ISSN 2148-838Xhttps://doi.org/10.13069/jacodesmath.1111720 J. Algebra Comb. Discrete Appl. 9(2) • 85–99 Received: 20 September 2021 Accepted: 16 February 2022 Journal of Algebra Combinatorics Discrete Structures and Applications A note on GDD(1,n,n,4;λ1,λ2) Research Article Dinesh G. Sarvate∗, Dinkayehu M. Woldemariam∗∗ Abstract: The present note is motivated by two papers on group divisible designs (GDDs) with the same block size three but different number of groups: three and four where one group is of size 1 and the others are of the same size n. Here we present some interesting constructions of GDDs with block size 4 and three groups: one of size 1 and other two of the same size n. We also obtain necessary conditions for the existence of such GDDs and prove that they are sufficient in several cases. For example, we show that the necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. 2010 MSC: 05B05, 05B30 Keywords: t-designs, Group divisible designs with unequal group sizes, BIBDs 1. Introduction Among all combinatorial designs, probably the most widely studied design is a Balanced Incomplete Block Design (BIBD). For definitions and and background please see Lindner and Rodger [6]. Definition 1.1. A Balanced Incomplete Block Design, BIBD(v,k,λ), is an arrangement of v distinct points into b proper subsets (called blocks) of size k each, such that every point appears in exactly r blocks and every pair of distinct points occurs together in exactly λ blocks. The numbers v,b,r,k and λ are parameters of the BIBD and satisfy the necessary conditions vr = bk and λ(v −1) = r(k −1) for the existence of a BIBD(v,k,λ). In 1961, Haim Hanani [4] proved that the necessary conditions are sufficient for the existence of BIBDs with block size three as well as four. Specifically he proved: ∗ This author thanks Professor Jurisich for support through departmental research funds. Dinesh G. Sarvate (Corresponding Author); College of Charleston, SC 29424, USA (email: SarvateD@cofc.edu). ∗∗ This author is supported by Adama Science and Technology University research grant number ASTU/SP- R/011/19. Dinkayehu M. Woldemariam; Adama Science and Technology University, Adama, Ethiopia (email: dinku- men@gmail.com). 85 https://orcid.org/0000-0003-3238-0389 https://orcid.org/0000-0001-7865-7430 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Theorem 1.2. A BIBD(v,4,λ) exists if and only if λ ≡ 1,5 (mod 6) and v ≡ 1,4 (mod 12); λ ≡ 2,4 (mod 6) and v ≡ 1 (mod 3); λ ≡ 3 (mod 6) and v ≡ 0,1 (mod 4); λ ≡ 0 (mod 6) and v ≥ 4. Group divisible designs defined below play a role in the construction of BIBDs as well as other designs. For example, in the construction of t-designs where instead of each pair occurring in λ blocks each t-tuple occurs in λ blocks. Definition 1.3. A group divisible design, GDD(n1,n2, ...,nm,k;λ1,λ2), is a triple (X,G,B), where X is a v-set, G is a partition of X into m subsets (called groups) of size n1,n2, ...,nm respectively and B is a collection of k-subsets of X (called blocks) such that each pair of points within the same group appear together in λ1 blocks, whereas each pair of points from different groups appear together in λ2 blocks. The points in the same group are called first associate of each other and elements not in the same group are called second associates of each other. Fu, Rodger and Sarvate [2, 3] obtained complete results on group divisible designs with m groups of size n and block size 3, namely GDD(n,n,...,n,3;λ1,λ2). In 1992, Colbourn, Hoffman and Rees [1] proved the sufficiency of the necessary conditions for the existence of a GDD(n,n,...,n,u,3; 0,1). In 2011, Pab- hapote and Punnim [8] studied all triples of positive integers (n1,n2,λ) for which a GDD(n1,n2,3;λ,1) exists. Later, Pabhapote [7] proved the existence of a GDD(n1,n2,3;λ1,λ2) for all n1 6= 2 and n2 6= 2 in which λ1 ≥ λ2. This note is specially motivated by the papers of Sakda and Uiyyasathian [9] and Lapchinda, Punnim and Pabhapote [5]. In 2014, Lapchinda, Punnim and Pabhapote [5] gave a complete solution for the existence of a group divisible design with block size 3 and 3 groups of sizes n, n and 1. In 2017, Sakda and Uiyyasathian, obtained complete result on group divisible designs with block size 3 and 4 groups of sizes n, n, n and 1, namely GDD(1,n,n,n,3;λ1,λ2). In this note we study the existence of a GDD(1,n,n,4;λ1,λ2) with three groups G1, G2, G3 of sizes 1, n, and n respectively. In general, when the number of groups is less than the block size the work is more involved and possibly making them harder to construct. It is well known that GDDs are used as a building block for BIBDs, but the converse is also true, for example, an easy observation is the following result. Theorem 1.4. If a BIBD(n1 + n2 + ... + nm,k,λ2) and a BIBD(ni,k,λ1) exist for i = 1,2, ...m, then a GDD(n1,n2, ...,nm,k;λ1 + λ2,λ2) exists. Corollary 1.5. If a BIBD(mn + 1,4,λ2) and a BIBD(n,4,λ) exist, then a GDD(1,n,n, ...,n,4;λ1 = λ2 + λ,λ2) exists. The converse of the above corollary is not true, for example, we show in Section 5 that a GDD(1,n,n,4;λ2 + λ,λ2) exist for n = 2 or n = 3 but clearly a BIBD(2,4,λ) or BIBD(3,4,λ) does not exist. One can find such examples for larger values of n by using the construction given in the next section. For example, the construction gives a GDD(1,7,7,4; 9,6) but a BIBD(7,4,3) does not exist. Another observation gives, Theorem 1.6. A GDD(n1,n2, ...,nm,k;λ1,0) exists if and only if a BIBD(ni,k, λ1) exist for i = 1,2, ...m. Corollary 1.7. A GDD(1,n,n, ...,n,k;λ1 = λ,0) exists if and only if a BIBD(n,k,λ) exists. One may notice that it is much easier to construct GDDs with λ1 ≥ λ2, specially when a BIBD(n,k,λ) exists. In the next section we present an important construction technique which produces GDDs where λ1 is less than λ2. 86 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 2. A new construction of a GDD(1,n,n,4;λ1, λ2) A Kn on Gi means the vertices of the complete graph Kn are labeled with the elements of Gi for i = 2,3. Let n be even. Then the complete graph Kn on G2 (respectively on G3) has a 1-factorization, say {E1,E2, ...,En−1} (respectively {F1,F2, ...,Fn−1}). For x = 1,2, ...,n−1, if Ex = {e1,e2, ...,en 2 } and Fx = {f1,f2, ...,fn 2 }, then we can form blocks el ∪fm of size 4, for 1 ≤ l,m ≤ n2 . On the other hand when n is odd, a Kn on G2 (respectively on G3) has a 2-factorization, say {E1,E2, ...,En−1 2 } (respectively {F1,F2, ...,Fn−1 2 }). For x = 1,2, ..., n−1 2 , if Ex = {e1,e2, ...,en} and Fx = {f1,f2, ...,fn}, then we can form blocks el ∪fm of size 4, for 1 ≤ l,m ≤ n. Now we define B4 = {el ∪fm : el ∈ Ex and fm ∈ Fx for x = 1,2, ...,n−1 and 1 ≤ l,m ≤ n2 } if n is even, and B4 = {el ∪fm : el ∈ Ex and fm ∈ Fx for x = 1,2, ..., n−12 and 1 ≤ l,m ≤ n } if n is odd. Theorem 2.1. Suppose a BIBD(2n,3,λ) and a BIBD(n,3,µ) exist. (a) Suppose n is even and there are nonnegative integers i, j, u and v such that i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + v(n−1). (1) Then there exists a GDD(1,n,n,4;iλ + jµ + unµ + vn/2, i(2n−1)λ+jµ(n−1) 2 ). (b) Suppose n is odd and there are nonnegative integers i, j, u and v such that i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + 2v(n−1). (2) Then there exists a GDD(1,n,n,4;iλ + jµ + unµ + vn, i(2n−1)λ+jµ(n−1) 2 ). Proof. Let G1 = {x}, G2 = {a1,a2, ...,an} and G3 = {b1,b2, ...,bn}. Then consider the following sets: • B1 = {G1 ∪B : B is a block of BIBD(2n,3,λ) on G2 ∪G3}; • B2 = {G1 ∪B : B is a block of BIBD(n,3,µ) on G2 and G3}; • B3 = {{a}∪B : a ∈ Gi and B is a block of BIBD(n,3,µ) on Gj for i,j = 2,3 and i 6= j}. In B1, every element from G2 ∪G3 comes with the point of G1 λ(2n−1) 2 times and every pair of elements from G2 ∪G3 comes λ times. In B2, every element from G2 and G3 comes with the point of G1 µ(n−1) 2 times and first associate pair from G2 and G3 comes µ times. In B3, first associate pair from G2 and G3 comes µn times and second associate pair from G2 and G3 comes µ(n−1) times. In B4, first and second associate pairs from G2 and G3 occur n and 2(n−1) times respectively if n is odd while first and second associate pairs from G2 and G3 occur n2 and n−1 times respectively if n is even. Suppose we have i copies of B1, j copies of B2, u copies of B3 and v copies of B4. Then the following matrix displays the replication number of each pair (a1,x), (a1,a2) and (a1,b1) in iB1, jB2, uB3 and vB4, where i, j, u and v are any nonnegative integers. For n even,   (a1,x) (a1,a2) (a1,b1) iB1 i(2n−1)λ 2 iλ iλ jB2 j(n−1)µ 2 jµ 0 uB3 0 unµ uµ(n−1) vB4 0 vn/2 v(n−1)   . 87 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 So we have a GDD(1,n,n,4;iλ + jµ + unµ + vn/2, i(2n−1)λ+jµ(n−1) 2 ) when i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + v(n−1). (3) For n odd,   (a1,x) (a1,a2) (a1,b1) iB1 i(2n−1)λ 2 iλ iλ jB2 j(n−1)µ 2 jµ 0 uB3 0 unµ uµ(n−1) vB4 0 vn 2v(n−1)   . So we have a GDD(1,n,n,4;iλ + jµ + unµ + vn, i(2n−1)λ+jµ(n−1) 2 ) when i(2n−1)λ + jµ(n−1) 2 = iλ + uµ(n−1) + 2v(n−1). (4) Theorem 2.2. (a) If a BIBD(2n,b,r,3,λ) exists for odd n and if r − λ = 2(n − 1)t, then a GDD(1,n,n,4;λ + nt,r) exists. (b) If a BIBD(2n,b,r,3,λ) exists for even n and if r−λ = (n−1)t, then a GDD(1,n,n,4;λ+ nt 2 ,r) exists. Proof. Let B = {G1 ∪B : B is a block of BIBD(2n,b,r,3,λ)}. Then t copies of B4 along with B give the required GDD. Example 2.3. As a BIBD(6,40,20,3,8) exists, r−λ = 12 = 4×3. We get a GDD(1,3,3,4; 8 + 3×3 = 17,20 = (8 + 3×4)) using 3 copies of B4. Essentially, B4 is a GDD on two groups of size