Super-simple pairwise balanced designs with block sizes 3 and 4

Discrete Mathematics 340 (2017) 236–242

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Super-simple pairwise balanced designs with block sizes 3 and 4 Guangzhou Chen a, *, Yong Zhang b , Kejun Chen c a

Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Sciences, Henan Normal University, Xinxiang, 453007, PR China b School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, PR China c Department of Mathematics, Taizhou University, Taizhou, 225300, PR China

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Article history: Received 28 January 2016 Received in revised form 13 August 2016 Accepted 16 August 2016

Keywords: Super-simple designs Pairwise balanced designs Balanced incomplete block designs Group divisible designs

a b s t r a c t Super-simple designs can be used to provide samples with maximum intersection as small as possible in statistical planning of experiments and can be also applied to cryptography and codes. In this paper, super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple (v, {3, 4}, λ)-PBD for 2 ≤ λ ≤ 6 are sufficient with three possible exceptions. © 2016 Elsevier B.V. All rights reserved.

1. Introduction A pairwise balanced design (or PBD) is a pair (X , A) such that X is a set of elements called points, and A is a set of subsets (called blocks) of X , each of cardinality at least two, such that every pair of points occurs in exactly λ blocks of A. If v is a positive integer and K is a set of positive integers, each of which is greater than one, then we say that (X , A) is a (v, K , λ)PBD if |X | = v , and |A| ∈ K for every A ∈ A. We denote B(K , λ) = {v : there exists a (v, K , λ)-PBD}. A set K is said to be PBD-closed if B(K , λ) = K . A PBD is resolvable if its blocks can be partitioned into parallel classes; a parallel class is a set of point-disjoint blocks whose union is the set of all points. The notation (v, K , λ)-RPBD is used for a resolvable PBD. When K = {k}, a (v, K , λ)-PBD is a balanced incomplete block design, the notations (v, k, λ)-BIBD and (v, k, λ)-RBIBD are sometimes used in this case. A design is said to be simple if it contains no repeated blocks. A design is said to be super-simple if the intersection of any two blocks has at most two elements. When k = 3, a super-simple design is just a simple design. When λ = 1, the designs are necessarily super-simple. A super-simple (v, K1 , λ)-PBD is also a super-simple (v, K2 , λ)-PBD if K1 ⊆ K2 . The concept of super-simple designs was introduced by Gronau and Mullin [26]. The existence of super-simple designs is an interesting problem by itself, but there are also useful applications. For example, such designs are used in constructing perfect hash families [34] and coverings [7], in the construction of new designs [6] and in the construction of superimposed codes [31]. In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Besides these, super-simple designs have also appeared as sub-orthogonal double covers of certain types of graphs (see, for example [24]). There are other useful applications in [21,29]. Super-simple pairwise balanced designs are powerful for the construction of other types of combinatorial structures, for example super-simple group divisible designs [10].

*

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

http://dx.doi.org/10.1016/j.disc.2016.08.021 0012-365X/© 2016 Elsevier B.V. All rights reserved.

