Matching divisible designs with block size four

Discrete Mathematics 339 (2016) 790–799

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Matching divisible designs with block size four Peter J. Dukes a,∗ , Tao Feng b , Alan C.H. Ling c a

Mathematics and Statistics, University of Victoria, Victoria, Canada

b

Mathematics, Beijing Jiao Tong University, Beijing, PR China

c

Computer Science, University of Vermont, Burlington, VT, USA

article

info

Article history: Received 14 May 2015 Received in revised form 4 September 2015 Accepted 7 October 2015 Available online 11 November 2015 Keywords: Group divisible design Graph factorization Edge-decomposition

abstract We consider edge-decompositions of the graph join of several equal-sized one-factors into cliques of a prescribed size. These objects are variants of group divisible designs and have applications to packings, coverings, and embeddings. Assuming block (clique) size four, we show that the obvious divisibility and counting conditions are sufficient for the existence of such designs. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Suppose T is a list of (simple, undirected) graphs G1 , G2 , . . . , Gu on a total of v vertices. Following [5], a graph divisible design of type T and block size k is a triple (V , Π , B ) so that

• V is a set of v points; • Π = {V1 , . . . , Vu } is a partition of V into groups with Vi = V (Gi ) for each i; and • B ⊆ Vk is a set of blocks such that (1) two points x ̸= y in the same Vi appear together in a block if and only if xy ∈ E (Gi ), (2) any two points from distinct groups appear together in exactly one block. A graph divisible design as above is equivalent to an edge-decomposition of the join G1 + · · · + Gu into cliques Kk . Note that this is not the same as a ‘group divisible covering design’ (see [6]), which has excess coverage across groups rather than within groups. Here, even the case of one group u = 1 is of considerable interest, and only partial existence results are known; see [2]. When all the Gi are edgeless graphs, one recovers as a special case a group divisible design. The type T can now be written as a list of integers, say g1 , . . . , gu , instead of graphs. In case all gi = 1 (and u = v ), we have a (v, k, 1)-balanced incomplete block design, or BIBD for short. Let us use the abbreviation GDD(T , k) (or simply GDD) when referring to both group and graph divisible designs. It is standard to use ‘exponential notation’ to compress the type T of a GDD. For instance, a uniform GDD, in which all groups u u have size g, has its type abbreviated g u . More generally, we can have type g1 1 · · · gt t , representing ui groups of size gi , u u i = 1, . . . , t. In the case of graph divisible designs, we use similar notation such as Gu or G11 · · · Gt t . Mixing integers and

∗

Corresponding author. E-mail addresses: [email protected] (P.J. Dukes), [email protected] (T. Feng), [email protected] (A.C.H. Ling).

http://dx.doi.org/10.1016/j.disc.2015.10.011 0012-365X/© 2015 Elsevier B.V. All rights reserved.

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graphs in the notation should cause no confusion. As one example, if L is a graph on l vertices (none of which are isolates) then a GDD(1v−l (L)1 , k) is also known as a (v, k, 1)-packing with leave L. There are numerical restrictions on the parameters of a GDD (or any edge-decomposition of graphs). Globally, the number of pairs of points to be covered must be a multiple of the number of pairs covered by each block. Locally, the degrees of each vertex must partition into (k − 1)-element neighbourhoods induced by each block. We state these conditions for the uniform case. Proposition 1.1 ([5]). The existence of a GDD(Gu , k) implies

|V (G)|(u − 1) + degG (x) ≡ 0 (mod k − 1) for all x ∈ V (G) and |V (G)|2 u(u − 1) + 2|E (G)|u ≡ 0 (mod k(k − 1)). In particular, note that all vertex degrees in G are in the same congruence class mod k − 1. Let Mn be a perfect matching (also known as a one-factor) on 2n vertices. The necessary conditions for existence of a uniform ‘matching divisible design’, or GDD(Mnu , k), are k − 1 | 2n(u − 1) + 1 and k(k − 1) | 4n2 u(u − 1) + 2nu. We see from the first of these conditions that k must be even. In the first case of interest, k = 4, the conditions simplify to n(u − 1) ≡ 1 (mod 3) and 2 | un. This leads to four separate classes for consideration:

• • • •

u u u u

≡ 0 (mod 6) and n ≡ 2 (mod 3); ≡ 2 (mod 6) and n ≡ 1 (mod 3); ≡ 3 (mod 6) and n ≡ 2 (mod 6); ≡ 5 (mod 6) and n ≡ 4 (mod 6).

It is possible to rule out the case u = 3. Proposition 1.2. There is no GDD(Mn3 , 4) for any n ≡ 2 (mod 6). Proof. Assume there is a GDD(Mn3 , 4) defined on points {1, 2, 3} × V (Mn ) with groups {i} × V (Mn ), i = 1, 2, 3. Each of its blocks must be of the form {(i, ∗), (i, ∗), (j, ∗), (k, ∗)} or {(i, ∗), (i, ∗), (j, ∗), (j, ∗)}, where i, j, k are distinct. Let the number of blocks of these forms be denoted by a and b, respectively. Then, from simple counting we have a + b = 2n2 + n/2 and a + 2b = 3n. So b = 5n/2 − 2n2 < 0, a contradiction.  In this article, we settle the existence question for GDD(Mnu , 4). The following is our main result. Theorem 1.3. A GDD(Mnu , 4) exists if and only if u ≥ 4, n(u − 1) ≡ 1 (mod 3) and 2 | un. In the next section, we consider holey group divisible designs and adaptations of some standard constructions, including ‘filling’ and ‘inflation’ constructions. This leads us to some partial results and important ingredients for our proof of Theorem 1.3. Section 3 combines these ideas and designs to complete the proof. Many small examples are needed and these are included in Appendix. As one consequence of our result, we obtain a wide variety of leaves for packings with block size four. This discussion occurs in Section 4. 2. HGDDs and constructions We assume a basic familiarity with the standard design-theoretic constructions (two references are Stinson’s book, [10], and an early paper of Wilson, [12]). On the other hand, what we use is intuitive from the inherently ‘additive’ nature of graph decompositions. For example, if there exists a Kk -decomposition of some graph G with an independent set I, and if I is the vertex set of another Kk -decomposable graph, say GI , then we obtain a Kk -decomposition of the (edge) union G ∪ GI by decomposing each separately. This is the key idea behind ‘filling’ constructions in design theory. As a warm-up and application of this principle, observe that we can fill GDDs with matching divisible designs to create larger ones. u

l t1 t2 ts Construction 2.1. Suppose there is a GDD(( 1 ) (2nu2 ) · · · (2nus ) , k). If there exists a GDD(Mn , k) for each 1 ≤ l ≤ s, 2nu s u then there exists a GDD(Mn , k), where u = l=1 ul tl .

It is helpful in what follows to consider a variation of group divisible designs. A holey group divisible design is a quadruple

(V , Π , Ξ , B ), where  • V is a set of u ti=1 mi points;  • Π = {V1 , . . . , Vu } is a partition of V into groups of size ti=1 mi ; • Ξ = {W1 , . . . , Wt } is a partition of V into holes, such that |Wi | = umi and |Wi ∩ Vj | = mi for any 1 ≤ j ≤ u; • B ⊆ Vk is a set of blocks which meet each group and each hole in at most one point; and • any two points from distinct groups and distinct holes appear together in exactly one block.

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P.J. Dukes et al. / Discrete Mathematics 339 (2016) 790–799 t

t

If Ξ contains tl holes of size uml , 1 ≤ l ≤ s, the above design is abbreviated as an HGDD of type u × m11 m22 · · · mtss . To t t indicate the block size, we write HGDD(u × m11 m22 · · · mtss , k). Note that a ‘uniform’ HGDD(u × mt , k) is equivalent upon u swapping groups and holes to an HGDD(t × m , k). There is a complete existence theory for uniform HGDDs in the case of block size four. Lemma 2.2 ([3]). There exists an HGDD(u × mt , 4) if and only if u, t ≥ 4 and (u − 1)(t − 1)m ≡ 0 (mod 3) except for m = 1 and {u, t } = {4, 6}. Because a vertex disjoint union of one-factors is another one-factor, we can fill the holes of an HGDD with matching divisible designs to create larger ones. Construction 2.3. Suppose there is an HGDD(u × (2n1 )t1 (2n2 )t2 · · · (2ns )ts , k). If there exists a GDD(Mnul , k) for each 1 ≤ l ≤ s,

then there exists a GDD(Mnu , k), where n =

s

l =1

nl t l .

