Existence of Steiner quadruple systems with an almost spanning block design

Discrete Mathematics xxx (xxxx) xxx

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Existence of Steiner quadruple systems with an almost spanning block design✩ Lijun Ji Department of Mathematics, Soochow University, Suzhou 215006, China

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Article history: Received 10 March 2019 Received in revised form 23 September 2019 Accepted 22 October 2019 Available online xxxx Keywords: Steiner quadruple system Almost spanning block design Candelabra quadruple system t-wise balanced design

a b s t r a c t A Steiner quadruple system of order v (SQS(v )) is said to be have an almost spanning block design and denoted by 1-AFSQS(v ) if it contains a subdesign S(2, 4, v − 1). In 1992, Hartman and Phelps posed a problem: Show that there exists a 1-AFSQS(v ) for each v ≡ 2 (mod 12). In this paper, we prove that the necessary condition for the existence of a 1-AFSQS(v ) is also sufficient with a definite exception v = 14 and possible exceptions v ∈ {86, 206, 374, 398}. © 2019 Elsevier B.V. All rights reserved.

1. Introduction A t-wise balanced design (t-BD) is a pair (X , B), where X is a finite set of points and B is a set of subsets of X , called blocks with the property that every t-element subset of X is contained in a unique block. If |X | = v and block sizes of B are all from K , we denote the t-BD by S(t , K , v ). When K = {k}, we simply write k for K . A 2-BD is usually called a pairwise balanced design (PBD). An S(t , k, v ) is called a Steiner system. An S(3, 4, v ) is called a Steiner quadruple system of order v , briefly denoted by SQS(v ). It is well known that an SQS(v ) exists if and only if v ≡ 2 or 4 (mod 6) [1]. In an SQS(v )(X , B), if there is a subset A ⊂ B and an element x ∈ X such that (X \ {x}, A) is an S(2, 4, v − 1), then such an S(2, 4, v − 1) is called an almost spanning block design (as in [5]). An SQS(v ) with an almost spanning block design is shortly denoted by 1-AFSQS(v ). It is clear that the necessary condition for the existence of a 1-AFSQS(v ) is v ≡ 2 (mod 12). Hartman and Phelps posed an open problem in [5]: Show that there exists a 1-AFSQS(v ) for each v ≡ 2 (mod 12). The existence problem has received some attention. In [11], it has been shown that if there exist both a 1-AFSQS(v ) containing a subdesign 1-AFSQS(w ) and a 1-AFSQS(u), then there also exists a 1-AFSQS((u − 1)(v − w ) + w ). For small orders, some 1-AFSQSs were listed in [11]. For example, there is a 1-AFSQS(v ) for v ∈ {26, 38}. Exhaustive search by computer shows that there does not exist a 1-AFSQS(14). In this paper, we almost determine the existence of 1-AFSQSs with an exception and four possible exceptions. For this purpose, the remainder of this paper is organized as follows. In Section 2, a recursive construction for 1-AFSQSs is given in Lemma 2.6. In Section 3, we show that for any integer u ≥ 15 and u ̸ = {17, 26, 27, 29, 31, 33}, there is a 1-FG(3, (M , N), u) a a of type g1 1 · · · gr r , where M = {4, 5, . . . , 13} ∪ {19}, N = {k ≥ 4: k is an integer} and gi ∈ Q = {2, 3, . . . , 14} \ {7} for 1 ≤ i ≤ r. An s-FG is a 3-BD containing s PBDs such that these s subdesigns share a common 1-BD. For the formal definition of an s-FG, see Section 2. In Section 4, we prove that there is a 3-FG(3, (3, 3, 4), 12 m) of type 12m for m ∈ M. In Section 5, we show that there exists a 1-AFSQS(12g + 2) for g ∈ Q . In the last section, we prove the main theorem of this paper. ✩ Research supported by NSFC Grants 11871363,11431003. E-mail address: [email protected]. https://doi.org/10.1016/j.disc.2019.111708 0012-365X/© 2019 Elsevier B.V. All rights reserved.

Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Theorem 1.1. There exists a 1-AFSQS(v ) for any integer v ≡ 2 (mod 12) with an exception v = 14 and possible exceptions v ∈ {86, 206, 374, 398}. 2. A recursive construction for 1-AFSQSs In this section, we give a recursive construction for 1-AFSQSs by using fan designs. The blocks of a 1-BD form a partition of its point set. We will shorten 1-wise balanced design to 1-design in the sequel. The type of a 1-design (X , G ) is defined to be the multiset {|G| : G ∈ G }. We sometimes use an exponential notation to a a a denote types: a type g1 1 g2 2 · · · gr r denotes ai occurrences of gi , 1 ≤ i ≤ r. An s-fan design (as in [4]) is an (s + 3)-tuple (X , G , B1 , . . . , Bs , T ), where X is a finite set of points, G , Bi (1 ≤ i ≤ s) and T are all⋃sets of subsets of X with the property that (X ,⋃ G ) is a 1-design, each (X , G ∪ Bi ) is a PBD for 1 ≤ i ≤ s, s s and (X , G ∪ ( i=1 Bi ) ∪ T ) is a 3-BD. The members of G and ( i=1 Bi ) ∪ T are called groups and blocks, respectively. Let a1 a2 ar the type of (X , G ) be g1 g2 · · · gr . If block sizes Ki (1 ≤ i ≤ s) and KT , respectively, then the s-fan ∑r of Bi and T are afrom a a design is denoted by s-FG(3, (K1 , . . . , Ks , KT ), i=1 ai gi ) of type g1 1 g2 2 · · · gr r . Specially, 0-FG is simply called a G-design. n A 0-FG(3, k, ng) of type g is simply denoted by G(n, g , k, 3). Lemma 2.1 ([13,18]). There exists a G(n, g , 4, 3) if and only if g = 1 and n ≡ 2, 4 (mod 6), or g is even and g(n − 1)(n − 2) ≡ 0 (mod 3). We use 3-FGs to state a recursive construction for 1-AFSQSs as follows. a

a

Lemma 2.2. Suppose that there exists a 3-FG(3, (3, 3, 4, 4), v ) of type g1 1 g2 2 · · · gr r . If there exists a 1-AFSQS(gi + 2) for any 1 ≤ i ≤ r, then there also exists a 1-AFSQS(v + 2). a

Proof. Let (X , G , B1 , B2 , B3 , T ) be a given 1-FG(3, (3, 3, 4, 4), v ). Suppose that {∞1 , ∞2 } ∩ X = ∅. For each G ∈ G , by assumption we can construct a 1-AFSQS(|G| + 2) on G ∪ {∞1 , ∞2 }, where an almost spanning block design is based on the set of the other blocks by FG . G ∪ {∞1 }. Denote the block set of the subdesign S(2, 4,⋃ |G| + 1) by FG′ and ⋃ For i = 1, 2, let Bi′ = {B ∪ {∞i }: B ∈ Bi }. Define F = G∈G FG and F ′ = G∈G FG′ . Then (X ∪ {∞1 , ∞2 }, F ′ ∪ B3 ∪ F ∪ B1′ ∪ B2′ ∪ T ) is the desired design, where (X ∪ {∞1 }, F ′ ∪ B3 ) is a subdesign S(2, 4, v + 1). □ This lemma shows that the 3-FG(3, (3, 3, 4, 4), v ) is useful in the construction of 1-AFSQSs. To obtain such 3-FGs, we state a fundamental construction for 3-designs, which is a special case of the fundamental construction of Hartman [4]. Let v be a non-negative integer, let t be a positive integer and K a set of positive integers. A group divisible t-design (or t-GDD) of order v and block sizes from K is a triple (X , G , B) such that (1) X is a set of cardinality v (called points), (2) G = {G1 , G2 , . . .} is a set of non-empty subsets (called groups) of X such that (X , G ) is a 1-design, (3) B is a family of subsets (called blocks) of X , each of cardinality from K , such that each block intersects any given group in at most one point, (4) each t-set of points from t distinct groups is contained in exactly one block. Such a design is denoted by GDD(t , K , v ). The type of a t-GDD is defined as the type of (X , G ). A GDD(t , K , mg) of type g m is also denoted by H(m, g , K , t). Mills [14] determined the existence of an H(m, g , 4, 3) design except when m = 5. Recently, the remaining case was solved by the author. Their results are summarized as follows. Theorem 2.3 ([8,14]). For m > 3, an H(m, g , 4, 3) design exists if and only if mg is even, g(m − 1)(m − 2) is divisible by 3 and (m, g) ̸ = (5, 2). a