G. Chen et al. / Discrete Mathematics 340 (2017) 236–242

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Let m denote the smallest integer in K . Define α (K ) = gcd{k − 1 : k ∈ K } and β (K ) = gcd{k(k − 1) : k ∈ K }. The necessary conditions for the existence of a super-simple (v, K , λ)-PBD are v ≥ λ(m − 2) + 2, λ(v − 1) ≡ 0 (mod α (K )) and λv (v − 1) ≡ 0 (mod β (K )). The existence of simple (v, 3, λ)-BIBDs was completely solved by Dehon [22], we state the result below. Lemma 1.1 ([22]). There exists a simple (v, 3, λ)-BIBD if and only if v > λ+ 2, λ(v− 1) ≡ 0 (mod 2) and λv (v− 1) ≡ 0 (mod 6). The necessary conditions for the existence of a super-simple (v, 4, λ)-PBD are v ≥ 2λ + 2, λ(v − 1) ≡ 0 (mod 3) and λv (v − 1) ≡ 0 (mod 12). For the existence of super-simple (v, 4, λ)-BIBDs, the necessary conditions are known to be sufficient for λ ∈ {2 − 6, 8, 9} (see [4,9,11,12,14,20,23,26,30,35]). Gronau and Mullin [26] solved the case for λ = 2, and the corrected proof appeared in [30]. The λ = 3 case was solved by Chen [11]. The λ = 4 case was solved independently by Adams, Bryant, and Khodkar [4] and Chen [12]. The case of λ = 5 was solved by Cao, Chen and Wei [9]. The case of λ = 6 was solved by Chen, Cao and Wei [14]. The case of λ = 8 was solved by Chen, Sun and Zhang [20]. The case of λ = 9 was solved by Zhang, Chen and Sun [35]. A survey on super-simple (v, 4, λ)-BIBDs with v ≤ 32 appeared in [8]. We summarize these known results in the following result. Lemma 1.2 ([4,9,11,12,14,20,23,26,30,35]). The necessary conditions of a super-simple (v, 4, λ)-BIBD for λ = 2, 3, 4, 5, 6, 8, 9 are sufficient. The necessary conditions for the existence of a super-simple (v, 5, λ)-BIBD are known to be sufficient for λ ∈ {2, 3, 4, 5} (see [1,15–17,23,25,33]). For more results on super-simple design we refer the reader to [2,5,8,13,18,28,32,33] and references therein. The existence of (v, {3, 4}, 1)-PBD was stated in [3,27]. Lemma 1.3 ([3,27]). There exists a (v, {3, 4}, 1)-PBD if and only if v ≡ 0, 1 (mod 3) and v ≥ 3. In this paper, the existence of a super-simple (v, {3, 4}, λ)-PBD for 2 ≤ λ ≤ 6 is investigated. The necessary conditions for the existence of such a super-simple design are v ≥ λ + 2 and λv (v − 1) ≡ 0 (mod 3). We shall use direct and recursive constructions to show that the necessary conditions are also sufficient with some possible exceptions. Specifically, we shall prove the following theorem. Theorem 1.4. The necessary conditions of a super-simple (v, {3, 4}, λ)-PBD for 2 ≤ λ ≤ 6 are sufficient except possibly for (v, λ) ∈ {(18, 5), (30, 5), (42, 5)}. The paper is organized as follows. Some recursive constructions are provided in Section 2. Some ingredient super-simple designs are given directly by computer programs in Section 3. The proof of our main theorem is given in Section 4. We present one research problem in Section 5. 2. Recursive constructions In this section, super-simple group divisible designs are used and some recursive constructions using group divisible designs are given, which will be needed in the sequel. A group divisible design (or GDD) is a triple (X , G , B) which satisfies the following properties: (i) G is a partition of a set X (of points) into subsets called groups. (ii) B is a set of subsets of X (called blocks) such that a group and a block contain at most one common point. (iii) Every pair of points from distinct groups occurs in exactly λ blocks. The group type (or type) of GDD is the multiset {|G| : G ∈ G }. We usually use an ‘‘exponential’’ notation to describe types: u u u so type g1 1 g2 2 · · · gk k denotes ui occurrences of gi , 1 ≤ i ≤ k, in the multiset. A GDD with block sizes from a set of positive integers K is called a (K , λ)-GDD. When λ = 1, we simply write K -GDD. When K = {k}, we simply write k for K . Taking the groups of a GDD as additional blocks yields a PBD, and taking a parallel class of blocks of a PBD as groups also yields a GDD. A (k, λ)-GDD of group type v k is called a transversal design and denoted by TDλ (k, v ) for short. The following result was stated in [28]. Lemma 2.1 ([28]). A super-simple TDλ (4, v ) exists if and only if λ ≤ v and (λ, v ) is neither (1, 2) nor (1, 6). We shall use the following standard recursive constructions in the proof of Theorem 1.4. For details of the following result, we refer the reader to [9,12,15–16], its proof is omitted here. Lemma 2.2 ([9,12,15–16]). (Weighting) Let (X , G , B) be a super-simple GDD with index λ1 , and let w : X → Z + ∪ {0} be a weight function on X , where Z + is the set of positive integers. Suppose that for each block ∑B ∈ B, there exists a super-simple (k, λ2 )-GDD of type {w (x) : x ∈ B}, then there exists a super-simple (k, λ1 λ2 )-GDD of type { x∈G w (x) : Gi ∈ G }. i

The following result can be regarded as a generalization of the construction of BIBD, see [9,12,15–16].