This generalizes the observation in [5], which treated only the case s = 1 and n1 = 1. There, the design being filled was simply a BIBD(2u, k). That special case and Lemma 2.2 with m = 2 are actually enough to completely settle existence in one of our four main classes. Lemma 2.4 ([5]). There exists a GDD(Mnu , 4) for all u ≡ 2 (mod 6), u ≥ 8, and n ≡ 1 (mod 3). There are two special cases of HGDDs which we define for later use. A modified group divisible design, or MGDD(u × t , k), is an HGDD(u × 1t , k). These are also known in the literature as ‘grid designs’. A holey transversal design, or HTD(k, mt ), is an HGDD(u × mt , k) with minimum number u = k of groups. This is equivalent to k − 2 mutually orthogonal latin squares of order n = mt missing t disjoint m × m subsquares on the diagonal. In particular, the case m = 1 leads to idempotent squares. The following is a straightforward generalization of [11, Construction 3.1]. t

t

Construction 2.5. Suppose that there exists a GDD(m11 m22 · · · mtss , k) and an HGDD(u × hk , k). Then there exists an HGDD(u × (hm1 )t1 (hm2 )t2 · · · (hms )ts , k). We use this to create some useful HGDDs with block size four and unequal hole sizes. Lemma 2.6. There exists an HGDD(u × 4t 101 , 4) for all u ≡ 0 (mod 6) and t ≡ 0 (mod 3), t ≥ 6 except possibly for (u, t ) = (6, 6). Proof. When u = 6, take a GDD(2t 51 , 4) for t ≡ 0 (mod 3) and t ≥ 9 (from [7, Theorem 4.9(1)]). Then apply Construction 2.5 with an HGDD(6 × 24 , 4). When u ≡ 0 (mod 6) and u ≥ 12, take a GDD(4t 101 , 4) for t ≡ 0 (mod 3) and t ≥ 6 (from Theorem 4.9(2) in [7]). Then apply Construction 2.5 with an MGDD(u × 4, 4).  Lemma 2.7. There exists an HGDD(u × 8t 201 , 4) for all u ≡ 5 (mod 6) and t ≡ 0 (mod 3), t ≥ 6. Proof. Take a GDD(8t 201 , 4) for t ≡ 0 (mod 3) and t ≥ 6 (from Theorem 3.10 in [9]). Then apply Construction 2.5 with an MGDD(u × 4, 4).  We close this section by constructing some HGDDs via certain holey latin squares. Lemma 2.8. There exists an HGDD(u × 2t 101 , 4) for all u ∈ {4, 10, 22} and t ≡ 0 (mod 6), t ≥ 12. Proof. For u = 4, such an HGDD is just an HTD and can be constructed from a holey self-orthogonal latin square of type 2t 101 , which can be found in [13]. For u = 10, 22, take a GDD(2u , 4), give weight t + 5 to each point, and replace blocks by copies of an HTD(4, 1t 51 ). For t ≥ 12, this HTD is similarly constructed from an incomplete self-orthogonal latin square; see [1].  3. Proof of Theorem 1.3 We consider first the two cases in which 3 | u. A natural starting point is n = 2, for which each group has size four with two disjoint edges. Lemma 3.1. There exists a GDD(M2u , 4) for all u ≡ 0 (mod 3) and u ≥ 6. Proof. When u = 6, 9, 12, 15, 18, 21, 27, see Appendix. Otherwise, we apply Construction 2.1. When u ≡ 0 (mod 6) and u ≥ 24, start from a GDD(24u/6 , 4) and fill in each group with a GDD(M26 , 4). When u ≡ 3 (mod 6) and u ≥ 33, take a GDD(6(u−9)/6 91 , 4) (from [7, Theorem 4.9(4)]) and give weight 4 to each point to obtain a GDD(24(u−9)/6 361 , 4); then fill in groups with GDD(M26 , 4) and GDD(M29 , 4).  With this, uniform HGDDs can be used to finish off another congruence class for u in a similar spirit as Lemma 2.4.

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Lemma 3.2. There exists a GDD(Mnu , 4) for all u ≡ 0 (mod 3), u ≥ 6 and n ≡ 2 (mod 6). Proof. The case n = 2 follows from Lemma 3.1. Assume that n ≡ 2 (mod 6) and n ≥ 8. Take an HGDD(u × 4n/2 , 4), which exists by Lemma 2.2, and fill in each hole with a GDD(M2u , 4).  With some more small examples and HGDDs, we obtain yet another of our four cases. Lemma 3.3. There exists a GDD(Mnu , 4) for all u ≡ 0 (mod 6) and n ≡ 5 (mod 6). Proof. When n = 5, 11 and u = 6, 12, 18, see Appendix. When n = 5, 11 and u ≡ 0 (mod 6), u ≥ 24, start from a GDD((12n)u/6 , 4) and fill in groups with GDD(Mn6 , 4). Suppose now that n ≥ 17. For (n, u) = (17, 6), see Appendix. In all other cases, take an HGDD(u × 4(n−5)/2 101 , 4) from Lemma 2.6 and fill in holes with GDD(M2u , 4) (from Lemma 3.1) and GDD(M5u , 4).  At this stage, the only remaining case is u ≡ 5 (mod 6), but this is perhaps the most challenging. Lemma 3.4. Let n ∈ {4, 10, 22}. There exists a GDD(Mnu , 4) for all u ≡ 5 (mod 6). Proof. For u ∈ {5, 11}, see Appendix. Otherwise, we write u = 6m + 5 for m ≥ 2. Take an HGDD(n × 26m 101 , 4) from Lemma 2.8 (beware of the change in notation) and fill each of its n groups with GDD(112m 101 , 4); these exist by [7, Theorem 4.8(1)]. The result is a graph divisible design GDD(Mn6m (10n)1 , 4), and the latter group can be filled with GDD(Mn5 , 4).  Now, from these base cases, we can extend to all admissible values of n. Lemma 3.5. There exists a GDD(Mnu , 4) for all u ≡ 5 (mod 6) and n ≡ 4 (mod 6). Proof. First, suppose n ≡ 4 (mod 12). The case n = 4 follows from Lemma 3.4. For n ≥ 16, take an HGDD(u × 8n/4 , 4) (from Lemma 2.2) and fill in each hole with a GDD(M4u , 4). Next, suppose n ≡ 10 (mod 12). The cases n = 10, 22 again follow from Lemma 3.4. For n ≥ 34, take an HGDD(u × 8(n−10)/4 201 , 4) (from Lemma 2.7) and fill holes using GDD(M4u , 4) u and GDD(M10 , 4).  Theorem 1.3 follows as a combination of Lemmas 2.4, 3.2, 3.3 and 3.5 and the remarks of Section 1. 4. Applications to K4 -decompositions and packings In [5], graph divisible designs were introduced to give explicit constructions for a few challenging edge-packing problems in which a prescribed leave graph is desired. For example, the graph divisible design GDD(1v−9
1 , 5) furnishes an optimal packing of edge-disjoint blocks of size 5 on v ≡ 13 (mod 20) points, where
denotes the Cartesian product K3 K3 (which is self-complementary). In a similar spirit, we discuss packings with block size four and obtain a variety of interesting leave graphs. First, we can easily get a noteworthy minimal leave in the case of (v, 4, 1)-packings with v ≡ 0 (mod 12). By Lemma 3.1, a GDD(M2u , 4) exists for all u ≡ 0 (mod 3), u ≥ 6. This implies a (4u, 4, 1)-packing whose leave is a disjoint union of u copies of C4 . The ‘standard’ construction of optimal packings in this case is different, starting from a (4u + 1, 4, 1)-BIBD and deleting a point to obtain a leave of triangles. In general, despite our complete knowledge of packing numbers for block size four, it seems to be a difficult problem to classify all leaves of such packings. This is already good evidence of the utility of matching divisible designs. Let us recall a familiar construction for group divisible designs. Here, points of a ‘master’ GDD get weighted and blocks get replaced with appropriate ‘ingredient’ GDDs. Construction 4.1 (Wilson’s Fundamental Construction, [12]). Take a group divisible design with points V , group partition

