 10

9 8 12 1

1 2 3 11

9 8 7 4

9 8 7 5

1 2 3 11 9 8 7 6

4 5 6 11

11 10 12 4

1 2 3 12

11 10 12 5

11 10 12 6

1 2 3 12 11 10 12 4

4 5 6 12 11 10 12 5

11 10 12 6

Theorem 4.1. The necessary conditions for the existence of GDD (6, 2, 4; 6s, 7s) are sufficient. 4.1. Construction for GDD(6, 2, 4; 12, 14) This can be done by either taking two copies of GDD(6, 2, 4; 6, 7), or by a direct construction as follows: There exists a GDD(3, 4, 4; 0, 1) on groups {1,2,3}, {4,5,6}, {7,8,9} and {10,11,12}. Eight copies of this form a GDD(3, 4, 4; 0, 8), which give 72 even blocks of the GDD(6, 2, 4; 12, 14). To get the remaining 72 odd blocks, we construct the following set of blocks. 1 2 3 7

1 5 6 7

4 2 6 7

4 3 5 7

1 2 3 8

1 4 6 8

5 2 4 8

5 3 6 8

1 2 3 9

1 4 5 9

6 2 5 9

6 3 4 9

1 2 3 7

4 5 6 7

1 2 3 8

4 5 6 8

1 2 3 8

4 5 6 8

The construction of these blocks is repeated with; 10, 11, and 12; and by relabeling {7,8,9,10,11,12} for {1,2,3,4,5,6} to get a total of 72 odd blocks. Skolem sequences and Hooked Skolem sequences are very useful tools for construction of combinatorial designs. Faruqi and Sarvate [4] have given the following information on these sequences from [16]. Definition 4.2. A Skolem sequence of order n is a sequence S = (s1 , s2 , . . . , s2n ) of 2n integers satisfying the conditions 1. for every k ∈ {1, 2, . . . , n} there exist exactly two elements si , sj ∈ S such that si = sj = k, and 2. if si = sj = k with i < j then j − i = k. Skolem sequences are also written as collections of ordered pairs {(ai , bi ) : 1 ≤ i ≤ n, bi − ai = i} with ∪ni=1 {ai , bi } =

{1, 2, . . . , 2n}.

Definition 4.3. An extended Skolem sequence of order n is a sequence ES = (s1 , s2 , . . . , s2n+1 ) of 2n + 1 integers satisfying conditions (1) and (2) of the definition and also satisfies the condition (3): there is exactly one si ∈ ES such that si = 0. The element si = 0 is called the hook of the sequence. A hooked Skolem sequence is an extended Skolem sequence of order n with s2n = 0. A hooked Skolem sequence is also represented as a collection of ordered pairs in an analogous way. Theorem 4.4. Let n be a positive integer. Then there exists a Skolem sequence of order n if n ≡ 0, 1(mod 4) and a hooked Skolem sequence of order n if n ≡ 2, 3(mod 4). Let the resultant sequence be denoted as a set of ordered pairs (ai , bi ). Then the set of triples (i, ai + n, bi + n) for 1 ≤ i ≤ n is such that it partitions {1, 2, . . . , 3n} in triples with either i + ai + n = bi + n or i + ai + bi + 2n ≡ 0(mod (6n + 1)). For each triple (a, b, c ) consider the difference set ⟨0, a, a + b⟩. These are the difference sets for a BIBD(6n + 1, 3, 1) which is a K3 -decomposition of K6n+1 whose vertex set is given by {0, 1, . . . , 6n}. Example 4.5. For n = 6, a Hooked Skolem sequence is (4, 2, 5, 2, 4, 3, 6, 5, 3, 1, 1, 0, 6) or equivalently {(10, 11), (2, 4), (6, 9), (1, 5), (3, 8), (7, 13)}. Note that (10, 11) indicates that 11 − 10 = 1 occurs at the 10th and the 11th position of the sequence. The triples are (1, 16, 17), (2, 8, 10), (3, 12, 15), (4, 7, 11), (5, 9, 14), (6, 13, 19) and the difference sets are ⟨0, 1, 17⟩, ⟨0, 2, 10⟩, ⟨0, 3, 14⟩, ⟨0, 4, 11⟩, ⟨0, 5, 14⟩, ⟨0, 6, 19⟩. These yield a K3 -decomposition of K37 . Faruqi and Sarvate [4] discuss interesting facts about the difference sets obtained by Skolem or hooked Skolem sequences. First, by definition, as bi ≤ 2m + 1, no element of a difference set is bigger than 3m + 1. Hence the maximum difference we may get without modulo 6m + 1 is 3m + 1. Note that it is achieved by the last difference set in Example 4.5. Further note that 3m + 1 as a difference will be achieved if and only if we use a hooked Skolem sequence to begin with, i.e. if and only if m ≡ 2, 3(mod 4). Hence if we wish to decompose K6m+i for i = 2, 4 and 5, we can get m difference sets which will take care of 3m differences modulo 6m + i, starting from 1 to 3m − 1 and then either 3m or 3m + 1. As is well known, BIBD(6m + i, 3, 1) does not exist, but this information implies that we can get m difference sets. Hence for 5t + 1, we can definitely get at least (t +1) (n−1) as the possible number of difference sets. This is because for n = 6m + i there are at least ⌊ 6 ⌋ difference sets; and 2 we have at least ⌊