Theorem 2.4 ([4]). Suppose that there exists a 1-FG(3, (K1 , K2 ), v ) of type g1 1 · · · gr r . If there exist an s-FG(3, (L1 , . . . , Ls , LT ), bk1 ) of type bk1 for any k1 ∈ K1 , and a GDD(3, LT , bk2 ) of type bk2 for any k2 ∈ K2 , then there exists an s-FG(3, (L1 , . . . , Ls , LT ), v b) of type (bg1 )a1 · · · (bgr )ar . a

Taking special input designs in Theorem 2.4 we have the following construction of 3-FG(3, (3, 3, 4, 4), v ). a

Lemma 2.5. Suppose that there exists a 1-FG(3, (K , N), v ) of type g1 1 · · · gr r with N containing all integers not less than 4. If there exists a 3-FG(3, (3, 3, 4, 4), 12k) of type 12k for any k ∈ K , then there also exists a 3-FG(3, (3, 3, 4, 4), 12v ) of type (12g1 )a1 · · · (12gr )ar . a

Proof. In Theorem 2.4, let s = 3 and b = 12, L1 = L2 = {3} and L3 = LT = {4}. Then the desired design is obtained by applying Theorem 2.4 with the known H(n, 12, 4, 3) in Theorem 2.3 and 3-FG(3, (3, 3, 4, 4), 12k) of type 12k by assumption. □ In the sequel, we denote the set {k : a ≤ k ≤ b and k is an integer} by [a, b]. Let U = {u: there is a 1-FG(3, a a (M , N), u) of type g1 1 · · · gr r , gi ∈ Q , 1 ≤ i ≤ r}, where M = {4, 5, . . . , 13} ∪ {19}, N = {k ≥ 4: k is an integer}, and Q = {2, 3, . . . , 14} \ {7}. Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Lemma 2.6. If there exist a 3-FG(3, (3, 3, 4, 4), 12m) of type 12m for any m ∈ M and a 1-AFSQS(12g + 2) for any g ∈ {2, 3 . . . , 14} \ {7}, there exists a 1-AFSQS(12u + 2) for any u ∈ U. a

Proof. For u ∈ U, there is a 1-FG(3, (M , N), u) of type g1 1 · · · gr r , gi ∈ Q for 1 ≤ i ≤ r. Apply Lemma 2.5 with the known 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m by assumption. A 3-FG(3, (3, 3, 4, 4), 12u) of type (12g1 )a1 · · · (12gr )ar is then obtained. Since there is a 1-AFSQS(12g + 2) for any g ∈ Q by assumption, we obtain a 1-AFSQS(12u + 2) by applying Lemma 2.2. □ a

3. 1-FG(3, (M , N ), u) In this section, we prove that the set U contains all integers u ≥ 15 with possible exceptions u ∈ {17, 26, 27, 29, 31, 33}. We first provide some constructions for 1-FG(3, (M , N), u). The method used is similar to those in [6,7,9]. c

c

Lemma 3.1 (Breaking Up Groups). Suppose that there exists a 1-FG(3, (M , N), u) of type m11 · · · mkk . If mi ∈ U ∪Q for 1 ≤ i ≤ k, then u ∈ U. c