Π = {V1 , . . . , Vu }, and blocks B . Let ω : V → Z≥0 , assigning nonnegative weights to each point. For B ∈ B , define TB := [ω(x) : x ∈ B] and assume that there exists a GDD(TB , K ) for every B ∈ B . Then there exists a GDD(T , K ), where    T = ω(x) : i = 1, . . . , u . x∈Vi

In fact, we have implicitly used this construction when weighting and replacing blocks, such as in Construction 2.5. In this case, though, we have used HGDDs as ingredients instead of GDDs. Something similar can be done with graph divisible designs, except that groups ‘fill up’ with copies of the graphs in the ingredients. Construction 4.2. Take a group divisible design with points V , group partition Π = {V1 , . . . , Vu }, and blocks B . Let ω : V → Z≥0 and, for every x ∈ V , assign a family of graphs {GBx : B ∈ x, B ∈ B } on a common set of ω(x) vertices. For x ∈ V , define Gx =

 B∈x

GBx .

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If, for each B ∈ B , there exists a GDD([GBx : x ∈ B], k), then there exists a GDD(T , k), where



  ˙ T = Gx : i = 1, . . . , u . x∈Vi

A proof sketch and generalization can be found in [5]. It is important to remark that there is great flexibility in forming the graph divisible design type, since we are free to permute copies of GBx at x. For instance, Construction 4.2 was used with a TD(8, n) and a GDD(Mn8 , 4) to provide efficient embeddings of partial designs into designs with block size four. Our main result now gives an improvement using fewer groups. In general, this construction is useful when the goal is a spanning leave having equal-sized components. Quite often, Construction 4.1 is coupled with the filling in groups construction, discussed and used earlier in Section 2. We draw attention to one standard variant, in which a ‘hole’ is pulled out of the designs on each group. Construction 4.3 (Filling in Groups and a Hole). Suppose there exists a GDD([g1 , . . . , gu ], k) and, for each i = 1, . . . , u, a GDD(1gi h1 , k). Then there exists a GDD(1a h1 , k), where a = g1 + · · · + gu . Now, instead of merging holes of size h, we can merge copies of a given graph. Construction 4.4. Suppose there exists a GDD([g1 , . . . , gu ], k) and, for each i, a GDD(1gi Hi1 , k), where the Hi are graphs on a common set of vertices V (H1 ). Then there exists a GDD(1a H 1 , k), where a = g1 + · · · + gu and H is the graph with V (H ) = V (H1 ) and E (H ) = ∪ui=1 E (Hi ). We offer an example application of the above construction in the case k = 4. Proposition 4.5. Let L be an r-regular graph on 2n vertices whose complement L is 1-factorable. Assume 3 | r and n ≡ 4 (mod 6), n ≥ 10. Then there exists a (v, 4, 1)-packing with leave L for v = 16n2 − 6n − 8nr. Proof. Start with a GDD(Mn5 , 4), which exists by Theorem 1.3. Fill four of its groups with copies of a GDD(2n , 4), aligning groups with 1-factors. The result is a GDD(18n Mn1 , 4). Now, put u := 2n − 1 − r ≡ 1 (mod 3); this is the degree of L. Working from a GDD((8n)u , 4), add a set V (L) of 2n points and apply Construction 4.4 with u copies of a GDD(18n Mn1 , 4). We ensure that 1-factors Mn are placed on V (L) based on a 1-factorization of L. We then have a GDD(18nu (L)1 , 4), and the total number of points is easily computed to be the given value of v . 

√

Remarks. The hypothesis that L be 1-factorable is guaranteed when r < (3 − 7)n − 1, using a result from [4,8]. That r be a multiple of three is necessary for packings with bounded leave. However, our conditions on n are used to facilitate the technique. To treat other congruence classes for n, one can use GDD(Mn8 , 4), which exists for all n ≡ 1 (mod 3), n ≥ 4. It is also possible to artificially add a small K4 -decomposable r-regular graph to L using a linear, r /3-regular, 4-uniform hypergraph (i.e. combinatorial configuration). This can shift n into other congruence classes. These operations increase the resulting value for v , but not substantially. We are not able to handle certain cases when L has an odd number of vertices, at least not directly. In the new paper [2], it is shown that edge-decompositions into K4 are possible for all very large K4 -divisible (number of edges a multiple of 6 and each degree a multiple of 3) graphs H of minimum degree at least 0.9996|V (H )|. In fact, that paper treats Kk -decompositions in general. Although for large r our graph Kv \L satisfies this minimum degree condition, there is no need for it to be very large. Our construction is also explicit, with no swapping of blocks. Unfortunately, most K4 -divisible graphs fall outside of our present technique. Enriching the technique is potentially of some interest when the goal is an explicit decomposition. Finally, we note that applying the construction in Proposition 4.5 with negative values of r (so that u ≥ 2n) can be used to construct various explicit coverings of Kv by blocks of size four with regular ‘excess graphs’. We omit the details but encourage further development of this technique. Acknowledgements Research of Peter Dukes is supported by NSERC grant 312595–2010; research of Tao Feng is supported by the NSFC grants 11271042 and 11471032. Appendix. Small examples For a positive integer m, we abbreviate {0, 1, . . . , m − 1} by Zm or [m], with the former indicating that a cyclic group of this order is acting.

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GDD(M26 , 4): V = Z3 × [8]; Π = {{(j, i), (j, i + 2), (j, i + 4), (j, i + 6)} : i = 0, 1, j ∈ Z3 }; the perfect matching is {{(j, i), (j, i + 2)}, {(j, i + 4), (j, i + 6)}} for i = 0, 1 and j ∈ Z3 ; B is generated by developing 14 base blocks by (+1 mod 3, −): {(0, 0), (1, 0), (0, 1), (0, 2)} {(0, 0), (1, 4), (0, 5), (2, 7)} {(0, 1), (1, 1), (2, 2), (0, 6)} {(0, 1), (2, 4), (1, 6), (1, 7)} {(0, 2), (2, 3), (1, 5), (2, 6)}

{(0, 0), (1, 1), (0, 3), (1, 3)} {(0, 0), (1, 5), (2, 5), (1, 6)} {(0, 1), (1, 3), (2, 5), (2, 7)} {(0, 2), (1, 2), (0, 5), (1, 7)} {(0, 3), (1, 4), (1, 6), (2, 6)}

{(0, 0), (1, 2), (2, 3), (2, 4)} {(0, 0), (2, 6), (0, 7), (1, 7)} {(0, 1), (0, 4), (1, 4), (1, 5)} {(0, 2), (0, 3), (2, 4), (2, 7)}

GDD(M29 , 4): V = Z9 × [4]; Π = {{(j, i) : i ∈ [4]} : j ∈ Z9 }; the perfect matching is {{(j, 0), (j, 1)}, {(j, 2), (j, 3)}} for j ∈ Z9 ; B is generated by developing 11 base blocks by (+1 mod 9, −): {(0, 0), (1, 0), (3, 0), (0, 1)} {(0, 0), (4, 1), (1, 3), (5, 3)} {(0, 0), (6, 2), (2, 3), (3, 3)} {(0, 1), (2, 1), (3, 2), (7, 3)}

{(0, 0), (4, 0), (2, 1), (5, 1)} {(0, 0), (3, 2), (5, 2), (8, 2)} {(0, 0), (7, 2), (6, 3), (8, 3)} {(0, 1), (4, 1), (6, 2), (8, 3)}