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G. Chen et al. / Discrete Mathematics 340 (2017) 236–242 u

u

t 1 Lemma 2.3 (Breaking up Groups). If there exists a super-simple ∑t (K , λ)-GDD of type h1 · · · ht and a super-simple (hi + η, K , λ)PBD for each i, 1 ≤ i ≤ t, then there exists a super-simple ( i=1 hi ui + η, K , λ)-PBD, where η = 0 or 1.

u

∑t

u

⋃u

Proof. Suppose that (X , G , B) is a super-simple (K , λ)-GDD of type h11 · · · ht t , where |X | = i=1 Gi , i=1 hi ui and G = u = u1 + u2 + · · · + ut , |Gi | = hi , 1 ≤ i ≤ u. Let Y = ∅ or {∞}, ∞ ̸ ∈ X . Then for each Gi ∈ G , we ⋃ construct a super-simple u (h i + η, K , λ)-PBD (Gi ∪ Y , Ci ), where η = 0 if Y = ∅, η = 1 if Y = ∞. It is obvious that (X ∪ Y , B ∪ i=1 Ci ) is a super-simple ∑ t ( i=1 hi ui + η, K , λ)-PBD. □ u

u

Lemma 2.4. If there exists a (K , 1)-GDD of type h11 · · · ht t , then there exists a ( η, h2 + η, . . . , ht + η} and η = 0 or 1.

∑t

i=1 hi ui

+ η, K1 , 1)-PBD, where K1 = K ∪ {h1 +

Proof. If η = 0, the conclusion holds by taking the groups of the GDD as blocks. If η = 1, the conclusion also holds by adding a point ∞ to each group of the GDD as blocks. □ The following results are obvious but very useful. Their proofs are omitted here. Lemma 2.5. If there exists a super-simple (v, K1 , λ)-PBD, then there exists a super-simple (v − m, K2 , λ)-PBD, where K1 = {k − i : k ∈ K1 and 0 ≤ i ≤ m}. Lemma 2.6. If there exists a super-simple (K1 , λ)-GDD of type h1 h2 h3 · · · hn−1 hn , then there exists a super-simple (K2 , λ)-GDD of type h1 h2 h3 · · · (hi − g1 )(hi+1 − g2 ) · · · (hi+m−1 − gm ) · · · hn−1 hn , 1 ≤ i ≤ n − m + 1, where K2 = {k − l : k ∈ K1 and 1 ≤ l ≤ m} and gj ≤ hi+j−1 , 1 ≤ j ≤ m.