c

Proof. Let (X , G , B, T ) be a given 1-FG(3, (M , N), u) of type m11 · · · mkk . For G ∈ G , by assumption |G| ∈ U ∪ Q . Denote G1 = {G ∈ G : |G| ∈ Q } and G2 = G \ G1 . For G ∈ G2 , construct a 1-FG(3, (M , N), |G|)(G, HG , CG , FG ) with group sizes from Q . Then (X , G1 ∪ (∪G∈G2 HG ), B ∪ (∪G∈G2 CG ), T ∪ (∪G∈G2 FG )) is a 1-FG(3, (M , N), u) with group sizes from Q . Thus u ∈ U. □ Let W = {w : there is a 1-FG(3, (M , N), w ) of type 1w }. Lemma 3.2 ([6, Lemma 3.9]). w ∈ W for any w ≥ 4 and w ̸ ∈ {14, 15, 18, 22, 23, 26, 27, 29}. Lemma 3.3. Suppose that there exists a 1-FG(3, (K1 , K2 ), uh) of type uh . Let m2 = minK2 . Let r, s⋃ be positive integers with s r + s = h and a1 , . . . , as be integers such that 0 ≤ a ≤ u, i = 1 , . . . , s. If m − 2s ≥ 4, i 2 i=0 (K1 − i) ⊂ W and ∑s {u, a1 , . . . , as } ⊂ U ∪ Q , then ru + i=1 ai ∈ U, where K1 − i = {k − i : k ∈ K1 }. Proof. Let (X , G , B, T ) be the given 1-FG(3, (K1 , K2 ), uh), where G = {G1 , . . . , Gh }. Delete all but ai points from the group Gr +i , i = 1, . . . , s. For any G ∈ G , we ⋃ have that |G ∩ and |G ∩ T | ≤ 2 for any T ∈ T . It follows ⋃B2s| ≤ 1 for any B ∈ ∑B s s that the obtained design is a 1-FG(3, ( i=0 (K1 − i), i=0 (K2 − i)), ru + ⋃ i=1 ai ) of type a1 ·∑ · · as ur , where the blocks in s s B with those points deleted form the block set of the subdesign GDD(2, i=0 (K1 − i), ru + i=1 ai ). Apply Theorem 2.4 ⋃2s with b = 1, L1 = M and LT = N. Since m − 2s ≥ 4, the set (K − i) is contained in N. By assumption, there is a 2 2 i=0 ⋃s ∑s 1-FG(3, (M , N), k) of type 1k for k ∈ i=0 (K1 − i). A 1-FG(3, (M , N), ru + i=1 ai ) of type a1 · · · as ur is then obtained by Theorem 2.4. The conclusion follows from Lemma 3.1 with the assumption {u, a1 , . . . , as } ⊂ U ∪ Q . □ Hanani gave some 1-FGs which we state below. Lemma 3.4 ([3, Theorem 5.1]). Let q be a prime-power. Then there exists an S(3, q + 1, q2 + 1) and a 1-FG(3, (q, q + 1), q2 ) of type qq . When the given 1-FG in Lemma 3.3 is a 1-FG(3, (q, q + 1), q2 ) of type qq from Lemma 3.4, we further have the following. Lemma 3.5. Let q be a prime-power and r, s be positive integers with r + s ≤ q. Let ∑ a1 , . . ., as be integers such that 0 ≤ ai ≤ q, s q+3 i = 1, . . . , s. If r ≥ ⌈ 2 ⌉, [r , r + s] ⊂ W and {q, a1 , . . . , as } ⊂ U ∪ Q , then qr + i=1 ai ∈ U. Proof. Let (X , G , B, T ) be a 1-FG(3, (q, q + 1), q2 ) of type qq from Lemma 3.4, where G = {G1 , . . . , Gq }. Let h = r + s. If h < q, delete all points from Gr +s+i , 1 ≤ i ≤ q − h. Since for each G ∈ G , we have that |G ∩ B| = 1 for any B ∈ B and |G ∩ T | ≤ 2 for any T ∈ T , we obtain a 1-FG(3, (h, {2h + 1 − q, . . . , q + 1}), qh) of type qh . If h = q, then the initial q+3 design is a 1-FG(3, (q, q + 1), q2 ) of type qq . In Lemma 3.3, let u = q, m2 = 2h + 1 − q and K1 = {h}. Since r ≥ ⌈ 2 ⌉ by assumption, m2 − 2s = 2r + 1 − q ≥ 4. Also by assumption, ∪si=0 (K1 − i) = [r , r + s] ⊂ W and {q, a1 , . . . , as } ⊂ U ∪ Q . The conclusion follows from Lemma 3.3. □ Remark. The derived design at a point of an S(3, q + 1, q2 + 1) is an S(2, q, q2 ), which must be an affine plane. So, there are q pairwise disjoint blocks in the subdesign GDD(2, q, q2 ) of the 1-FG(3, (q, q + 1), q2 ) of type qq . When we delete all points from Gr +s+i , 1 ≤ i ≤ q − h and consider the groups as blocks and the q pairwise disjoint blocks with those points deleted as groups, we obtain a 1-FG(3, ({h, q}, {2h + 1 − q, . . . , q + 1}), qh) of type hq . Similar to Lemma 3.5, applying Lemma 3.3, we can obtain other 1-FGs. Now, we consider u ≤ 2809. Lemma 3.6 ([6, Lemma 3.10]). {15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 28, 30, 32} ⊂ U. Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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L. Ji / Discrete Mathematics xxx (xxxx) xxx Table 1 1-FG(3, (M , N), u) for u ∈ [48, 247] \ {49, 210}. u