{(0, 0), (3, 1), (1, 2), (2, 2)} {(0, 0), (4, 2), (4, 3), (7, 3)} {(0, 1), (1, 1), (5, 2), (3, 3)}

GDD(M212 , 4): V = Z12 × [4]; Π = {{(j, i), (j + 6, i), (j, i + 1), (j + 6, i + 1)} : i = 0, 2, j ∈ [6]}; the perfect matching is {{(j, i), (j + 6, i)}, {(j, i + 1), (j + 6, i + 1)}} for i = 0, 2 and j ∈ [6]; B is generated by developing 16 base blocks by (+1 mod 12, −), where each base block with a star only generates 6 distinct blocks: {(0, 0), (6, 0), (0, 2), (6, 2)}∗ {(0, 0), (2, 0), (5, 0), (7, 1)} {(0, 0), (8, 1), (10, 1), (1, 3)} {(0, 0), (4, 2), (3, 3), (7, 3)} {(0, 1), (3, 1), (3, 2), (5, 2)} {(0, 1), (6, 2), (7, 2), (8, 3)}

{(0, 1), (6, 1), (0, 3), (6, 3)}∗ {(0, 0), (4, 0), (5, 2), (1, 1)} {(0, 0), (11, 1), (0, 3), (8, 2)} {(0, 0), (2, 3), (4, 3), (9, 2)} {(0, 1), (4, 1), (2, 3), (11, 3)}

{(0, 0), (1, 0), (6, 3), (11, 3)} {(0, 0), (3, 1), (4, 1), (2, 2)} {(0, 0), (3, 2), (7, 2), (10, 2)} {(0, 0), (11, 2), (8, 3), (9, 3)} {(0, 1), (5, 1), (1, 2), (9, 3)}

GDD(M215 , 4): V = Z15 × [4]; Π = {{(j, i) : i ∈ [4]} : j ∈ Z15 }; the perfect matching is {{(j, 0), (j, 1)}, {(j, 2), (j, 3)}} for j ∈ Z15 ; B is generated by developing 19 base blocks by (+1 mod 15, −): {(0, 0), (1, 0), (12, 2), (3, 3)} {(0, 0), (4, 0), (1, 3), (13, 1)} {(0, 0), (7, 0), (9, 2), (3, 1)} {(0, 0), (6, 2), (10, 3), (7, 2)} {(0, 1), (2, 1), (9, 1), (10, 2)} {(0, 1), (2, 2), (9, 2), (12, 2)} {(0, 2), (6, 2), (8, 3), (11, 3)}

{(0, 0), (2, 0), (5, 2), (1, 2)} {(0, 0), (5, 0), (0, 1), (11, 3)} {(0, 0), (6, 1), (8, 3), (4, 3)} {(0, 0), (8, 2), (10, 2), (5, 3)} {(0, 1), (3, 1), (7, 2), (7, 3)} {(0, 1), (11, 2), (9, 3), (10, 3)}

{(0, 0), (3, 0), (4, 1), (5, 1)} {(0, 0), (6, 0), (14, 1), (4, 2)} {(0, 0), (7, 1), (12, 1), (13, 3)} {(0, 0), (7, 3), (9, 3), (14, 3)} {(0, 1), (4, 1), (3, 2), (12, 3)} {(0, 1), (13, 2), (5, 3), (14, 3)}

GDD(M218 , 4): V = Z36 × [2]; Π = {{(18i + j, l) : i, l ∈ [2]} : j ∈ [18]}; the perfect matching is {{(j, l), (18 + j, l)} : l ∈ [2]} for j ∈ [18]; B is generated by developing 13 base blocks by (+1 mod 36, −), where each base block with a star only generates 9 distinct blocks: {(0, 0), (9, 0), (18, 0), (27, 0)}∗ {(0, 0), (1, 0), (35, 1), (29, 0)} {(0, 0), (5, 0), (10, 1), (21, 1)} {(0, 0), (14, 0), (25, 1), (28, 1)} {(0, 1), (2, 1), (16, 1), (21, 1)}

{(0, 1), (9, 1), (18, 1), (27, 1)}∗ {(0, 0), (2, 0), (19, 1), (31, 1)} {(0, 0), (11, 0), (2, 1), (15, 1)} {(0, 0), (15, 0), (22, 1), (23, 1)}

{(6, 0), (10, 0), (7, 1), (0, 1)} {(0, 0), (3, 0), (13, 0), (19, 0)} {(0, 0), (12, 0), (24, 1), (32, 1)} {(0, 0), (3, 1), (9, 1), (13, 1)}

GDD(M221 , 4): V = Z21 × [4]; Π = {{(j, i) : i ∈ [4]} : j ∈ Z21 }; the perfect matching is {{(j, 0), (j, 1)}, {(j, 2), (j, 3)}} for j ∈ Z21 ; B is generated by developing 27 base blocks by (+1 mod 21, −): {(0, 0), (1, 0), (9, 3), (14, 1)} {(0, 0), (6, 0), (1, 2), (20, 3)} {(0, 0), (9, 0), (19, 1), (11, 3)} {(0, 0), (5, 1), (15, 2), (11, 1)} {(0, 0), (9, 1), (2, 2), (5, 2)} {(0, 0), (11, 2), (13, 3), (16, 3)} {(0, 1), (2, 1), (7, 1), (5, 2)} {(0, 1), (9, 1), (15, 2), (3, 3)} {(0, 1), (7, 2), (20, 2), (17, 3)}

{(0, 0), (2, 0), (20, 1), (18, 0)} {(0, 0), (7, 0), (1, 1), (10, 3)} {(0, 0), (10, 0), (7, 3), (6, 3)} {(0, 0), (6, 1), (7, 1), (5, 3)} {(0, 0), (4, 2), (8, 2), (10, 2)} {(0, 0), (17, 2), (18, 2), (4, 3)} {(0, 1), (3, 1), (5, 3), (10, 3)} {(0, 1), (10, 1), (18, 2), (11, 3)} {(0, 2), (10, 2), (0, 3), (6, 3)}

{(0, 0), (4, 0), (0, 1), (13, 2)} {(0, 0), (8, 0), (14, 2), (3, 1)} {(0, 0), (4, 1), (1, 3), (12, 3)} {(0, 0), (8, 1), (12, 1), (3, 2)} {(0, 0), (7, 2), (12, 2), (19, 2)} {(0, 0), (20, 2), (15, 3), (19, 3)} {(0, 1), (8, 1), (9, 2), (12, 3)} {(0, 1), (2, 2), (6, 3), (14, 3)} {(0, 2), (1, 3), (13, 3), (15, 3)}

GDD(M227 , 4): V = Z27 × [4]; Π = {{(j, i) : i ∈ [4]} : j ∈ Z27 }; the perfect matching is {{(j, 0), (j, 1)}, {(j, 2), (j, 3)}} for j ∈ Z27 ; B is generated by developing 35 base blocks by (+1 mod 27, −): {(0, 0), (1, 3), (17, 2), (14, 1)} {(0, 0), (3, 0), (9, 3), (12, 2)} {(0, 0), (6, 0), (26, 2), (13, 2)} {(0, 0), (9, 0), (6, 1), (14, 2)} {(0, 0), (12, 0), (20, 1), (5, 3)} {(0, 0), (9, 1), (11, 3), (15, 1)} {(0, 0), (3, 2), (12, 3), (21, 2)} {(0, 0), (17, 3), (19, 3), (23, 3)} {(0, 1), (10, 1), (5, 2), (25, 2)} {(0, 1), (2, 2), (17, 2), (24, 3)} {(0, 1), (10, 2), (13, 2), (13, 3)} {(0, 2), (5, 2), (11, 2), (25, 3)}