(5t ) 6

⌋ difference sets and

(t +1) 2

<

5t , 6

for t > 1.

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Theorem 4.6. The necessary conditions are sufficient for the existence of GDD(5t + 1, 2, 4; 5t + 1, 7t ) for all t ≡ 0(mod 2). Proof. We need 2t difference sets for 5t + 1 so that each distinct difference occurs exactly once. For example, the difference set {1, 2, 4} gives differences 1, 2, and 3 exactly once for 5t + 1 = 11. Taking 2t difference sets of G1 and developing 5t + 1

times gives pairs with 3t2 differences (3 differences per difference set) occurring (5t + 1)t times. Adding each of the 5t + 1 elements of G2 in the 5t + 1 developments of each difference set (of G1 ) gives 2t (5t + 1)2 odd blocks of size four with one element from G2 . Similarly, develop the difference sets of G2 5t + 1 times and add each element of G1 to have 2t (5t + 1)2 blocks of size four with one element from G1 . As a result, first associate pairs occur 5t + 1 times except for the pairs of the unused t differences while second associate pairs occur 3t times. As the first associate pairs with 5t2 − 3t2 = t differences have not occurred yet and for each difference there are 5t + 1 pairs, we have t (5t + 1) pairs each occurring 5t + 1 times. So for even blocks we have t (5t + 1)2 pairs from G1 and same number of pairs from G2 . Therefore taking each pair of G1 5t + 1 times and uniting them with 5t + 1 pairs of a difference of G2 gives t (5t + 1)2 even blocks. The first associate pairs considered here occur 5t + 1 times while the second associate pairs occur 4t times more. All together from both even and odd blocks, the first associate pairs occur 5t + 1 times while the second associate pairs occur 3t + 4t = 7t times.  Theorem 4.7. The necessary conditions are sufficient for the existence of GDD(5t + 1, 2, 4; 2(5t + 1), 14t ) for all t. Proof. For t even, we already have the result from Theorem 4.6. Let t be odd, we will use t +2 1 difference sets which exist

according to the description before Theorem 4.6. We use two copies of t −2 1 difference sets and one difference set only once. Develop the difference sets and write each development 5t + 1 times. Each difference set developed has 5t + 1 triples and (t −1) so total number of triples is 2 2 (5t + 1)(5t + 1) + (5t + 1)(5t + 1) = t (5t + 1)2 . We construct odd blocks by adding an element from the second group on each of the 5t + 1 developed difference sets to get the required t (5t + 1)2 odd blocks with three elements from the first group. Similarly we construct t (5t + 1)2 odd blocks using the difference sets from the second group. Here the first associate pairs with differences from the t −2 1 difference sets occur 2(5t + 1) times while second associates pairs occur 6t times. Also in these odd blocks, pairs with 3 differences from the ( t +2 1 )th difference set occur (5t + 1) 3(t −1) 2