3. Direct constructions In this section, direct constructions are used and some super-simple (v, {3, 4}, λ)-PBDs for small values of v are obtained, which will be used as master designs or input designs in the recursive constructions. All these designs are obtained by computer. Usually, it is difficult to find all the blocks of a design directly. So, a technique of ‘‘+d (mod v )’’ is used, which means that we try to find a subset S ⊆ B and an element d ∈ Zv such that {B + kd : B ∈ S , k ∈ Z } = B. The blocks of S are called base blocks. The ‘‘+d’’ is omitted when d = 1. Sometimes S is divided into parts: P and R, and we try to find an element m ∈ Zv and an integer s such that there is ⋃s−two 1 a subset P1 ⊆ P satisfying i=0 {mi B : B ∈ P1 } = P . Here m is a partial multiplier of order s of the design. In this note, m is taken to be some unit of the ring Zv , i.e., m satisfies that gcd(m, v ) = 1. Further, the founded base blocks of S are shuffled when the program takes too long time to find a design. Most of these ideas come from the previous papers such as [15,16–18]. Lemma 3.1. There exists a super-simple (v, {3, 4}, 3)-PBD for v ∈ {10, 14, 18, 22, 26, 30}. Proof. For v ∈ {10, 14, 18}, we take point set X = Zv , the base blocks are listed below and all the required blocks can be generated from them by +2 (mod v ). v = 10 : {0, 2, 4}, {0, 1, 2, 5}, {0, 1, 4, 7}, {0, 1, 6, 9}, {0, 5, 7, 9}. v = 14 : {1, 3, 7}, {0, 1, 2, 3}, {0, 1, 4, 6}, {0, 2, 5, 8}, {0, 3, 7, 10}, {0, 4, 9, 13}, {0, 5, 7, 13}. v = 18 : {1, 3, 7}, {0, 3, 9, 11}, {0, 4, 12, 13}, {0, 1, 2, 12}, {0, 1, 4, 16}, {0, 7, 11, 17}, {0, 4, 15, 17}, {0, 3, 7, 16}, {0, 5, 8, 13}. For v ∈ {22, 26, 30}, we take the point set as X = Zv . With a computer program we found the required base blocks, which are divided into two parts, P and R, where P consists of some base blocks with a partial multiplier m of order s (i.e., each base block of P has to be multiplied by mi for 0 ≤ i ≤ s − 1), and R is the set of the remaining base blocks. We list P , m, s and R below. The desired super-simple design is generated by developing the base blocks +2 (mod v ). v = 22 : P : {0, 1, 2, 3}, (m, s) = (3, 5), R : {0, 4, 12}, {0, 2, 4, 10}, {0, 11, 17, 21}, {0, 7, 19, 21}, {0, 11, 13, 19}, {0, 6, 13, 17}. v = 26 : P : {0, 1, 2, 3}, (m, s) = (7, 6), R : {0, 6, 12}, {1, 4, 8, 12}, {0, 13, 19, 21}, {1, 3, 7, 17}, {1, 5, 13, 14}, {0, 2, 11, 12}, {0, 2, 5, 18}. v = 30 : P : {0, 1, 2, 3}, {0, 1, 4, 6}, (m, s) = (7, 3), R : {1, 7, 15}, {1, 11, 18, 23}, {0, 11, 12, 20}, {1, 10, 27, 28}, {1, 5, 7, 17}, {0, 4, 13, 15}, {0, 10, 15, 27}, {0, 5, 9, 15}, {0, 13, 20, 26}. □ Lemma 3.2. There exists a super-simple (v, {3, 4}, 5)-PBD for v ∈ {10, 22}.

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Proof. For v ∈ {10, 22}, we take point set X = Zv . The desired super-simple design is generated by developing the base blocks +2 (mod v ). v = 10 : {0, 1, 2, 6}, {0, 2, 4}, {0, 1, 7}, {0, 1, 3}, {0, 5, 9}, {0, 7, 9}, {1, 3, 7}, {0, 3, 8}, {0, 1, 9}, {0, 3, 6}, {0, 1, 4}, {0, 2, 7}, {1, 3, 8}, {1, 2, 5}. v = 22 : {0, 7, 10, 19}, {0, 12, 14, 15}, {1, 8, 9, 10}, {0, 3, 14, 16}, {1, 15, 18, 20}, {0, 5, 6, 7}, {0, 7, 11, 14}, {1, 5, 17, 19}, {1, 2, 12, 16}, {0, 4, 13, 15}, {0, 10, 11, 17}, {0, 5, 18, 21}, {1, 5, 7, 13}, {0, 4, 5, 17}, {0, 5, 9, 14}, {1, 4, 10, 20}, {0, 2, 17, 18}, {1, 3, 15}. □ 4. The Proof of Theorem 1.4 In this section, we shall give the proof of Theorem 1.4. For λ = 2, 4, 6, the necessary conditions for the existence of a super-simple (v, {3, 4}, λ)-PBD are the same as the necessary conditions for the existence of a simple (v, 3, λ)-BIBD. Thus the existence problem of a super-simple (v, {3, 4}, λ)-PBD has been solved for λ = 2, 4, 6 already. To prove Theorem 1.4, we shall divide it into 2 cases according to the remaining values of λ = 3, 5. Case 1. λ = 3 When λ = 3, the necessary conditions of a super-simple (v, {3, 4}, 3)-PBD become v ≥ 5. We shall prove that there exists a super-simple (v, {3, 4}, 3)-PBD for any v ≥ 5 and v ̸ = 6. Lemma 4.1. There exists a super-simple (v, {3, 4}, 3)-PBD for v ≡ 0, 1, 3 (mod 4) and v ≥ 5. Proof. For v = 5, let X = Z5 , B consists of all of the 10 triples on Z5 , that is, B = {{0, 1, 2}, {0, 1, 3}, {0, 1, 4}, {0, 2, 3}, {0, 2, 4}, {0, 3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}}.