{(0, 0), (1, 0), (11, 2), (4, 1)} {(0, 0), (4, 0), (8, 2), (7, 3)} {(0, 0), (7, 0), (22, 2), (1, 1)} {(0, 0), (10, 0), (23, 1), (4, 3)} {(0, 0), (13, 0), (2, 1), (25, 1)} {(0, 0), (10, 1), (17, 1), (2, 2)} {(0, 0), (24, 2), (14, 3), (25, 2)} {(0, 1), (1, 1), (11, 3), (19, 1)} {(0, 1), (11, 1), (6, 3), (20, 3)} {(0, 1), (4, 2), (16, 3), (25, 3)} {(0, 1), (16, 2), (24, 2), (26, 3)} {(0, 2), (10, 2), (15, 3), (23, 3)}

{(0, 0), (2, 0), (10, 3), (18, 2)} {(0, 0), (5, 0), (0, 1), (6, 2)} {(0, 0), (8, 0), (19, 1), (7, 1)} {(0, 0), (11, 0), (2, 3), (24, 3)} {(0, 0), (5, 1), (19, 2), (23, 2)} {(0, 0), (18, 1), (22, 3), (25, 3)} {(0, 0), (15, 3), (16, 3), (26, 3)} {(0, 1), (2, 1), (5, 1), (1, 2)} {(0, 1), (13, 1), (3, 3), (18, 3)} {(0, 1), (9, 2), (11, 2), (15, 3)} {(0, 1), (20, 2), (1, 3), (21, 3)}

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GDD(M45 , 4): V = Z10 × [4]; Π = {{(j, l), (j + 5, l) : l ∈ [4]}, j ∈ [5]}; the perfect matching is {{(j, 0), (j, 1)}, {(j + 5, 0), (j + 5, 1)}, {(j, 2), (j, 3)}, {(j + 5, 2), (j + 5, 3)}} for j ∈ [5]; B is generated by developing 11 base blocks by (+1 mod 10, −): {(0, 0), (1, 0), (3, 0), (0, 1)} {(0, 0), (4, 1), (7, 3), (8, 3)} {(0, 0), (1, 3), (9, 2), (9, 3)} {(0, 1), (2, 2), (6, 3), (9, 3)}

{(0, 0), (4, 0), (2, 1), (1, 2)} {(0, 0), (6, 1), (2, 2), (3, 3)} {(0, 0), (4, 3), (6, 2), (8, 2)} {(0, 1), (4, 2), (7, 2), (8, 2)}

{(0, 0), (1, 1), (3, 1), (4, 2)} {(0, 0), (3, 2), (2, 3), (6, 3)} {(0, 1), (1, 1), (4, 1), (2, 3)}

GDD(M411 , 4): V = Z22 × [4]; Π = {{(j, l), (j + 11, l) : l ∈ [4]}, j ∈ [11]}; the perfect matching is {{(j, 0), (j, 1)}, {(j + 11, 0), (j + 11, 1)}, {(j, 2), (j, 3)}, {(j + 11, 2), (j + 11, 3)}} for j ∈ [11]; B is generated by developing 27 base blocks by (+1 mod 22, −): {(0, 0), (1, 0), (20, 1), (21, 2)} {(0, 0), (4, 0), (17, 1), (14, 1)} {(0, 0), (7, 0), (8, 2), (9, 3)} {(0, 0), (10, 0), (20, 3), (14, 3)} {(0, 0), (18, 1), (2, 2), (4, 2)} {(0, 0), (13, 2), (16, 3), (18, 3)} {(0, 1), (4, 1), (7, 3), (17, 3)} {(0, 1), (7, 2), (9, 3), (16, 2)} {(0, 1), (8, 3), (13, 2), (20, 2)}

{(0, 0), (2, 0), (21, 3), (2, 1)} {(0, 0), (5, 0), (12, 3), (13, 3)} {(0, 0), (8, 0), (14, 2), (12, 1)} {(0, 0), (5, 1), (15, 3), (9, 2)} {(0, 0), (5, 2), (10, 2), (1, 3)} {(0, 0), (17, 2), (18, 2), (17, 3)} {(0, 1), (10, 1), (2, 3), (15, 3)} {(0, 1), (1, 3), (4, 3), (18, 3)} {(0, 1), (12, 2), (16, 3), (20, 3)}

{(0, 0), (3, 0), (19, 2), (6, 3)} {(0, 0), (6, 0), (9, 1), (7, 1)} {(0, 0), (9, 0), (12, 2), (15, 1)} {(0, 0), (8, 1), (16, 1), (21, 1)} {(0, 0), (7, 2), (5, 3), (15, 2)} {(0, 1), (1, 1), (7, 1), (10, 2)} {(0, 1), (5, 2), (17, 2), (21, 2)} {(0, 1), (6, 3), (14, 2), (21, 3)} {(0, 1), (15, 2), (18, 2), (12, 3)}

GDD(M56 , 4): V = Z15 × [4]; Π = {{(3i + j, l), (3i + j, l + 2) : i ∈ [5]} : j ∈ [3], l ∈ [2]}; the perfect matching is {(3i + j, l), (3i + j, l + 2) : i ∈ [5]} for j ∈ [3] and l ∈ [2]; B is generated by developing 17 base blocks by (+1 mod 15, −): {(0, 0), (0, 1), (1, 0), (2, 1)} {(0, 0), (1, 3), (2, 3), (5, 0)} {(0, 0), (4, 3), (11, 1), (12, 1)} {(0, 0), (6, 1), (10, 1), (10, 2)} {(0, 1), (1, 2), (7, 1), (11, 3)} {(0, 1), (5, 1), (8, 2), (10, 2)}

{(0, 0), (2, 0), (0, 2), (0, 3)} {(0, 0), (4, 0), (3, 3), (8, 1)} {(0, 0), (5, 2), (8, 3), (9, 1)} {(0, 0), (7, 1), (7, 3), (9, 3)} {(0, 1), (1, 3), (5, 3), (12, 2)} {(0, 2), (1, 3), (9, 3), (11, 2)}

{(0, 0), (1, 2), (2, 2), (3, 1)} {(0, 0), (5, 1), (7, 0), (11, 2)} {(0, 0), (5, 3), (8, 2), (10, 3)} {(0, 0), (6, 3), (7, 2), (14, 2)} {(0, 1), (2, 2), (7, 2), (13, 3)}

GDD(M512 , 4): V = Z60 × [2]; Π = {{(12i + j, l) : i ∈ [5], l ∈ [2]} : j ∈ [12]}; the perfect matching is {{(12i + j, 0), (12i + j, 1)} : i ∈ [5]} for j ∈ [12]; B can be divided into two parts: the first part is generated by developing 10 base blocks by (+4 mod 60, −): {(0, l), (13, l), (58, l), (59, l)} {(0, l), (1, l), (14, l), (31, l)}

{(0, l), (29, l), (30, l), (43, l)} {(0, l), (15, l), (17, l), (46, l)}

{(0, l), (2, l), (45, l), (47, l)}

where l ∈ [2], and the second part is generated by developing 16 base blocks by (+1 mod 60, −): {(6, 0), (10, 0), (1, 0), (43, 0)} {(0, 0), (7, 0), (6, 1), (3, 1)} {(0, 0), (11, 0), (1, 1), (51, 1)} {(0, 0), (21, 0), (4, 1), (30, 1)} {(0, 0), (26, 0), (28, 1), (34, 1)} {(0, 0), (22, 1), (29, 1), (54, 1)}

{(0, 0), (3, 0), (49, 1), (41, 1)} {(0, 0), (8, 0), (28, 0), (0, 1)} {(0, 0), (16, 0), (21, 1), (58, 1)} {(0, 0), (22, 0), (15, 1), (33, 1)} {(0, 0), (7, 1), (18, 1), (45, 1)}

{(0, 0), (6, 0), (19, 1), (23, 1)} {(0, 0), (10, 0), (57, 1), (37, 1)} {(0, 0), (19, 0), (35, 1), (44, 1)} {(0, 0), (25, 0), (20, 1), (39, 1)} {(0, 0), (10, 1), (26, 1), (31, 1)}