q

r

s

[48, 64] \ {49, 63} [56, 81] [79, 121] [104, 169] [162, 208] [209, 247] \ {210}

8 9 11 13 16 19

6 6 7 8 10 11

2 3 4 5 3 2

Table 2 1-FG(3, (M , N), u) for u ∈ [247, 2809] \ {369}.

Lemma 3.7.

u

1 − FG

type

r

s

[247, 343] [338, 392] \ {343, 353, 362, 363, 365, 367, 369} [345, 392] \ {350, 360, 369, 370, 372, 374, 376} [386, 512] [488, 729] [695, 891] [849, 1331] [1232, 1681] [1592, 2809]

1 − FG(3, (7, 8), 343) 1 − FG(3, (7, [7, 9]), 392) 1 − FG(3, ([7, 8][7, 9]), 392) 1 − FG(3, (8, 9), 512) 1 − FG(3, (9, 10), 729) 1 − FG(3, (9, [8, 12]), 891) 1 − FG(3, (11, 12), 1331) 1 − FG(3, (41, 42), 1681) 1 − FG(3, (53, 54), 2809)

497 567 498 648 819 999 12111 4141 5353

5 6 7 6 6 7 7 30 30

2 1 1 2 3 2 4 11 23

[34, 247] ⊂ U.