It is easy to see that (X , B) is a super-simple (5, {3, 4}, 3)-PBD. For v ≡ 0, 1 (mod 4) and v > 5, there exists a super-simple (v, {3, 4}, 3)-PBD since there exists a super-simple (v, 4, 3)BIBD by Lemma 1.2. For v ≡ 3 (mod 4) and v > 5, we can easily get a super-simple (v, {3, 4}, 3)-PBD by deleting a point from the point set of a super-simple (v + 1, 4, 3)-BIBD coming from Lemma 1.2. □ Lemma 4.2. There does not exist a super-simple (6, {3, 4}, 3)-PBD. Proof. Suppose that (V , B) is a super-simple (6, {3, 4}, 3)-PBD. For any point a ∈ V , let x = |{B : a ∈ B, B ∈ B, |B| = 3}| and y = |{B : a ∈ B, B ∈ B, |B| = 4}|. It is obvious that 2x + 3y = 15. We list all these possible values for x and y in the following : x = 0, y = 5; x = 3, y = 3; x = 6, y = 1. It is easy to see that every point appears in at least one block of size four since y ≥ 1. Next, we ⋃3prove y = 1. In fact, if y > 1, then there exist at least three blocks Bi of size four with 1 ≤ i ≤ 3 such that a ∈ Bi . Then | i=1 (Bi \ {a})| = 9. Since all these 9 points come from the point set V − {a}, there exist at least two points b and c which appear in at least two blocks of Bi for 1 ≤ i ≤ 3. This contradicts super-simplicity. So every point should appear in exactly one block of size four. It is impossible since 6 is not divisible by 4. This completes the proof. □ Lemma 4.3. There exists a super-simple (v, {3, 4}, 3)-PBD for v ∈ M = {34, 38, 42, 46, 50, 54, 58, 62, 66}. Proof. For v ∈ M, let v = 3g + m. The two parameters g and m are listed in the following table. We start from a super-simple TD3 (4, g) by Lemma 2.1. Removing g − m points from the last group of the super-simple TD3 (4, g), we get a super-simple ({3, 4}, 3)-GDD of group type g 3 m1 by Lemma 2.6. By Lemma 2.3 with η = 0, we get a super-simple (3g + m, {3, 4}, 3)-PBD, where the input super-simple (g , {3, 4}, 3)-PBD and (m, {3, 4}, 3)-PBD come from Lemma 3.1 and Lemma 4.1 respectively. v = 3g + m

g

m

v = 3g + m

g

m

34 38 42 46 50

9 10 14 12 14

7 8 0 10 8

54 58 62 66

14 16 16 18

12 10 14 12

□

Lemma 4.4. There exists a super-simple (v, {3, 4}, 3)-PBD for v ≡ 2 (mod 4) and v ≥ 70. Proof. For each v ≡ 2 (mod 4) and v ≥ 70, it can be written as v = 12t + m, where t ≥ 5 and m ∈ {10, 14, 18}. By Lemma 2.1 there exists a super-simple TD3 (4, 4t). Removing 4t − m points from the last group, we get a super-simple ({3, 4}, 3)-GDD of group type (4t)3 m1 by Lemma 2.6. By Lemma 2.3 with η = 0, we get a super-simple (12t + m, {3, 4}, 3)-PBD, where the input super-simple (4t , {3, 4}, 3)-PBD and (m, {3, 4}, 3)-PBD come from Lemma 4.1 and Lemma 3.1 respectively. □