differences have occurred 2(5t + 1) times and pairs with 3 differences from )th difference set have occurred 5t + 1 times. As t is odd, let 5t + 1 equals 10s + 6, there are 5s + 3 differences (the difference 5s + 3 is special and is not included in t +2 1 difference sets used). Note that there are 10s + 6 pairs for each difference except for the difference 5s + 3 for which there are 5s + 3 pairs. For the remaining unused 2s differences, 2s − 1 differences have altogether (2s − 1)(10s + 6) pairs, for the difference of 5s + 3 there are 5s + 3 pairs. These (2s − 1)(10s + 6) + 5s + 3 pairs need to come 2(10s + 6) times and the 3(10s + 6) pairs of 3 differences from the ( t +2 1 )th difference set need to come (10s + 6) times in the remaining even blocks. The total number of pairs for each group is 2(10s + 6)((2s − 1)(10s + 6)+ 5s + 3)+ 3(10s + 6)2 = 2(10s + 6)(20s2 + 22s + 6) = 2(10s + 6)(10s + 6)(2s + 1) = 2t (5t + 1)2

times and the first group pairs with

(

t +1 2

which is exactly the number of even blocks needed. The way we construct even blocks is the same as in Theorem 4.6. A pair from the first group is repeated 5t + 1 times and is combined with the 5t + 1 pairs of a difference from the second group. For the difference of 5s + 3 we use the pairs twice to get 5t + 1 blocks. (Recall t is odd so, t + 1 is even). The number of differences and the available pairs match perfectly. There are 2s − 1 differences coming 2(5t + 1) times, difference 5s + 3 coming 2(5t + 1) times and 3 differences of ( t +2 1 )th difference set coming 5t + 1 times. In all there are (5t + 1) + (2s − 1)2(5t + 1) = (4s + 2)(5t + 1) = 2t (5t + 1) differences (not necessarily distinct), counting with multiplicities.  5. GDD(n, 2, 4; 5n, 7(n − 1)) This case is the easiest among all the cases as one can construct such GDDs by summing even and odd GDDs which are discussed in [11]. Though the following theorem and example are not needed for the conclusion of this section, it may be instructive to see the construction and its proof. Theorem 5.1. The necessary conditions are sufficient for the existence of GDD(12s + 8, 2, 4; 10(6s + 4), 7(12s + 7)) for all s ≥ 0. Proof. The required number of blocks b is equal to 8(6s + 4)2 (12s + 7). Hence 4(6s + 4)2 (12s + 7) even blocks are required. Let H1 = {1, 2, . . . , 6s + 4}, H2 = {6s + 5, 6s + 6, . . . , 12s + 8}, H3 = {12s + 9, 12s + 10, . . . , 18s + 12} and H4 = {18s + 13, 18s + 14, . . . , 24s + 16} where G1 = H1 ∪ H2 and G2 = H3 ∪ H4 . Taking 4(12s + 7) copies of TD(6s + 4, 4) on groups H1 , H2 , H3 and H4 gives 4(6s + 4)2 (12s + 7) even blocks which is exactly equal to the required number of even blocks. Here all the pairs occur 4(12s + 7) times except those with points from Hi′ s, i = 1, 2, 3, 4 which do not occur at all.