Proof. For any u ∈ [48, 247] \ {49, 210}∑ , we can obtain a 1-FG(3, (M , N), u) by applying Lemma 3.5 with appropriate q, r, s s and ai (1 ≤ i ≤ s) such that u = qr + i=1 ai , [r , r + s] ⊂ W and {q, a1 , a2 , . . . , as } ⊂ U ∪ Q . We list the u ∈ U, q, r and s in Table 1. Note that {15, 16, 18, 19} ⊂ U by Lemma 3.6. For u = 210, by the remark of Lemma 3.5, deleting all points from seven groups and all but one point from another group of a 1-FG(3, (19, 20), 361) of type 1919 yields a 1-FG(3, ({11, 12, 19}, [4, 20]), 210) of type 1118 121 , thereby, 210 ∈ U. For u ∈ [34, 48] \ {35}, fix a block A in an S(3, 8, 50). From the proof of [7, Lemma 3.5] there exists a block B disjoint from A. Take two points x and y from A, deleting x gives a 1-FG(3, (7, 8), 49) of type 149 , where all blocks containing x with x deleted form the block set of a subdesign S(2, 7, 49). Further, delete aB points of B and aA points including y of A \ {x} from this 1-FG such that u = 49 − aB − aA , where aB = 8 or aB ≤ 4 and aA ̸ = 6. Then we obtain a 1-FG(3, ([4, 7], [4, 8]), u), where the blocks containing y in the subdesign S(2, 7, 49), with those points deleted, form a 1-design with group sizes in Q . Thus u ∈ U. For u = 35, by the remark of Lemma 3.5, deleting all points from two groups of a 1-FG(3, (7, 8), 49) of type 77 yields there is a 1-FG(3, ({5, 7}, [4, 8]), 35) of type 57 , thereby 35 ∈ U. For u = 49, similar to the proof of 210 ∈ U, let B1 , . . . , B8 be pairwise disjoint blocks of size 8 in a 1-FG(3, (8, 9), 64) of type 88 . Delete seven points from the first two groups each and a point from the third group of a 1-FG(3, (8, 8), 64) of type 88 such that no points of B1 are deleted. Consider groups as blocks and the eight pairwise disjoint blocks with those points deleted as groups. Then we obtain a 1-FG(3, ([5, 8], [4, 9]), 49) of type 81 51 66 , therefore 49 ∈ U. □ Lemma 3.8. [247, 2809] ⊂ W . Proof. We start with a 1-FG of type g h in Table 2, where the first seven 1-FGs were given in the proof of [7, Lemma 3.10] ∑ and the others exist by Lemma 3.4. Apply Lemma 3.3 with appropriate r, s in Table 2 and ai ∈ U ∪ Q . Then gr + 1≤i≤s ai ∈ U since g ∈ U by Lemma 3.7. We list u ∈ U, the 1-FG and the corresponding r, s in Table 2. For u = 369, u ∈ U comes from the proof of [6, Lemma 3.13]. □ For any integer u ≥ 2810, we shall prove that u also belongs to U by induction. The following result on the distribution of primes is due to Sylvester and quoted by Hanani in [2]. Lemma 3.9. Let x ≥ 33. Then there exists a prime between x and 8x/7. Theorem 3.10. Let u ≥ 15 be an integer and u ̸ ∈ {17, 26, 27, 29, 31, 33}. Then there exists a 1-FG(3, (M , N), u) with group sizes from [2, 14] \ {7}, where M = {4, 5, . . . , 13, 19} and N = {k : k is an integer and k ≥ 4}. Proof. By Lemmas 3.6–3.8 the given values of u < 34 have been taken care of and also we have [34, 532 ] ⊂ U. Our proof will be based on induction. Let q be a prime and q ≥ 53. We make our assumption that [34, q2 ] ⊂ U for the induction. Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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5 q′ +3

By Lemma 3.9, there always exists a prime q′ such that q + 1 ≤ q′ ≤ 8(q + 1)/7. Apply Lemma 3.5 with r = ⌈ 2 ⌉, s = q′ − r and ai ∈ U ∪ Q , where r ≥ 30 and s ≥ 3 since q′ > q ≥ 53. Also, by assumption and q′ < q2 we have ′ [34, q′ ] ⊂ U. From Lemma 3.5 we know that u ∈ U for any integer u ∈ [q′ ⌈ q 2+3 ⌉, q′2 ]. Since q ≥ 53 and q′ ≤ 8(q + 1)/7, we have q′ ⌈

q′ +3 2

⌉ < q2 . It follows that [34, q′2 ] ⊂ U. By induction, u ∈ U for any integer u ≥ 34. □

4. 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m In this section we show that there is a 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m for m ∈ M = {4, 5, . . . , 13, 19}. Lemma 4.1. There is a 3-FG(3, (3, 3, 4, 4), 16) of type 44 . Proof. Let G = (GF (4), +), where the primitive polynomial f (x) = x2 + x + 1 over Z2 is used to generate the field. Let ξ be a primitive element of the field. Denote the field elements 0, 1, ξ , ξ 2 by 0, 1, 2, 3. We construct a 3-FG(3, (3, 3, 4, 4), 16) on G × Z4 having groups G × {j}, 0 ≤ j ≤ 3. First, construct two GDD(2, 3, 16)’s of type 44 . The blocks are generated by the following base blocks under (G, −), where the first eight base blocks generate the block set of a GDD(2, 3, 16), so do the other base blocks, and the symbol ab stands for (a, b). Here, a base block {ab , cd , ef } under (G, −) yields blocks {(a + x)b , (c + x)d , (e + x)f }, x ∈ G. 00 01 02 00 01 12