n, where the indices depend on n odd or even. For n even, B4= GDD(n,n,4;λ1 = n2 ,λ2 = n − 1) and for n odd, B4= GDD(n,n,4;λ1 = n,λ2 = 2(n − 1)). Now the replication number r for a BIBD(2n,3,λ) is λ(2n−1) 2 . If we wish r − λ to be a multiple of (n−1) , say s(n−1) when n is even (respectively 2s(n−1) when n is odd), then λ = 2s(n−1) 2n−3 (respectively λ = 4s(n−1) 2n−3 ). For s = 2n−3, λ = 2(n−1) for n even (respectively λ = 4(n−1) for n odd). Example 2.4. For n = 4, we have GDD(1,4,4,4; 16,21) by using blocks of a BIBD(8,b,21,3,6) and s = 5 copies of B4 =GDD(4,4,4; 2,3). Example 2.5. For n = 5, we have GDD(1,5,5,4; 51,72) by using blocks of a BIBD(10,b,72,3,16) and s = 7 copies of B4 =GDD(5,5,4; 5,8). In general : Theorem 2.6. For n even, using a BIBD(2n,3,2(n−1)) and 2n−3 copies of a GDD(n,n,4; n 2 ,n−1), a GDD(1,n,n,4; 2n 2+n−4 2 ,(n−1)(2n−1)) and for n odd, using a BIBD(2n,3,4(n−1)) and 2n−3 copies of a GDD(n,n,4;n,2(n−1)), a GDD(1,n,n,4; 2n2 + n−4,2(n−1)(2n−1)) exists. In the next section, we obtain some necessary conditions for the existence of a GDD(1,n,n,4;λ1,λ2). Towards this aim, assuming a GDD(1,n,n,4;λ1,λ2) exists, we count the number of blocks, ri, containing a given element x of Gi for i = 1,2,3, and the required number of blocks, say b, for the GDD. 88 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 3. Necessary conditions Suppose a GDD(1,n,n,4;λ1,λ2) exists with groups G1, G2, G3 of size 1, n, n respectively. Let ri be the replication number of each element of Gi for i = 1,2,3. As the size of G2 is equal to the size of G3, r2 = r3. Then by counting argument, r1 = 2nλ23 and r2 = r3 = λ1(n−1)+λ2(n+1) 3 . Let b be the required number of blocks for a GDD(1,n,n,4;λ1,λ2) if it exists. Since 4 × b = r1 × 1 + r2 × (n + n), we have b = λ1(n 2−n)+λ2(n2+2n) 6 . As r1 and r2 must be integers, we have the following. • If n ≡ 0 (mod 3), then λ1 ≡ λ2 (mod 3). • If n ≡ 1 (mod 3), then λ2 ≡ 0 (mod 3). • If n ≡ 2 (mod 3), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3). Since b must be an integer, we have the following. • If n ≡ 0,4 (mod 6), then no restriction on λ1 and λ2. • If n ≡ 1,3 (mod 6), then λ2 ≡ 0 (mod 2). • If n ≡ 2 (mod 6), then λ1 + λ2 ≡ 0 (mod 3). • If n ≡ 5 (mod 6), then 2λ1 + 5λ2 ≡ 0 (mod 6). Hence some basic necessary conditions for the existence of a GDD(1,n,n,4;λ1,λ2) are • If n ≡ 0 (mod 6), then λ1 ≡ λ2 (mod 3), • If n ≡ 1 (mod 6), then λ2 ≡ 0 (mod 6), • If n ≡ 2 (mod 6), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3), • If n ≡ 3 (mod 6), then λ1 ≡ λ2 (mod 3), λ2 ≡ 0 (mod 2), • If n ≡ 4 (mod 6), then λ2 ≡ 0 (mod 3), and • If n ≡ 5 (mod 6), then λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 6). Above necessary conditions are summarized in Table 1, where “None” means the design does not exist for any n. λ1 is given in modulo 3 and λ2 is given in modulo 6. λ1\λ2 0 1 2 3 4 5 0 all n None None n even None None 1 n ≡ 1 (mod 3) n ≡ 0 (mod 6) None n ≡ 4 (mod 6) n ≡ 0 (mod 3) None 2 n ≡ 1 (mod 3) None n ≡ 0 (mod 3) n ≡ 4 (mod 6) None n ≡ 0 (mod 6) Table 1. The necessary conditions for GDD(1,n,n,4;λ1,λ2) A side application of the table is that for n ≡ 3 (mod 6) instead of constructing three families: GDD(1,6m + 3,6m + 3,4; 3t,6s), GDD(1,6m + 3,6m + 3,4; 3t + 1,6s + 4) and GDD(1,6m + 3,6m + 3,4; 3t + 2,6s + 2), one needs to construct just one family GDD(1,6m + 3,6m + 3,4; 3t,6s). Then the family GDD(1,6m + 3,6m + 3,4; 3t + 2,6s + 2) can be obtained by taking the blocks of a GDD(1,6m + 89 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 3,6m + 3,4; 3t,6s) and the blocks of a BIBD(12m + 7,4,2). Similarly, a GDD(1,6m+3,6m+3,4; 3t+1,6s+4) can be obtained by taking the blocks of a GDD(1,6m+ 3,6m + 3,4; 3(t−1),6s) and the blocks of a BIBD(12m + 7,4,4) where m is any nonnegative integer. As a GDD(1,n,n,4;λ1,λ2) has 3 groups and blocks of size 4, each block contains at least one associate pair. Then b ≤ [ ( n 2 ) + ( n 2 ) ]λ1 = n(n − 1)λ1. Now substituting the value of b, we have the following theorem: Theorem 3.1. A necessary condition for the existence of a GDD(1,n,n,4;λ1,λ2) is λ2 ≤ 5(n−1) n + 2 λ1. Corollary 3.2. For the existence of a GDD(1,n,n,4,λ1,λ2), λ2 ≤ 5λ1. The blocks of a GDD(1,n,n,4;λ1,λ2), if exists, have (n2 + 2n)λ2 second associate pairs. There can be at most r1 blocks of type (1,1,2) which account for 5r1 second associate pairs, we have b − r1 ≥ (n2+2n)λ2−5r1 4 as all other blocks can have at the most 4 second associate pairs. Thus, we have the following: Theorem 3.3. A necessary condition for the existence of a GDD(1,n,n,4;λ1,λ2) is b ≥ (n2+2n)λ2−r1 4 . Corollary 3.4. A necessary condition for the existence of a GDD(1,n,n,4;λ1,λ2) is λ2 ≤ 2(n−1) n λ1 < 2λ1. Proof. Substituting the values of b and r1 in b ≥ (n2+2n)λ2−r1 4 , we have λ2 ≤ 2(n−1) n λ1 < 2λ1. As a consequence of the above corollary, we have Corollary 3.5. For any nonnegative integer t and any positive integers x and s, where x ≤ s following GDDs do not exist. 