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To construct odd blocks, take (2s + 2) copies of BIBD(12s + 8, 3, 6) on G2 . Note that BIBD(12s + 8, 3, 6) has 12s + 7 3-resolvable classes [1]. So total number of 3-resolvable classes of BIBD(12s + 8, 3, (2s + 2) · 6) on G2 is (12s + 7)(2s + 2) = 24s2 + 38s + 14. For each point from 1 to 6s + 2 from G1 , we take s + 1 3-resolvable classes of BIBD(12s + 8, 3, 6) on G2 and construct (s + 1)(6s + 2)(12s + 8) odd blocks in which points 1 to 6s + 2 occurs with the points of G2 3(s + 1) times. Until now we have used (6s + 2)(s + 1) = 6s2 + 8s + 2 3-resolvable classes. Now we use (3s + 2) 3-resolvable classes with each of the 6s + 6 specific points of G1 , namely 6s + 3, . . . , 12s + 8 to get (3s + 2)(6s + 6)(12s + 8) odd blocks with each of the points from G2 occurring with each of the specific points of G1 3(3s + 2) times. We have used (3s+2)(6s+6) = 18s2 +30+12 3-resolvable classes, making a total of (18s2 +30+12)+(6s2 +8s+2) = 24s2 + 38s + 14 with first associate pairs occurring 6(2s + 2) times. Repeat these steps for BIBD(12s + 8, 3, 6) on G1 using points from G2 to get a total of 4(s + 1)(12s + 8)(12s + 7) odd blocks. Altogether second associate pairs occur 6(3s + 2) times except pairs with the points 1, . . . , 6s + 2 and points 12 + 9, . . . , 18s + 10 which occur less by 6s + 3. Also we will use (24s + 14) copies of Bi = BIBD(6s + 4, 3, 2) on Hi , i = 1, 2, 3, 4. Recall, the replication number on Bi is 6s + 3. For each x in H3 , from each block B of B1 , construct 3 copies of odd block B ∪ {x}. For H4 and B2 , H2 and B3 , and H1 and B4 , we proceed similarly and obtain 12(2s + 1)(6s + 4)2 odd blocks in total. Hence we have used up 18s + 12 copies of B1 with elements of H3 , similarly 18s + 12 copies of B2 with elements of H4 , 18s + 12 copies of B3 with elements of H2 and 18s + 12 copies of B4 with elements of H1 ; with second associate pairs occurring 3(6s + 3) times. We construct the remaining 4(6s + 2)(2s + 1)(6s + 4) odd blocks by taking a union of each of the remaining (6s + 2) copies of B3 , B4 with points 1, . . . , 6s + 2 of G1 and taking a union of each of the remaining (6s + 2) copies of B1 , B2 with points 12s + 9, . . . , 18s + 10 of G2 as shown below. B3 B4 1

B3 B4 2

B1 B2 12s + 9

··· ···

B3 B4 6s + 2

B1 B2 12s + 10

where column, say



B3 B4 1



··· ···

B1 B2 18s + 10

, means that we construct blocks of size 4 by taking union of each blocks of B3 and B4 with 1. Here