GDD(M518 , 4): V = Z180 ; Π = {{18i + j : i ∈ [10]} : j ∈ [18]}; the perfect matching is {{18i + j, 18i + 90 + j} : i ∈ [5]} for each j ∈ [18]; B can be divided into two parts: the first part is generated by developing 5 base blocks by +4 mod 180: {0, 133, 178, 179}

{0, 89, 90, 43}

{0, 45, 2, 47}

{0, 1, 134, 91}

{0, 137, 46, 135}

and the second part is generated by developing 13 base blocks by +1 mod 180: {6, 10, 1, 43} {0, 10, 128, 70} {0, 23, 73, 102}

{0, 3, 109, 41} {0, 11, 28, 160} {0, 24, 75, 100}

{0, 6, 92, 65} {0, 13, 57, 97} {0, 26, 56, 95}

{0, 7, 19, 165} {0, 14, 49, 81}

{0, 8, 127, 63} {0, 16, 82, 103}

5 GDD(M10 , 4): V = Z5 × Z20 ; Π = {{(j, i) : i ∈ Z20 } : j ∈ Z5 }; the perfect matching is {{(j, 2l), (j, 2l + 1)} : l ∈ [10]} for j ∈ Z5 ; B is generated by developing 27 base blocks by (+1 mod 5, +4 mod 20):

{(0, 0), (1, 0), (2, 8), (3, 13)} {(0, 0), (2, 1), (1, 13), (4, 13)} {(0, 0), (3, 2), (4, 5), (2, 7)} {(0, 0), (2, 3), (3, 3), (4, 4)} {(0, 0), (2, 6), (1, 9), (3, 15)} {(0, 0), (4, 9), (3, 10), (1, 17)} {(0, 0), (1, 11), (2, 15), (4, 18)} {(0, 1), (1, 3), (4, 14), (2, 15)} {(0, 1), (3, 10), (4, 11), (2, 19)}

{(0, 0), (2, 0), (1, 7), (0, 1)} {(0, 0), (4, 1), (1, 2), (3, 5)} {(0, 0), (4, 2), (2, 18), (3, 18)} {(0, 0), (1, 4), (4, 8), (2, 14)} {(0, 0), (2, 9), (4, 11), (3, 14)} {(0, 0), (2, 10), (1, 14), (4, 17)} {(0, 1), (1, 1), (4, 6), (3, 17)} {(0, 1), (4, 3), (2, 6), (1, 19)} {(0, 1), (4, 10), (1, 11), (3, 11)}

{(0, 0), (1, 1), (2, 5), (4, 15)} {(0, 0), (2, 2), (1, 19), (3, 8)} {(0, 0), (1, 3), (3, 16), (2, 19)} {(0, 0), (1, 6), (4, 6), (2, 11)} {(0, 0), (3, 9), (1, 15), (2, 17)} {(0, 0), (4, 10), (3, 17), (1, 18)} {(0, 1), (1, 2), (2, 10), (4, 15)} {(0, 1), (4, 7), (1, 15), (3, 19)} {(0, 2), (0, 3), (4, 10), (2, 14)}

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11 GDD(M10 , 4): V = Z55 × [4]; Π = {{(11i + j, l) : i ∈ [5], l ∈ [4]} : j ∈ [11]}; the perfect matching is {{(11i + j, 0), (11i + j, 1)}, {(11i + j, 2), (11i + j, 3)} : i ∈ [5]} for j ∈ [11]; B is generated by developing 67 base blocks by (+1 mod 55, −):

{(0, 0), (4, 0), (6, 3), (24, 3)} {(0, 0), (8, 0), (37, 0), (18, 2)} {(0, 0), (0, 1), (36, 3), (35, 3)} {(0, 0), (3, 1), (38, 1), (32, 2)} {(0, 0), (7, 1), (19, 2), (13, 0)} {(0, 0), (2, 2), (29, 1), (45, 2)} {(0, 0), (5, 3), (46, 1), (47, 1)} {(0, 0), (14, 1), (30, 1), (48, 1)} {(0, 0), (12, 2), (35, 1), (37, 2)} {(0, 0), (16, 2), (31, 1), (37, 1)} {(0, 0), (21, 2), (12, 3), (26, 2)} {(0, 0), (19, 3), (27, 2), (27, 3)} {(0, 0), (42, 1), (43, 2), (37, 3)} {(0, 1), (2, 1), (5, 1), (1, 3)} {(0, 0), (1, 0), (9, 3), (24, 1)} {(0, 0), (5, 2), (40, 3), (43, 3)} {(0, 0), (25, 1), (31, 3), (41, 3)} {(0, 1), (9, 2), (26, 1), (45, 2)} {(0, 1), (7, 3), (15, 1), (24, 3)} {(0, 1), (14, 2), (37, 2), (45, 3)} {(0, 1), (21, 3), (25, 3), (52, 3)} {(0, 2), (8, 2), (21, 2), (37, 3)} {(0, 2), (9, 3), (28, 3), (34, 3)}

{(0, 0), (5, 0), (12, 0), (53, 2)} {(0, 0), (9, 0), (19, 1), (10, 3)} {(0, 0), (1, 1), (3, 3), (26, 1)} {(0, 0), (4, 1), (27, 1), (19, 0)} {(0, 0), (9, 1), (45, 1), (28, 0)} {(0, 0), (3, 2), (4, 2), (34, 1)} {(0, 0), (21, 0), (53, 1), (34, 2)} {(0, 0), (15, 1), (30, 2), (43, 1)} {(0, 0), (14, 2), (29, 3), (28, 2)} {(0, 0), (17, 2), (23, 0), (15, 3)} {(0, 0), (13, 3), (48, 3), (25, 3)} {(0, 0), (29, 2), (41, 1), (54, 1)} {(0, 0), (35, 2), (39, 2), (53, 3)} {(0, 1), (7, 1), (3, 2), (5, 3)} {(0, 0), (2, 0), (23, 3), (54, 2)} {(0, 0), (8, 2), (14, 3), (28, 1)} {(0, 1), (10, 1), (5, 2), (18, 3)} {(0, 1), (10, 2), (26, 2), (41, 2)} {(0, 1), (12, 1), (38, 3), (43, 3)} {(0, 1), (18, 2), (23, 3), (46, 2)} {(0, 1), (27, 2), (37, 3), (53, 2)} {(0, 2), (4, 3), (25, 3), (38, 2)}

{(0, 0), (6, 0), (45, 3), (18, 1)} {(0, 0), (10, 0), (6, 1), (40, 0)} {(0, 0), (2, 1), (23, 2), (25, 2)} {(0, 0), (5, 1), (17, 3), (52, 1)} {(0, 0), (1, 2), (16, 0), (7, 2)} {(0, 0), (9, 2), (4, 3), (52, 3)} {(0, 0), (13, 1), (28, 3), (30, 3)} {(0, 0), (16, 1), (24, 2), (20, 1)} {(0, 0), (15, 2), (50, 1), (34, 3)} {(0, 0), (20, 2), (24, 0), (50, 3)} {(0, 0), (18, 3), (50, 2), (47, 2)} {(0, 0), (31, 2), (39, 1), (38, 2)} {(0, 0), (42, 2), (38, 3), (54, 3)} {(0, 1), (9, 1), (6, 2), (3, 3)} {(0, 0), (3, 0), (17, 0), (49, 3)} {(0, 0), (7, 3), (20, 0), (16, 3)} {(0, 1), (7, 2), (10, 3), (17, 2)} {(0, 1), (4, 3), (17, 1), (31, 1)} {(0, 1), (13, 2), (31, 2), (30, 3)} {(0, 1), (19, 3), (34, 3), (48, 3)} {(0, 1), (29, 3), (39, 2), (48, 2)} {(0, 2), (7, 3), (20, 3), (35, 2)}

6 GDD(M11 , 4): V = Z132 ; Π = {{6i + j : i ∈ [22]} : j ∈ [6]}; the perfect matching is {{12i + j, 12i + 66 + j} : i ∈ [11]} for j ∈ [6]; B can be divided into two parts: the first part is generated by developing 5 base blocks by +4 mod 132:

{0, 97, 130, 131}

{0, 65, 66, 31}

{0, 33, 2, 35}

{0, 1, 98, 67}

{0, 101, 34, 99}

and the second part is generated by developing 8 base blocks by +1 mod 132: {0, 13, 3, 17} {0, 20, 46, 93}

{0, 5, 16, 37} {0, 22, 62, 91}

{0, 7, 45, 88} {0, 23, 50, 75}

{0, 8, 61, 76}

{0, 9, 28, 83}

12 GDD(M11 , 4): V = Z132 × [2]; Π = {{(12i + j, l) : i ∈ [11], l ∈ [2]} : j ∈ [12]}; the perfect matching is {{(12i + j, 0), (12i + j, 1)} : i ∈ [11]} for j ∈ [12]; B can be divided into two parts: the first part is generated by developing 10 base blocks by (+4 mod 132, −):

{(0, l), (97, l), (130, l), (131, l)} {(0, l), (1, l), (67, l), (98, l)}

{(0, l), (31, l), (65, l), (66, l)} {(0, l), (34, l), (99, l), (101, l)}

{(0, l), (2, l), (33, l), (35, l)}

where l ∈ [2], and the second part is generated by developing 38 base blocks by (+1 mod 132, −): {(6, 0), (10, 0), (1, 0), (120, 1)} {(0, 0), (7, 0), (37, 0), (45, 1)} {(0, 0), (11, 0), (52, 1), (85, 0)} {(0, 0), (16, 0), (105, 1), (51, 1)} {(0, 0), (19, 0), (61, 1), (106, 1)} {(0, 0), (22, 0), (43, 1), (3, 1)}

{(0, 0), (3, 0), (131, 1), (17, 1)} {(0, 0), (8, 0), (30, 1), (50, 0)} {(0, 0), (13, 0), (75, 1), (23, 1)} {(0, 0), (17, 0), (37, 1), (32, 1)} {(0, 0), (20, 0), (118, 1), (59, 1)} {(0, 0), (23, 0), (127, 1), (124, 1)}

{(0, 0), (27, 0), (86, 0), (25, 1)} {(0, 0), (38, 0), (27, 1), (31, 1)} {(0, 0), (43, 0), (87, 0), (50, 1)} {(0, 0), (62, 0), (16, 1), (67, 1)} {(0, 0), (40, 1), (63, 1), (79, 1)} {(0, 0), (49, 1), (76, 1), (91, 1)} {(0, 1), (10, 1), (38, 1), (79, 1)}

{(0, 0), (28, 0), (94, 1), (83, 1)} {(0, 0), (39, 0), (80, 0), (29, 1)} {(0, 0), (56, 0), (26, 1), (69, 1)} {(0, 0), (64, 0), (70, 1), (92, 1)} {(0, 0), (44, 1), (65, 1), (115, 1)} {(0, 0), (57, 1), (74, 1), (82, 1)} {(0, 1), (14, 1), (46, 1), (76, 1)}

{(0, 0), (6, 0), (0, 1), (55, 0)} {(0, 0), (10, 0), (64, 1), (63, 0)} {(0, 0), (14, 0), (29, 0), (107, 1)} {(0, 0), (18, 0), (9, 1), (2, 1)} {(0, 0), (21, 0), (109, 1), (75, 0)} {(0, 0), (25, 0), (51, 0), (4, 1)} {(0, 0), (32, 0), (100, 1), (129, 1)} {(0, 0), (40, 0), (11, 1), (58, 1)} {(0, 0), (61, 0), (80, 1), (117, 1)} {(0, 0), (33, 1), (46, 1), (90, 1)} {(0, 0), (47, 1), (53, 1), (73, 1)} {(0, 1), (9, 1), (58, 1), (77, 1)}

18 GDD(M11 , 4): V = Z396 ; Π = {{18i + j : i ∈ [22]} : j ∈ [18]}; the perfect matching is {{18i + j, 18i + 198 + j} : i ∈ [11]} for j ∈ [18]; B can be divided into two parts: the first part is generated by developing 5 base blocks by +4 mod 396:

{0, 97, 394, 395}

{0, 197, 198, 295}

{0, 297, 2, 299}

{0, 1, 98, 199}

{0, 101, 298, 99}

and the second part is generated by developing 30 base blocks by +1 mod 396: {0, 17, 105, 96} {0, 7, 73, 165} {0, 16, 150, 173} {0, 29, 100, 184} {0, 37, 113, 254} {0, 47, 130, 186}

{0, 3, 65, 55} {0, 8, 148, 19} {0, 20, 111, 189} {0, 30, 154, 194} {0, 38, 80, 281} {0, 48, 118, 171}

{0, 4, 25, 181} {0, 12, 39, 149} {0, 22, 174, 68} {0, 32, 117, 192} {0, 43, 103, 190} {0, 51, 133, 271}

{0, 5, 183, 109} {0, 13, 28, 135} {0, 24, 170, 93} {0, 33, 161, 196} {0, 44, 94, 253} {0, 57, 121, 188}

{0, 6, 120, 151} {0, 14, 182, 63} {0, 26, 112, 193} {0, 34, 95, 136} {0, 45, 172, 230} {0, 59, 175, 264}

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6 GDD(M17 , 4): V = Z204 ; Π = {{6i + j : i ∈ [34]} : j ∈ [6]}; the perfect matching is {{6i + j, 6i + 102 + j} : i ∈ [17]} for j ∈ [6]; B can be divided into two parts: the first part is generated by developing 5 base blocks by +4 mod 204:

{0, 49, 202, 203}

{0, 101, 102, 151}

{0, 153, 2, 155}

{0, 1, 50, 103}

{0, 53, 154, 51}

and the second part is generated by developing 13 base blocks by +1 mod 204: {0, 3, 43, 11} {0, 10, 79, 137} {0, 17, 63, 122}

{0, 4, 185, 26} {0, 13, 94, 65} {0, 25, 86, 117}

{0, 5, 100, 170} {0, 14, 35, 142} {0, 37, 75, 130}

{0, 7, 140, 27} {0, 15, 88, 121}

{0, 9, 157, 89} {0, 16, 44, 163}

5 GDD(M22 , 4): V = Z55 × [4]; Π = {{(j + 5i, l) : i ∈ [11], l ∈ [4]}, j ∈ [5]}; the perfect matching is {{(j + 5i, 0), (j + 5i, 1)}, {(j + 5i, 2), (j + 5i, 3)} : i ∈ [11]} for j ∈ [5]; B is generated by developing 59 base blocks by (+1 mod 55, −):

{(0, 0), (12, 0), (26, 1), (54, 2)} {(0, 0), (11, 1), (13, 1), (2, 3)} {(0, 0), (17, 0), (9, 2), (13, 3)} {(0, 0), (39, 1), (6, 3), (17, 2)} {(0, 0), (22, 2), (23, 2), (24, 3)} {(0, 0), (2, 0), (48, 3), (31, 3)} {(0, 0), (1, 1), (37, 2), (53, 1)} {(0, 0), (4, 1), (31, 1), (28, 3)} {(0, 0), (9, 0), (38, 2), (6, 1)} {(0, 0), (9, 1), (18, 2), (31, 2)} {(0, 0), (18, 1), (22, 1), (41, 1)} {(0, 0), (26, 0), (12, 3), (34, 3)} {(0, 0), (32, 1), (4, 2), (36, 3)} {(0, 1), (7, 1), (21, 1), (3, 2)} {(0, 1), (13, 1), (31, 1), (12, 3)} {(0, 1), (7, 2), (23, 2), (41, 3)} {(0, 1), (13, 3), (14, 3), (16, 2)} {(0, 1), (17, 3), (41, 2), (53, 2)} {(0, 1), (23, 3), (39, 3), (42, 3)} {(0, 2), (8, 3), (24, 2), (51, 3)}