1. A GDD(1,6t + 4,6t + 4,4; 3x + 1,6s + 3). 2. A GDD(1,3t,3t,4; 3x + 1,6s + 4). 3. A GDD(1,6t,6t,4; 3x + 2,6s + 5). 4. A GDD(1,n,n,4; 3x,6s). 4. Existence of families of GDD(1,n,n,4;λ1, λ2) Unless otherwise stated in this section we are assuming λ1 ≥ λ2. Remark 4.1. A GDD(1,n,n,4; 0,λ2) does not exist as the number of groups is less than the block size. As a consequence of Theorem 1.4 we have following theorem where unless otherwise stated λ, s and t are nonnegative integers and n > 1. Theorem 4.2. If a BIBD(2n+1,4,λ2) and a BIBD(n,4,λ) exist, then a GDD(1,n,n,4;λ1 = λ2 +λ,λ2) exists. In particular, we have 1. A GDD(1,n,n,4; 6t,6s) exists for all n ≥ 4, where t ≥ s. 2. A GDD(1,n,n,4; 6s + 3t,6s) exists for n ≡ 0,1 (mod 4). 90 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 3. A GDD(1,n,n,4; 6t + 3s,3s) exists for n ≡ 0 (mod 2). 4. A GDD(1,n,n,4; 6t + 2s,2s) exists when n ≡ 0 (mod 3). 5. A GDD(1,n,n,4; 3t + s,s) exists for n ≡ 0 (mod 12). 6. A GDD(1,n,n,4; 3t + 2s,2s) exists for n ≡ 9 (mod 12). 7. A GDD(1,n,n,4; 6t + λ,λ) exists for n ≡ 0 (mod 6). 8. A GDD(1,n,n,4; 3t,3s) exists for n ≡ 0 (mod 4), where t ≥ s. 9. A GDD(1,n,n,4; 2t + 6s,6s) exists for n ≡ 1 (mod 6). 10. A GDD(1,n,n,4; 2t + 3s,3s) exists for n ≡ 4 (mod 6). 11. A GDD(1,n,n,4; 6s + λ,6s) exists for n ≡ 1 (mod 12). 12. A GDD(1,n,n,4; 3s + λ,3s) exists for n ≡ 4 (mod 12). Case 1: λ2 ≡ 0 (mod 6) From Theorem 4.2(1) we have Corollary 4.3. Necessary conditions are sufficient for the existence of a GDD(1, n,n,4;λ1,λ2) for λ1 ≡ 0 (mod 6) and λ2 ≡ 0 (mod 6). From Theorem 4.2(2) we have Corollary 4.4. Necessary conditions are sufficient for the existence of a GDD(1, n,n,4;λ1,λ2) for n ≡ 5 (mod 12). In the above family λ1 −λ2 ≡ 3 (mod 6). From the necessary conditions, when λ2 ≡ 3 (mod 6) and λ1 ≡ 0 (mod 3), n has to be even. From Theorem 4.2 (3), a GDD(1,n,n,4; 6t + 3,6s + 3) exists for any even n and any nonnegative integers s and t, where t ≥ s. From Theorem 4.2(8), a GDD(1,n,n,4; 6t,6s+3) exists when n ≡ 0 (mod 4) for any nonnegative integers s and t, where t > s. Hence we have Corollary 4.5. Necessary conditions are sufficient for the existence of a GDD(1, n,n,4;λ1,λ2) for n ≡ 0 (mod 4), λ1 ≡ 0 (mod 3) and λ2 ≡ 0 (mod 3). When λ1 ≡ 2 (mod 3) and λ2 ≡ 0 (mod 6), from the necessary conditions, we have n ≡ 1 (mod 3). For n ≡ 1 (mod 3), a BIBD(n,4,2) and a BIBD(2n + 1,4,6) exist for n ≥ 4. From Theorem 4.2(9) and (10), we have Lemma 4.6. A GDD(1,n,n,4; 6t+ 2,6s) exists when n ≡ 1 (mod 3) for any nonnegative integers s and t, where t ≥ s. For n ≡ 1,4 (mod 12), a BIBD(n,4,5) and a BIBD(2n + 1,4,6) exist. From Theorem 4.2(11) and (12), we have Lemma 4.7. A GDD(1,n,n,4; 6t + 5,6s) exists when n ≡ 1,4 (mod 12) for any nonnegative integers s and t, where t ≥ s. For λ2 ≡ 0 (mod 6), and λ1 ≡ 1,2 (mod 3) from the necessary conditions, we have n ≡ 1 (mod 3). From Theorem 4.2 ( 9) and (10), we have 91 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Lemma 4.8. A GDD(1,n,n,4; 6t+ 4,6s) exists when n ≡ 1 (mod 3) for any nonnegative integers s and t, where t ≥ s. From Theorem 4.2 (11) and (12), we have the following: Lemma 4.9. A GDD(1,n,n,4; 6t + 1,6s) exists for n ≡ 1,4 (mod 12), where s and t are nonnegative integers such that t ≥ s. For n ≡ 9 (mod 12), a BIBD(n,4,3) and BIBD(2n + 1,4,6) exist. So we have Lemma 4.10. A GDD(1,n,n,4;λ1 = 3t,λ2 = 6s) exists when n ≡ 9 (mod 12) for any nonnegative integers s and t, where λ1 ≥ λ2. Hence, we have : Corollary 4.11. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,λ2) for λ1 ≥ λ2 for n ≡ 1,4 (mod 12), λ1 ≡ 1 (mod 3) and λ2 ≡ 0 (mod 6). Case 2: λ2 ≡ 1 (mod 6) In this case from the necessary conditions, we have λ1 ≡ 1 (mod 3) and n ≡ 0 (mod 6). From Theorem 4.2 (7): Lemma 4.12. A GDD(1,n,n,4; 6t + 1,6s + 1) exists when n ≡ 0 (mod 6) for any nonnegative integers s and t, where t ≥ s. From Theorem 4.2 (5): Lemma 4.13. A GDD(1,n,n,4; 6t+ 4,6s+ 1) exists when n ≡ 0 (mod 12) for any nonnegative integers s and t, where t ≥ s. Corollary 4.14. Necessary conditions are sufficient for the existence of a GDD(1,n, n,4;λ1,λ2) for n ≡ 0 (mod 12), and λ2 ≡ 1 (mod 6). Case 3: λ2 ≡ 2 (mod 6) In this case, λ1 ≡ 2 (mod 3) and n ≡ 0 (mod 3). From Theorem 4.2(4) we have the following lemma. Lemma 4.15. A GDD(1,n,n,4; 6t + 2,6s + 2) exists when n ≡ 0 (mod 3) for any nonnegative integers s and t, where t ≥ s. From Theorem 4.2(5) and (6), we have Lemma 4.16. A GDD(1,n,n,4; 6t + 5,6s + 2) exists when n ≡ 0 (mod 12) for any nonnegative integer s and t, where t ≥ s. Hence we have: Corollary 4.17. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,λ2) for n ≡ 0,9 (mod 12), and λ2 ≡ 2 (mod 6). Case 4: λ2 ≡ 3 (mod 6) In this case from the necessary conditions, when λ1 ≡ 1,2 (mod 3), n ≡ 4 (mod 6). For n ≡ 4 (mod 6), a BIBD(n,4,2), a BIBD(n,4,4) and a BIBD(2n+1,4,3) exist. From Theorem 4.2(10), we have the following lemma: 92 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Lemma 4.18. When n ≡ 4 (mod 6), a GDD(1,n,n,4; 6t+ 1,6s+ 3) exists for any nonnegative integers s and t, where t > s and a GDD(1,n,n,4; 6t + 5,6s + 3) exists where t ≥ s. For n ≡ 4 (mod 12), a BIBD(n,4,1) and a BIBD(2n + 1,4,3) exist. Hence we have a GDD(1,n,n, 4; 3,3), a GDD(1,n,n,4; 4,3) and a GDD(1,n,n,4; 5,3). From Theorem 4.2 (12), we have Corollary 4.19. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,6s+3) when n ≡ 4 (mod 12) for any nonnegative integers s and λ1, where λ1 ≥ 6s + 3. When λ1 ≡ 0 (mod 3), n ≡ 0 (mod 2), from Theorem 4.2 (8), we have Corollary 4.20. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,6s+3) when n ≡ 0 (mod 4) for any nonnegative integers s and λ1, where λ1 ≥ 6s + 3. Case 5: λ2 ≡ 4 (mod 6) In this case from the necessary conditions, we have λ1 ≡ 1 (mod 3) and n ≡ 0 (mod 3). For n ≡ 0 (mod 12), a BIBD(2n + 1,4,1), and a BIBD(n,4,3) exist. Hence, a GDD(1,n,n,4; 3x + 1,4) exists for x > 1. Hence Corollary 4.21. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1 = 3t+1,λ2 = 6s + 4) for n ≡ 0 (mod 12). Similarly from Theorem 4.2 (6), we have a GDD(1,n,n,4;λ1 = 3x + 4,λ2 = 4) for a nonnegative integer x, hence Corollary 4.22. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1 = 3t+1,λ2 = 6s + 4) for n ≡ 9 (mod 12) where λ1 > λ2. For n ≡ 0 (mod 3), a BIBD(2n + 1,4,2), a BIBD(2n + 1,4,4) and a BIBD(n,4,6) exist for n ≥ 4. From Theorem 4.2(4), we have Lemma 4.23. A GDD(1,n,n,4; 6t + 4,6s + 4) exists when n ≡ 0 (mod 3) for any nonnegative integers s and t, where t ≥ s. Case 6: λ2 ≡ 5 (mod 6) Here, from the necessary conditions, we have λ1 ≡ 2 (mod 3) and n ≡ 0 (mod 6). But for n ≡ 0 (mod 12), a BIBD(2n + 1,4,5) and a BIBD(n,4,3) exist. From Theorem 4.2(5) and (6), we have Lemma 4.24. A GDD(1,n,n,4; 6t+ 2,6s+ 5) exists when n ≡ 0 (mod 12) for any nonnegative integers s and t, where t > s. From Theorem 4.2(7), we have Lemma 4.25. A GDD(1,n,n,4; 6t + 5,6s + 5) exist when n ≡ 0 (mod 6) for any nonnegative integers s and t, where t ≥ s. Hence we have Corollary 4.26. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1 = 3t+2,λ2 = 6s + 5) for n ≡ 0 (mod 12) where λ1 > λ2. We have summarized main results from this section in Table 2. From Table 2, we have Theorem 4.27. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. 93 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 n ≡ The existence is not known for 0 (mod 12) Corollary 4.3, Corrolary 4.5, Corol- lary 4.14, Corollary 4.17, Corollary 4.20, Corollary 4.21, Corollary 4.26, Lemma 4.12, Lemma 4.13, Lemma 4.15, Lemma 4.16, Lemma , Lemma 4.23, Lemma 4.24 1 (mod 12) Corollary 4.3, Corollary 4.11, Lemma 4.6, Lemma 4.7, Lemma 4.8, Lemma 4.9, Theorem 4.2 (11) 2 (mod 12) Corollary 4.3 GDD(1,n,n,4; 3t,6s + 3) GDD(1,n,n,4; 6t + 3,6s) 3 (mod 12) Corollary 4.3, Lemma 4.15, Lemma 4.23 GDD(1,n,n,4; 6t + 3,6s) GDD(1,n,n,4; 6t + 1,6s + 4) GDD(1,n,n,4; 6t + 5,6s + 2) 4 (mod 12) Corollary 4.3, Corollary 4.5, Corol- lary 4.11, Corollary 4.19, Corol- lary 4.20, Lemma 4.6, Lemma 4.7, Lemma 4.8, Lemma 4.9, Lemma 4.18 5 (mod 12) Corollary 4.3, Corollary 4.4 6 (mod 12) Corollary 4.3, Lemma 4.12, Lemma 4.15, Lemma 4.25 GDD(1,n,n,4; 6t + 3,6s) GDD(1,n,n,4; 3t,6s + 3) GDD(1,n,n,4; 6t + 4,6s + 1) GDD(1,n,n,4; 6t + 5,6s + 2) GDD(1,n,n,4; 6t + 1,6s + 4) GDD(1,n,n,4; 6t + 2,6s + 5) 7 (mod 12) Corollary 4.3, Lemma 4.6, Lemma 4.8 GDD(1,n,n,4; 6t + 1,6s) GDD(1,n,n,4; 6t + 3,6s) GDD(1,n,n,4; 6t + 5,6s) 8 (mod 12) Corollary 4.3, Corollary 4.5, Corol- lary 4.20, 9 (mod 12) Corollary 4.3, Corollary 4.17, Corollary 4.22, Lemma 4.10, Lemma 4.15, Lemma 4.23 10 (mod 12) Corollary 4.3, Lemma 4.6, Lemma 4.8, Lemma 4.18 GDD(1,n,n,4; 6t + 3,6s) GDD(1,n,n,4; 6t + 1,6s) GDD(1,n,n,4; 6t + 5,6s) GDD(1,n,n,4; 6t + 2,6s + 3) 11 (mod 12) Corollary 4.3 GDD(1,n,n,4; 6t + 3,6s) Table 2. For the existence of a GDD(1,n,n,4;λ1,λ2), λ1 ≥ λ2 5. Specific GDDs In this section, we study the existence of GDD(1,n,n,4;λ1,λ2) for specific values of the parameters. 5.1. λ1 = 1 Theorem 5.1. Necessary conditions are sufficient for the existence of a GDD(1,n, n,4; 1,λ2). Specif- ically, a GDD(1,n,n,4; 1,λ2) exists when λ2 = 1 and n ≡ 0 (mod 6) and when λ2 = 0 and n ≡ 1,4 (mod 12). Proof. For λ1 = 1, by Corollary 3.4, λ2 < 2, hence λ2 can only be 0 or 1. A GDD(1,n,n,4; 1,1) exists for n ≡ 0 (mod 6) as a BIBD(2n + 1,4,1) on G1 ∪ G2 ∪ G3 exists. A GDD(1,n,n,4; 1,0) exists for n ≡ 1,4 (mod 12) as a BIBD(n,4,1) on Gi for i = 2,3, where G1, G2, G3 are groups of size 1, n, n 94 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 respectively. 