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By Lemmas 4.1–4.4 and Lemma 3.1, we have the following result. Theorem 4.5. There exists a super-simple (v, {3, 4}, 3)-PBD for v ≥ 5, except for v = 6. Case 2. λ = 5 When λ = 5, the necessary conditions of a super-simple (v, {3, 4}, 5)-PBD become v ≡ 0, 1 (mod 3) and v ≥ 7. We shall prove that there exists a super-simple (v, {3, 4}, 5)-PBD for v ≡ 0, 1 (mod 3), v ≥ 7, except possibly for v ∈ {18, 30, 42}. Lemma 4.6. There exists a super-simple (v, {3, 4}, 5)-BIBD for any v ≡ 0, 1, 3, 4, 7, 9 (mod 12) and v ≥ 7. Proof. For v ≡ 0, 3 (mod 12) and v ≥ 12, the conclusion holds by deleting a point from the point set of a super-simple (v + 1, 4, 5)-BIBD coming from Lemma 1.2. For v ≡ 1, 4 (mod 12) and v ≥ 13, there exists a super-simple (v, 4, 5)-BIBD by Lemma 1.2, so does a super-simple (v, {3, 4}, 5)-PBD. For v ≡ 7, 9 (mod 12) and v ≥ 7, there exists a super-simple (v, {3, 4}, 5)-PBD since there exists a super-simple (v, 3, 5)BIBD by Lemma 1.1. The proof is completed. □ Lemma 4.7. There exists a super-simple (v, {3, 4}, 5)-PBD for v ∈ {34, 54, 66}. Proof. For v = 34, by Lemma 2.6 we can get a super-simple ({3, 4}, 5)-GDD of group type 93 71 by deleting 2 points from the last group of a super-simple TD5 (4, 9) coming from Lemma 2.1. By Lemma 2.3 with η = 0, we get a super-simple (34, {3, 4}, 5)-PBD, where the input super-simple (9, {3, 4}, 5)-PBD and (7, {3, 4}, 5)-PBD come from Lemma 4.6. For v = 54, starting from a super-simple TD5 (4, 15) by Lemma 2.1, we delete 6 points from the last group to get a supersimple ({3, 4}, 5)-GDD of group type (15)3 91 . By Lemma 2.3 with η = 0, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (15, {3, 4}, 5)-PBD and (9, {3, 4}, 5)-PBD come from Lemma 4.6. For v = 66, we can obtain a super-simple ({3, 4}, 5)-GDD of group type (19)3 91 by deleting 10 points from the last group of a super-simple TD5 (4, 19) coming from Lemma 2.1. By Lemma 2.3 with η = 0, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (19, {3, 4}, 5)-PBD and (9, {3, 4}, 5)-PBD come from Lemma 4.6. □ Lemma 4.8. There exists a super-simple (v, {3, 4}, 5)-BIBD for any v ≡ 10 (mod 12) and v > 34. Proof. For v ≡ 10 (mod 12) and v > 34, it can be written as v = 12t + 10, where t > 2. When t ≡ 0, 1 (mod 3), then 4t ≡ 0, 4 (mod 12). We can obtain a super-simple ({3, 4}, 5)-GDD of group type (4t)3 (10)1 by deleting 4t − 10 points from the last group of the super-simple TD5 (4, 4t) coming from Lemma 2.1. By Lemma 2.3 with η = 0, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (4t , {3, 4}, 5)-PBD and (10, {3, 4}, 5)-PBD come from Lemma 4.6 and Lemma 3.2 respectively. When t ≡ 2 (mod 3), then 4t + 1 ≡ 9 (mod 12). We can obtain a super-simple ({3, 4}, 5)-GDD of group type (4t)3 (9)1 by deleting 4t − 9 points from the last group of the super-simple TD5 (4, 4t) coming from Lemma 2.1. By Lemma 2.3 with η = 1, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (4t + 1, {3, 4}, 5)-PBD and (9 + 1, {3, 4}, 5)-PBD come from Lemma 4.6 and Lemma 3.2 respectively. □ Lemma 4.9. There exists a super-simple (v, {3, 4}, 5)-PBD for v ∈ M = {78, 90, 114, 126, 150, 186}. Proof. For v ∈ M, let v = 3g + m. The two parameters g and m are listed in the following table. We start from a super-simple TD3 (4, g) by Lemma 2.1. Removing g − m points from the last group of the super-simple TD3 (4, g), we get a super-simple ({3, 4}, 3)-GDD of group type g 3 m1 by Lemma 2.6. By Lemma 2.3 with η = 0, we get a super-simple (3g + m, {3, 4}, 3)PBD, where the input super-simple (g , {3, 4}, 3)-PBD and (m, {3, 4}, 3)-PBD come from Lemmas 4.6–4.8 and Lemma 3.2 respectively. v = 3g + m