the points from G1 and from G2 occur with points from G2 and G1 6s + 3 times respectively and the first associate pairs of points from Hi , i = 1, 2, 3, 4 occur 2(24s + 14) times.  Following example illustrates the construction given in the above theorem. Example 5.2. Construction for GDD(8, 2, 4; 40, 49). The required number of blocks b is equal to 2 · 8 · 8 · 7. Hence the required number of even blocks and odd blocks is 8 · 8 · 7. Let H1 = {1, 2, 3, 4}, H2 = {5, 6, 7, 8}, H3 = {9, 10, 11, 12} and H4 = {13, 14, 15, 16} where G1 = H1 ∪ H2 and G2 = H3 ∪ H4 . We construct even blocks by taking 28 copies of TD(4, 4) on the groups H1 , H2 , H3 and H4 and get 28 · 4 · 4 = 8 · 8 · 7 blocks which is exactly equal to the required number of even blocks. Due to this, all pairs occur 28 times except pairs of elements from Hi , i = 1, 2, 3, 4 which do not occur at all. We construct odd blocks by taking 2 copies of BIBD(8, 3, 6) on G1 which yield fourteen 3-resolvable classes on G1 . For each points 9 and 10 from G2 , we take one 3-resolvable class of BIBD(8, 3, 6) on G1 and construct 1 · 2 · 8 odd blocks. Due to this each of the points (9 and 10) used from G2 occurs with the points of G1 3 times and we have used two 3-resolvable classes. Now we use two 3-resolvable classes with each of the remaining points 11, 12, 13, 14, 15, 16 from G2 to get 2 · 6 · 8 odd blocks with each of the points used from G2 (11, 12, . . . , 16) occurring with each point of G1 6 times. Until now, all fourteen 3-resolvable classes have been used and first associate pairs occur 12 times. These steps are repeated with two copies of BIBD(8, 3, 6) on G2 to get a total of 4 · 8 · 7 odd blocks in which second associate pairs occur 12 times except pairs containing points 1, 2, 9, 10 which occur less by 3 times. Also we use 14 copies of Bi = BIBD(4, 3, 2) on Hi , i = 1, 2, 3, 4. For each x in H3 , from each block B of B1 , construct 3 copies of odd block B ∪ {x}; repeat the same with each point from H4 with B2 , H2 with B3 and H1 with B4 . This gives 6 · 4 · 8 odd blocks with each point from G1 occurring with each point from G2 9 times. The remaining 2 copies of B′i s are used to construct the remaining blocks by taking a union of each block of B3 , B4 with 1 and 2 and B1 , B2 with 9 and 10. This yields 4 · 8 odd blocks, with the points 1, 2 and 9, 10 occurring with G2 and G1 3 times respectively. Here the first associate pairs of elements from Hi , i = 1, 2, 3, 4 occur together 28 times. The total number of blocks and the values of the indices obtained here are exactly as required. Thus necessary conditions are sufficient for the existence of GDD(8, 2, 4; 40s, 49s) for all s ≥ 1. Now we provide another way to construct the same GDD as in Example 5.2. Example 5.3. GDD(8, 2, 4; 40, 49). The required number of blocks for this design is b = 2·647·49 = 896. Let H1 = {1, 2, 3, 4}, H2 = {5, 6, 7, 8}, H3 = {9, 10, 11, 12} and H4 = {13, 14, 15, 16} where G1 = H1 ∪ H2 and G2 = H3 ∪ H4 , we take twenty eight copies of transversal design, TD(4, 4) = GDD(4, 4, 4; 0, 1) with groups Hi , i = 1, 2, 3, 4

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to obtain 448 even blocks. TD(4, 4) has the following blocks; 1 5 9 13

2 6 10 14

3 7 11 15

4 8 12 16

1 6 11 16

2 5 12 15

3 8 9 14

4 7 10 13

1 7 12 14

2 8 11 13

3 5 10 16

4 6 9 15

1 8 10 15

2 7 9 16

3 6 12 13

4 5 11 14

Also take two copies of a BIBD(8, 3, 6) on G1 and two copies of a BIBD(8, 3, 6) on G2 to get fourteen 3-resolvable classes of BIBDs on G1 as well as on G2 . Use two 3-resolvable classes on G1 with each of 9, 10, 11, 12, 13, 14, 15 and two 3-resolvable classes on G2 with each of 1, 2, 3, 4, 5, 6, 7. Take 14 copies of the BIBD(4, 3, 2) on each of the set H1 , H2 , H3 and H4 ; each yielding sets of blocks B1 , B2 , B3 and B4 respectively. Note that 2 copies of B1 and B2 form two 3-resolvable classes of {1, 2, 3, 4, 5, 6, 7, 8} on G1 while 2 copies of B3 and B4 form two 3-resolvable classes of {9, 10, 11, 12, 13, 14, 15, 16} on G2 . Use two copies of B1 , B2 with 16 to get the blocks {1, 2, 3, 16}, {1, 2, 4, 16}, {1, 3, 4, 16}, {2, 3, 4, 16}, {5, 6, 7, 16}, {5, 6, 8, 16}, {5, 7, 8, 16} and {6, 7, 8, 16}, repeat same for B3 , B4 with 8. The remaining 192 odd blocks are constructed using the remaining copies of B′i s i = 1, 2, 3, 4 as follows. B1 9

B1 9

B1 9

B1 10

B1 10

B1 11

B1 11

B1 11

B1 11

B1 12

B1 12

B1 12

Similarly, B2 , B3 , B4 are used with {13, 14, 15, 16}, {5, 6, 7, 8} and {1, 2, 3, 4} respectively; where column of say