{(0, 0), (13, 0), (42, 1), (6, 2)} {(0, 0), (12, 1), (28, 1), (4, 3)} {(0, 0), (18, 0), (12, 2), (16, 2)} {(0, 0), (7, 3), (14, 2), (21, 2)} {(0, 0), (27, 2), (33, 2), (33, 3)} {(0, 0), (3, 0), (51, 0), (19, 1)} {(0, 0), (2, 1), (11, 3), (28, 2)} {(0, 0), (6, 0), (27, 3), (53, 3)} {(0, 0), (7, 1), (24, 1), (18, 3)} {(0, 0), (11, 0), (7, 2), (49, 3)} {(0, 0), (23, 0), (44, 1), (32, 3)} {(0, 0), (27, 0), (14, 3), (46, 2)} {(0, 0), (37, 1), (11, 2), (43, 2)} {(0, 1), (8, 1), (2, 2), (1, 3)} {(0, 1), (2, 3), (11, 2), (44, 2)} {(0, 1), (8, 3), (16, 3), (42, 2)} {(0, 1), (17, 2), (28, 3), (46, 2)} {(0, 1), (19, 3), (37, 3), (51, 3)} {(0, 1), (26, 3), (32, 3), (38, 2)} {(0, 2), (14, 2), (26, 3), (47, 3)}

{(0, 0), (14, 0), (47, 1), (48, 1)} {(0, 0), (16, 0), (2, 2), (13, 2)} {(0, 0), (19, 0), (3, 3), (16, 3)} {(0, 0), (17, 3), (19, 3), (26, 3)} {(0, 0), (23, 3), (1, 0), (22, 0)} {(0, 0), (0, 1), (49, 1), (8, 2)} {(0, 0), (3, 1), (24, 2), (36, 1)} {(0, 0), (8, 0), (51, 1), (44, 2)} {(0, 0), (8, 1), (54, 1), (37, 3)} {(0, 0), (17, 1), (44, 3), (46, 1)} {(0, 0), (24, 0), (1, 2), (3, 2)} {(0, 0), (27, 1), (38, 1), (39, 2)} {(0, 0), (26, 2), (43, 3), (54, 3)} {(0, 1), (12, 1), (4, 2), (18, 3)} {(0, 1), (3, 3), (7, 3), (34, 3)} {(0, 1), (13, 2), (31, 2), (34, 2)} {(0, 1), (18, 2), (21, 3), (54, 2)} {(0, 1), (24, 2), (33, 3), (52, 2)} {(0, 2), (8, 2), (17, 2), (24, 3)}

11 GDD(M22 , 4): V = Z121 × Z4 ; Π = {{(11i + j, l) : i ∈ [11], l ∈ Z4 } : j ∈ [11]}; the perfect matching is {{(11i + j, 0), (11i + j, 2)}, {(11i + j, 1), (11i + j, 3)} : i ∈ [11]} for j ∈ [11]; B can be divided into two parts: the first part is generated by developing 13 base blocks by (+1 mod 121, −):

{(0, 0), (17, 1), (0, 2), (17, 3)} {(0, 0), (117, 1), (2, 2), (118, 3)} {(0, 0), (120, 1), (3, 2), (5, 3)} {(0, 0), (3, 1), (5, 2), (120, 3)} {(0, 0), (6, 1), (4, 2), (1, 3)}

{(0, 0), (115, 1), (116, 2), (119, 3)} {(0, 0), (118, 1), (117, 2), (2, 3)} {(0, 0), (1, 1), (6, 2), (4, 3)} {(0, 0), (4, 1), (1, 2), (6, 3)}

{(0, 0), (116, 1), (119, 2), (115, 3)} {(0, 0), (119, 1), (115, 2), (116, 3)} {(0, 0), (2, 1), (118, 2), (117, 3)} {(0, 0), (5, 1), (120, 2), (3, 3)}

and the second part is generated by developing 67 base blocks by (+1 mod 121, +2 mod 4): {(0, 0), (64, 2), (1, 0), (76, 0)} {(0, 0), (4, 0), (101, 2), (19, 3)} {(0, 0), (9, 0), (56, 3), (70, 0)} {(0, 0), (13, 0), (114, 1), (89, 2)} {(0, 0), (17, 0), (48, 2), (40, 3)} {(0, 0), (20, 0), (7, 3), (71, 1)} {(0, 0), (24, 0), (49, 2), (14, 1)} {(0, 0), (30, 0), (95, 1), (70, 1)} {(0, 0), (34, 0), (8, 2), (65, 3)} {(0, 0), (38, 0), (9, 3), (111, 3)} {(0, 0), (53, 0), (100, 1), (91, 1)} {(0, 0), (85, 1), (27, 3), (53, 3)} {(0, 0), (58, 0), (13, 3), (30, 3)} {(0, 0), (27, 1), (48, 1), (97, 3)} {(0, 0), (48, 0), (65, 2), (41, 3)} {(0, 0), (114, 0), (60, 3), (102, 3)} {(0, 0), (13, 1), (74, 3), (89, 3)} {(0, 0), (53, 1), (16, 3), (39, 3)} {(0, 0), (20, 1), (89, 1), (62, 3)} {(0, 0), (52, 1), (64, 1), (24, 3)} {(0, 0), (80, 1), (82, 1), (59, 3)} {(0, 1), (3, 1), (8, 1), (70, 1)} {(0, 1), (24, 1), (7, 3), (65, 3)}

{(0, 0), (2, 0), (43, 0), (52, 2)} {(0, 0), (5, 0), (74, 0), (40, 2)} {(0, 0), (10, 0), (94, 1), (84, 2)} {(0, 0), (14, 0), (73, 1), (63, 3)} {(0, 0), (18, 0), (75, 3), (50, 1)} {(0, 0), (21, 0), (93, 1), (62, 1)} {(0, 0), (25, 0), (79, 3), (10, 2)} {(0, 0), (16, 0), (48, 3), (28, 3)} {(0, 0), (36, 0), (78, 1), (79, 1)} {(0, 0), (49, 0), (67, 2), (26, 1)} {(0, 0), (56, 0), (70, 2), (91, 3)} {(0, 0), (59, 0), (100, 2), (42, 3)} {(0, 0), (19, 1), (34, 3), (81, 3)} {(0, 0), (28, 1), (104, 1), (36, 3)} {(0, 0), (8, 0), (45, 3), (106, 3)} {(0, 0), (12, 0), (93, 0), (30, 1)} {(0, 0), (31, 1), (78, 3), (113, 3)} {(0, 0), (12, 1), (105, 1), (85, 3)} {(0, 0), (23, 1), (36, 1), (107, 1)} {(0, 0), (60, 1), (103, 1), (112, 3)} {(0, 0), (106, 1), (71, 3), (87, 3)} {(0, 1), (7, 1), (36, 1), (89, 1)}

{(0, 0), (3, 0), (105, 2), (78, 2)} {(0, 0), (6, 0), (92, 0), (85, 2)} {(0, 0), (37, 1), (83, 1), (108, 2)} {(0, 0), (15, 0), (53, 2), (96, 1)} {(0, 0), (19, 0), (7, 1), (42, 0)} {(0, 0), (39, 0), (74, 1), (10, 1)} {(0, 0), (26, 0), (64, 3), (8, 3)} {(0, 0), (31, 0), (76, 1), (100, 3)} {(0, 0), (37, 0), (58, 3), (46, 1)} {(0, 0), (50, 0), (75, 1), (101, 3)} {(0, 0), (68, 1), (102, 1), (52, 3)} {(0, 0), (15, 1), (29, 1), (56, 1)} {(0, 0), (57, 0), (27, 2), (18, 3)} {(0, 0), (32, 0), (92, 2), (61, 3)} {(0, 0), (54, 0), (108, 1), (82, 2)} {(0, 0), (8, 1), (59, 2), (83, 3)} {(0, 0), (49, 1), (87, 1), (20, 3)} {(0, 0), (16, 1), (46, 3), (94, 3)} {(0, 0), (34, 1), (68, 3), (72, 3)} {(0, 0), (61, 1), (67, 1), (98, 2)} {(0, 0), (14, 3), (95, 3), (105, 3)} {(0, 1), (18, 1), (31, 3), (103, 3)}

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