5.2. λ1 = 2 Theorem 5.2. A GDD(1,n,n,4; 2,λ2) exists for λ2 ≤ 2, specifically when λ2 = 2 and n ≡ 0 (mod 3) and when λ2 = 0 and n ≡ 1,4 (mod 6). Proof. For λ1 = 2, by Corollary 3.4, λ2 < 4, hence λ2 can be 0, 1, 2 and 3. A GDD(1,n,n,4; 2,0) exists for n ≡ 1 (mod 3) as a BIBD(n,4,2) on Gi for i = 2,3, where G1, G2, and G3 are groups of size 1, n, and n respectively. A GDD(1,n,n,4; 2,1) does not exists for any n from the necessary conditions. A GDD(1,n,n,4; 2,2) exists for n ≡ 0 (mod 3) as a BIBD(2n + 1,4,2) on G1 ∪G2 ∪G3 exists. As a GDD(1,n,n,4; 2,3) exists for n = 4 (see Example 5.13) and for n = 10 (see Example 6.8), we have Theorem 5.3. Necessary conditions are sufficient for the existence of a GDD(1,n,n,4; 2,λ2) except possibly for n ≡ 4 (mod 6), n 6= 4, 10 and λ2 = 3. 5.3. λ1 = 3 Theorem 5.4. Necessary conditions are sufficient for the existence of a GDD(1,n, n,4; 3,λ2). Specif- ically, a GDD(1,n,n,4; 3,λ2) exists when λ2 = 3 and n ≡ 0 (mod 2) and when λ2 = 0 and n ≡ 0,1 (mod 4). Proof. For λ1 = 3, by Corollary 3.4, λ2 < 6. Hence λ2 can be 0 and 3. A GDD(1,n,n,4; 3,0) exists for n ≡ 0,1 (mod 4) as a BIBD(n,4,3) on Gi for i = 2,3 exists. But a GDD(1,n,n,4; 3,0) does not exist for n ≡ 2,3 (mod 4) by Corollary 1.7. A GDD(1,n,n,4; 3,3) exists for n ≡ 0 (mod 2) as a BIBD(2n+1,4,3) on G1 ∪G2 ∪G3 exists. 5.4. n=2 When n = 2, both λ1 and λ2 are 0 modulo 3. In a GDD(1,2,2,4;λ1,λ2), there are no blocks of type (0,4) and (1,3). Hence, a GDD(1,2,2,4;λ1,0) does not exist. Let b1 and b2 be the number of blocks of type (1,1,2) and (2,2) for a GDD(1,2,2, 4;λ1,λ2) if it exists. Then b1 + 2b2 = 2λ1 and 5b1 + 4b2 = 8λ2. Hence 3b1 = 8λ2 −4λ1 As b1 = r1 = 43λ2, we have Lemma 5.5. A necessary condition for the existence of a GDD(1,2,2,4;λ1,λ2) is λ1 = λ2. As a GDD(1,n,n,k;λ,λ) is a BIBD(2n + 1,k,λ) and as a BIBD(5,4,3) exists by Theorem 1.2, we have Theorem 5.6. Necessary conditions are sufficient for the existence of a GDD(1,2, 2,4;λ1,λ2). 5.5. n=3 Let G1 = {x}, G2 = {a,b,c} and G3 = {1,2,3}. In a GDD(1,3,3,4;λ1,λ2), there is no block of type (0,4). Let b1, b2 and b3 be the number of blocks of type (1,1,2), (1,3) and (2,2) respectively. Then b1 + 3b2 + 2b3 = 6λ1 and 5b1 + 3b2 + 4b3 = 15λ2. As 6λ1 = b1 + 3b2 + 2b3 ≤ 5b1 + 3b2 + 4b3 = 15λ2, we have λ1 ≤ 52λ2. Also, by Corollary 3.4, λ2 ≤ 4 3 λ1. Hence, 95 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Lemma 5.7. A necessary condition for the existence of a GDD(1,3,3,4;λ1,λ2) is 34λ2 ≤ λ1 ≤ 5 2 λ2. In other words, 2 5 λ1 ≤ λ2 ≤ 43λ1. Remark 5.8. A GDD(1,3,3,4;λ1, 0) does not exist as there are no blocks of type (0,4). Case 1. λ1 < λ2. For n = 3, λ1 ≡ λ2 (mod 3). Let λ2 −λ1 =3s for some positive integer s. Let λ1 = 3t + i, for i = 0,1,2. Then from λ2 ≤ 4λ13 , λ2 ≤ 4t+i. Therefore the difference λ2−λ1 = 3s is less than or equal to t. Using λ2 = 3s + λ1 and λ2 ≤ 4λ13 , the smallest λ1 will be 9s and λ2 will be 12s. Hence the smallest GDD where λ2 −λ1 = 3s is a GDD(1,3,3,4; 9s,12s). For s = 1, we construct a GDD(1,3,3,4; 9,12) as follows. A relabeling construction Let X = {0,1,2,3,4,5,6}, G1 = {0}, G2 = {1,2,3} and G3 = {4,5,6}. Then B = {{2,4,5,6},{3,5, 6,0},{4,6,0,1},{5,0,1,2}, {6,1,2,3},{0,2,3,4}, {1,3,4,5}} is a collection of blocks of a BIBD(7,4, 2) on G1 ∪G2 ∪G3 = X. We relabel elements of these blocks using different permutations (say α) on X to get following six isomorphic BIBDs. • α(0) = 0, α(1) = 1, α(2) = 3, α(3) = 2, α(4) = 4, α(5) = 6 and α(6) = 5. Then, when we relabel the points of X using α, the blocks of B become B1 = {{3,4,6,5},{2,6,5,0},{4,5,0,1},{6, 0,1,3},{5,1,3,2},{0,3,2,4},{1,2,4,6}}. • α(0) = 0, α(1) = 2, α(2) = 1, α(3) = 3, α(4) = 6, α(5) = 5 and α(6) = 4. Then the blocks of B become B2 = {{1,6,5,4},{3,5,4,0}, {6,4,0,2},{5,0,2,1},{4,2,1,3},{0,1,3,6},{2,3,6,5}}. • α(0) = 2, α(1) = 1, α(2) = 0, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 6. Then the blocks of B become B3 = {{0,4,5,6},{3,5,6,2}, {4,6,2,1},{5,2,1,0},{6,1,0,3},{2,0,3,4},{1,3,4,5}}. • α(0) = 2, α(1) = 1, α(2) = 0, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 6. Then the blocks of B become B4 = {{0,4,5,6},{3,5,6,2}, {4,6,2,1},{5,2,1,0},{6,1,0,3},{2,0,3,4},{1,3,4,5}}. • α(0) = 6, α(1) = 1, α(2) = 2, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 0. Then the blocks of B become B5 = {{2,4,5,0},{3,5,0,6}, {4,0,6,1},{5,6,1,2},{0,1,2,3},{6,2,3,4},{1,3,4,5}}. • α(0) = 6, α(1) = 1, α(2) = 2, α(3) = 3, α(4) = 4, α(5) = 5 and α(6) = 0. Then the blocks of B become B6 = {{2,4,5,0},{3,5,0,6}, {4,0,6,1},{5,6,1,2},{0,1,2,3},{6,2,3,4},{1,3,4,5}}. So B∪B1 ∪ ...