g

m

v = 3g + m

g

m

78 90 114

22 27 34

12 9 12

126 150 186

39 46 58

9 12 12

□

Lemma 4.10. There exists a super-simple (v, {3, 4}, 5)-PBD for v ∈ {102, 138, 162, 174, 198, 210}. Proof. For v ∈ {102, 138, 162, 174, 198, 210}, it can be written as v = 3m, where m ∈ {34, 46, 54, 58, 66, 70}. Starting from a super-simple TD5 (4, m) by Lemma 2.1, we delete the last group to get a super-simple (3, 5)-GDD of group type m3 . By Lemma 2.3 with η = 0, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (m, {3, 4}, 5)-PBD comes from Lemmas 4.7–4.8 respectively. □

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Lemma 4.11. There exists a super-simple (v, {3, 4}, 5)-BIBD for any v ≡ 6 (mod 12) and v ≥ 222. Proof. For v ≡ 6 (mod 12) and v ≥ 222, it can be written as v = 12t + 6, where t ≥ 18. When t ≡ 1, 2 (mod 3), then 4(t − 4) ≡ 0, 4 (mod 12). We can obtain a super-simple ({3, 4}, 5)-GDD of group type (4(t − 4))3 (54)1 by deleting 4(t − 4) − 54 points from the last group of the super-simple TD5 (4, 4(t − 4)) coming from Lemma 2.1. By Lemma 2.3 with η = 0, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (4(t − 4), {3, 4}, 5)-PBD and (54, {3, 4}, 5)-PBD come from Lemmas 4.6–4.7 respectively. When t ≡ 0 (mod 3), then 4(t − 4) + 1 ≡ 9 (mod 12). We can obtain a super-simple ({3, 4}, 5)-GDD of group type (4(t − 4))3 (53)1 by deleting 4(t − 4) − 53 points from the last group of the super-simple TD5 (4, 4(t − 4)) coming from Lemma 2.1. By Lemma 2.3 with η = 1, we get a super-simple (v, {3, 4}, 5)-PBD, where the input super-simple (4(t − 4) + 1, {3, 4}, 5)-PBD and (53 + 1, {3, 4}, 5)-PBD come from Lemmas 4.6–4.7 respectively. □ By Lemmas 4.6–4.11 and Lemma 3.2, we have the following result. Theorem 4.12. There exists a super-simple (v, {3, 4}, 5)-PBD for v ≥ 7, v ≡ 0, 1 (mod 3), except possibly for v ∈ {18, 30, 42}. The proof of Theorem 1.4. For λ = 2, 4, 6, the necessary conditions for the existence of a super-simple (v, {3, 4}, λ)-PBD are sufficient since there exists a simple (v, 3, λ)-BIBD by Lemma 1.1. For λ = 3, 5, we can get the results by Theorems 4.5 and 4.12. □ 5. Concluding remarks In the last section, we list one research problem of super-simple (v, {3, 4}, λ)-PBD. There exists a super-simple (v, {3, 4}, λ)-PBD for even index λ since there exists a simple (v, 3, λ)-BIBD by Lemma 1.1. We have the following open question. 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