  B1 9

means

that we construct blocks of size 4 by taking union of each blocks of B1 with 9. Note that Theorem 5.1 gives sub designs which are even and odd. Hurd and Sarvate have studied these designs and have obtained the following results: Theorem 5.4 ([11]). (a) Even GDD (n, 2, 4; ns, 2(n − 1)s) exist for n odd and even GDD (n, 2, 4; ns , (n − 1)s) exist for n even, s ≥ 2 1. (b) The necessary conditions are sufficient for the existence of odd GDD (n, 2, 4; n, n − 1) for n ≡ 0, 1, 2, 3, 4(mod 6). (c) The necessary conditions are sufficient for the existence of odd GDD (n, 2, 4; 3n, 3(n − 1)) when n ≡ 5(mod 6). If we take the union of the collection of blocks of an even GDD(n, 2, 4; 2n, 4(n − 1)) and an odd GDD(n, 2, 4; 3n, 3(n − 1)) we obtain the collections of blocks for GDD(n, 2, 4; 5n, 7(n − 1)). Note the number of blocks in even GDD is equal to the number of blocks in the odd GDD which is n2 (n − 1). The existence of even GDD(n, 2, 4; 2n, 4(n − 1)) and odd GDD(n, 2, 4; 3n, 3(n − 1)) is given in Theorem 5.4 (a, b and c). Therefore we have: Theorem 5.5. Necessary conditions are sufficient for the existence of GDD (n, 2, 4; 5n, 7(n − 1)) for all n. Since we have constructed GDD(n, 2, 4; 5n, 7(n − 1)) for all n ≥ 2, as a corollary we have the required GDDs for Case 1 where gcd(n − 1, 5) = 1 and 7 ̸ |n. 6. Conclusion To complete the problem on the existence of GDD(n, 2, 4; λ1 , λ2 ) with equal number of even and odd blocks two families are needed: GDD(5t + 1, 2, 4; 5t + 1, 7t ) for t ≡ 1(mod 2), t ≥ 3 and the case of gcd(n − 1, 5) = 5 and 7|n that is GDD(35s + 21, 2, 4; 5s + 3, 7s + 4), s ≥ 1. For s = 0, we get GDD(21, 2, 4; 3t , 4t ) from Theorem 2.4. Other than presenting the construction of all other families, an interesting family of GDD(n, 2, 4, 3, 4) has also been constructed in this paper. Acknowledgments Authors thank the Council for International Exchange of Scholars (CIES) and the United States Department of States Bureau of Educational and Cultural Affairs for granting Dinesh Sarvate a Fulbright core fellowship and College of Charleston for granting a sabbatical which made this collaboration possible. Authors also thank the referee for his/her very useful comments and careful reading of the proofs given in the paper. References [1] R. Abel, R. Julian, G. Ge, J. Yin, Resolvable and near-resolvable designs, in: C.J. Colbourn, J.H. Dinitz (Eds.), The Handbook of Combinatorial Designs, second ed., Chapman/CRC Press, Boca Raton, FL, 2007, pp. 124–134. [2] A.E. Brouwer, A. Schrijver, H. Hanani, Group divisible designs with block size four, Discrete Math. 20 (1) (1977–1978) 1–10. [3] C.J. Colbourn, A. Rosa, Triple System, Oxford University Press Inc., New York, 1999. [4] S. Faruqi, D.G. Sarvate, On Perfect MRDs, manuscript. [5] H.L. Fu, C.A. Rodgers, Group divisible designs with two associate classes: n = 2 or m = 2, J. Combin. Theory Ser. A 83 (1998) 94–117. [6] H.L. Fu, C.A. Rodgers, D.G. Sarvate, The existence of group divisible designs with first associate, having block size 3, Ars Combin. 54 (2000) 33–50. [7] F. Gao, G. Ge, A complete generalization of clatworthy group divisible designs, SIAM J. Discrete Math. 25 (4) (2011) 1547–1561. [8] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (3) (1975) 255–369.

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