∪B6 gives a BIBD(7,4,14) which contains 49 blocks. Removing the ten blocks containing all three points 1,2,3 and all three points 4,5,6, we have a GDD(1,3,3,4; 14−5 = 9,14−2 = 12). In general, as a BIBD(7,4,14t) exists for any positive integer t, we have the following result. Lemma 5.9. A GDD(1,3,3,4; 14t−5s,14t−2s) for s = 0,1, · · · , t exists for any positive integer t where s = 0 gives a BIBD. For example, when t = 1, we have a GDD(1,3,3,4; 14 − 5 = 9,14 − 2 = 12), and hence by using BIBD(7,4,2) repeatedly, we have GDD(1,3,3,4; 9 + 2m,12 + 2m), specifically we are in- terested in GDD(1,3,3,4; 13,16), GDD(1,3,3,4; 15,18), GDD(1,3,3,4; 17,20), GDD(1,3,3,4; 19,22), GDD(1,3,3,4; 21,24). Lemma 5.10. Necessary conditions are sufficient for the existence of a GDD(1,3,3,4;λ1,λ2) for λ1 ≤ λ2. 96 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Proof. A GDD(1,3,3,4; 9,12) and a BIBD(7,4,2) exist. The smallest GDD(1,3,3,4;λ1,λ2) when the difference λ2 − λ1 = 3s is GDD(1,3,3,4; 9s,12s). s copies of GDD(1,3,3,4; 9,12) and m copies of BIBD(7,4,2) together give all required GDD(1,3,3,4; 9s + 2m,12s + 2m) where λ2 − λ1 = 3s. Recall that for n = 3, λ2 is always even. When λ1 = λ2, necessary conditions are the same as the conditions for the existence of a BIBD(7,4,λ1). Case 2 λ1 > λ2. Since a BIBD(6,3,2) and a BIBD(3,3,1) exist, we have the following from Theorem 2:   (1,x) (1,2) (1,a) iB1 5i 2i 2i jB2 j j 0 uB3 0 3u 2u vB4 0 3v 4v   So when 5i + j = 2i + 2u + 4v, then we have a GDD(1,3,3,4; 2i + j + 3u + 3v,5i + j). (5) Let λ1 − λ2 = 3s for some nonnegative integer s. As λ1 ≤ 5λ22 , 3s ≤ 1.5λ2. Hence when the dif- ference λ1 − λ2 = 3s, smallest value of λ2 is 2s and corresponding smallest parameter GDD will be GDD(1,3,3,4; 5s,2s). For s = 1, the required GDD is GDD(1,3,3,4; 5,2) which can be constructed from 5 by letting i = 0, j = 2, u = 1, v = 0. Lemma 5.11. Necessary conditions are sufficient for the existence of a GDD(1,3,3,4;λ1,λ2) for λ1 ≥ λ2. Proof. A GDD(1,3,3,4; 5,2) and a BIBD(7,4,2) exist. The smallest GDD(1,3,3,4;λ1,λ2) when the difference λ1−λ2 = 3s is GDD(1,3,3,4; 5s,2s). Note that s copies of GDD(1,3,3, 4; 5,2) and m copies of BIBD(7,4,2) together give all required GDD(1,3,3,4; 5s + 2m,2s + 2m) where the λ1 −λ2 = 3s. Recall that for n = 3, λ2 is always even. When λ1 = λ2, necessary conditions are the same as the conditions for the existence of a BIBD(7,4,λ1). Lemma 5.10 and Lemma 5.11 together complete the case for n = 3 and we have Theorem 5.12. Necessary conditions are sufficient for the existence of a GDD(1,3,3,4;λ1,λ2). 5.6. n = 4 Example 5.13. A GDD(1,4,4,4; 2,3) with G1 = {x}, G2 = {a,b,c,d} and G3 = {1,2,3,4}. The blocks are given below in columns. x x x x x x x x a c a c a b a b a d b c b a a b b d b d c d c d 1 1 2 2 c d d c 1 1 3 3 1 1 2 2 4 4 3 3 1 2 3 4 2 2 4 4 3 3 4 4 For n = 4, from Theorem 3.3, λ2 ≤ 32λ1. Hence, we have the following corollary: Corollary 5.14. A necessary condition for the existence of a GDD(1,4,4,4;λ1,λ2) is λ1 ≥ 23λ2. From Table 1, we need to construct two families: GDD(1,4,4,4;λ1,6s) and GDD(1,4,4,4;λ1,6s+3) where s and λ1 are nonnegative integers. For the first family, by Corollary 5.14, λ1 ≥ 4s. Using 2s copies 97 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 of the GDD(1,4,4,4; 2,3) and λ copies of BIBD(4,4,1), we have a GDD(1,4,4,4; 4s + λ,6s), for any λ. To construct a GDD(1,4,4,4;λ1,6s + 3), we observe that by Corollary 5.14, λ1 ≥ 4s + 2. Hence using 2s+1 copies of the GDD(1,4,4,4; 2,3) and λ copies of a BIBD(4,4,1), we have GDD(1,4,4,4; 4s+ 2 + λ,6s + 3), for any nonnegative integer λ. Hence we have: Theorem 5.15. Necessary conditions are sufficient for the existence of a GDD(1,4,4,4;λ1,λ2). 6. Difference families constructions The aim of this section is to construct some examples of GDDs with a difference family. In the process we make some comments to show sufficiency in certain cases. 6.1. n = 5 Recall for n = 5, from the necessary conditions, λ2 ≡ 0 (mod 6). So let λ2 = 6t where t is a nonnegative integer. Now from the necessary condition, we have λ1 < λ2 < 2λ1. Let λ2 −λ1 =3s. Then using λ2 < 2λ1, 3s < λ1. If smallest possible value of λ1 is 3s + 3, then λ2 = 6s + 3 not 0 mod 6. Hence the smallest possible value of λ1 has to be 3s + 6 and λ2 = 6s + 6. For example, when s = 1 the smallest possible GDD will be GDD(1,5,5,4; 9,12). We present a difference family construction for GDD(1,5,5,4; 9,12). Example 6.1. Let the groups be G1 = {∞}, G2 = {1,3,5,7,9} and G3 = {0,2,4, 6,8}. Difference family is {{∞,0,1,3},{∞,0,1,4},{∞,0,3,4},{∞,0,3,6},{0,5,1,2},{0,5,1,3},{0,5,1, 4},{0,5,2,3},{0,5,2,4},{0,5,3,4}}. Hence, we also have GDD(1,5,5,4; 6m + 9t,6m + 12t) for nonnegative integers m and t. Theorem 6.2. The necessary conditions for the existence of a GDD(1,5,5,4; 6t,6s) are sufficient for t ≥ s and the necessary conditions for the existence of a GDD(1,5,5,4; 6t + 3,6s) are sufficient for t ≥ s−1 On the other hand, let λ1 −λ2 =3s, as λ2 ≡ 0 (mod 6), a GDD(1,5,5,4; 6t,6t) and a BIBD(5,4,3) exist, hence a GDD(1,5,5,4; 6t + 3s,6t) exists. Theorem 6.3. The necessary conditions for the existence of a GDD(1,5,5,4;λ1,λ2) are sufficient for λ1 ≥ λ2. 6.2. n = 6 Let G1 = {∞}, G2 = {1,3,5,7,9,11} and G3 = {0,2,4,6,8,10} be groups. Example 6.4. The following multiset {{∞,0,2,4},{0,1,4,5},{0,2,7,8},{0,2,4,6},{0,3,6,9},{0, 3,6,9}} is a difference family for a GDD(1,6,6,4; 6,3). Note that {0,3,6,9} is a short difference set and gives only three blocks. These bocks cover difference 3 and 6 pairs only once. Hence, we also have a GDD(1,6,6,4;m + 6t,m + 3t) for any nonnegative integers m and t. Example 6.5. The following multiset {{∞,0,1,5},{∞,0,2,3},{∞,0,3,4},{0,1,2,5},{0,1,4,5}, {0,1,7,10},{0,2,7,8},{0,3,5,10},{0,3,6,9},{0,3,6,9}} is a difference family for a GDD(1,6,6,4; 6,9). Hence, we also have a GDD(1,6,6,4;m + 6t,m + 9t) for any nonnegative integers m and t. Example 6.6. A GDD(1,6,6,4; 4,1) can be constructed by difference family: {{∞,0, 4,8},{0,1,4,6}, {0,2,4,6}}. 98 D. G. Sarvate, D. M. Woldemariam / J. Algebra Comb. Discrete Appl. 9(2) (2022) 85–99 Example 6.7. The difference family for a GDD(1,6,6,4; 5,2) is {{∞,0,4,8},{∞,0,4,8},{0,1,6,7}, {0,2,4,5},{0,2,4,10},{0,3,6,9} where G1={∞}, G2={1,3,5,7,9,11} and G3={0,2,4,6,8,10}. So we have a GDD(1,6,6,4; 3t + 1,3s + 1) for t > s + 1, GDD(1,6,6,4; 3t + 2,3s + 2) for t > s + 1, GDD(1,6,6,4; 6t + 1,6s + 1), GDD(1,6,6,4; 6t + 1,6s + 1) and GDD(1,6,6,4; 6t + 3,6s + 3) for t ≥ s. 6.3. n = 10 Example 6.8. G1 = {∞}, G2 = {1,3,5, ...,19} and G3 = {0,2,4, ...,18}. The difference family {{∞,0,3,6},{0,1,3,7},{0,1,8,9},{0,5,7,16},{0,5,10,15}, {0,5,10,15}} provides a GDD(1,10,10,4; 2,3). 7. Summary We used interesting construction techniques to construct specific examples for GDDs and obtained an important general construction for GDDs with three groups of sizes 1,n,n with block size 4. We obtained necessary conditions for the existence of these GDDs and proved that they are sufficient for specific values of n, specific values of λ1 and for the existence of a GDD(1,n,n,4;λ1,λ2) for n ≡ 0,1,4,5,8,9 (mod 12) when λ1 ≥ λ2. The work leads to several open problems including questions on the existence of unknown families of GDDs. Acknowledgment: The first author thanks Professor Jurisich for support through departmental research funds. The second author was supported by Adama Science and Technology University research grant number ASTU/SP-R/011/19. References [1] C. J. Colbourn, D. G. Hoffman, R. Rees, A new class of group divisible designs with block size three, J. Combin. Theory Ser. A 59(1) (1992) 73–89. [2] H. L. Fu, C. A. Rodger, Group divisible designs with two associate cases: n = 2 or m = 2, J. Combin. Theory Ser. A 83(1) (1998) 94–117. [3] H. L. Fu, C. A. Rodger, D. G. Sarvate, The existence of group divisible designs with first and second associates having block size three, Ars Combin. 54 (2000) 33–50. [4] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11 (1975) 225–369. [5] W. Lapchinda, N. Punnim, N. Pabhapote, GDDs with two associate classes with three groups of sizes 1, n and n, Australas. J. Combin. 58(2) (2014) 292–303. [6] C. C. Lindner, C. A. Rodger, Design theory, 2nd Edition, Chapman & Hall/CRC, New York (2008). [7] N. Pabhapote, Group divisible designs with two associate classes and with two unequal groups, Int. J. Pure Appl. Math. 81(1) (2012) 191–198. [8] N. Pabhapote, N. Punim, Group divisible designs with two associate classes and λ2 = 1, Int. J. Math. Sci. (2011) 1–10. [9] A. Sakda, C. Uiyyasathian, Group divisible designs GDD(n,n,n,1;λ1,λ2), Australas. J. Comb. 69(1) (2017) 18–28. 99 https://doi.org/10.1016/0097-3165(92)90099-g https://doi.org/10.1016/0097-3165(92)90099-g https://doi.org/10.1006/jcta.1998.2868 https://doi.org/10.1006/jcta.1998.2868 https://mathscinet.ams.org/mathscinet-getitem?mr=1742405 https://mathscinet.ams.org/mathscinet-getitem?mr=1742405 https://doi.org/10.1016/0012-365X(75)90040-0 https://mathscinet.ams.org/mathscinet-getitem?mr=3211784 https://mathscinet.ams.org/mathscinet-getitem?mr=3211784 https://doi.org/10.1201/9781315107233 https://ijpam.eu/contents/2012-81-1/16/16.pdf https://ijpam.eu/contents/2012-81-1/16/16.pdf https://www.hindawi.com/journals/ijmms/2011/148580/ https://www.hindawi.com/journals/ijmms/2011/148580/ https://mathscinet.ams.org/mathscinet-getitem?mr=3684654 https://mathscinet.ams.org/mathscinet-getitem?mr=3684654 Introduction A new construction of a GDD(1,n,n,4;1, 2) Necessary conditions Existence of families of GDD(1,n,n,4;1, 2) Specific GDDs Difference families constructions Summary References