00 11 22 00 11 32

00 21 03 00 21 13

00 31 33 00 31 23

00 12 13 00 02 03

00 32 23 00 22 33

01 12 33 01 02 23

01 22 13 01 32 03

Then we construct the other blocks of size 4. They are generated by the following base blocks under (G, −), where the first four base blocks generate the block set of a GDD(2, 4, 16) of type 44 . 00 00 00 00 01

01 10 30 21 21

22 22 03 22 03

03 13 23 23 33

00 00 00 00 01

11 20 01 31 31

12 01 32 12 02

23 11 13 03 22

00 00 00 00 01

21 20 01 31 31

32 02 23 22 03

33 32 33 32 23

00 00 00 01 02

31 20 11 11 12

02 03 02 02 03

13 33 33 13 23

00 00 00 01 02

10 30 11 11 22

01 01 03 22 03

31 21 13 23 33

00 00 00 01 02

10 30 21 21 32

02 02 02 02 03

23 22 12 32 13

□

When we construct a 3-FG(3, (3, 3, 4, 4), 48) of type 124 , we need a special 3-BD of order 12. Let (X , G ) be a 1-design of type 34 where G = {G0 , G1 , G2 , G3 }. An S(3, {4, 6}, 12)(X , C ) is denoted by S∗ (3, {4, 6}, 12) if it satisfies that (1) Gi ∪ Gi+1 , i = 0, 2, are the only two blocks of size six; (2) there is a subset D ⊂ C such that (X , G , D) is a GDD(2, 4, 12). From the proof of [7, Lemma 4.10], an S∗ (3, {4, 6}, 12) exists. We also need the following lemma, which is proved by Stern and Lenz in [16]. Lemma 4.2 ([16]). Let G be a graph with vertex set Z2k and let L be a set of integers in the range 1, 2, . . . , k such that {a, b} is an edge of G if and only if |b − a| ∈ L, where |b − a| = b − a if 0 ≤ b − a ≤ k, and |b − a| = a − b if k < b − a < 2k. Then G has a one-factorization if and only if 2k/gcd(j, 2k) is even for some j ∈ L. Lemma 4.3. There is a 3-FG(3, (3, 3, 4, 4), 48) of type 124 . Proof. Let (X , G ) be a 1-design of type 34 where G = {G0 , G1 , G2 , G3 }. Let (X , T ) be an S∗ (3, {4, 6}, 12), which exists from the proof of [7, Lemma 4.10]. Let its two disjoint blocks of size six be Gi ∪ Gi+1 , i = 0, 2. Also, there is a subset B ⊂ T such that (X , G , B) is a GDD(2, 4, 12). We shall construct the desired design on X × Z4 with groups Gi × Z4 , 0 ≤ i ≤ 3. For each block B ∈ T \ B with |B| = 4, by Theorem 2.3 we can construct an H(4, 4, 4, 3) on B × Z4 having groups {x} × Z4 , x ∈ B. Denote the block set by CB . For each block B ∈ B, construct a 3-FG(3, (3, 3, 4, 4), 16) on B × Z4 having groups {x} × Z4 , x ∈ B. Such a 3-FG exists from Lemma 4.1. Denote its block sets of the two subdesigns GDD(2, 3, 16)’s by DB1 and DB2 . Denote the block set of the subdesign GDD(2, 4, 16) by DB3 and the set of other blocks by DB′ . For 0 ≤ i ≤ 3, we consider the multipartite graph Γi on Gi × Z4 that contains all edges {(x, s), (x′ , s′ )} with x ̸ = x′ . Such a graph can be thought as a graph on Z12 having all edges {a, b} with |a − b| ∈ L = {1, 2, 4, 5}. By Lemma 4.2 Γi has a one-factorization F i = {F1i , . . .⋃ , F8i }. For j = 0, 2, let ⋃ Aj = {{a, b, c , d}: {a⋃ , b} ∈ Fkj , {c , d} ∈ Fkj+1 , 1 ≤ k ≤ 8}. i ′ i ′ 0 2 For 1 ≤ i ≤ 3, let D = B∈B DB . Let D = ( B∈T \B,|B|=4 CB ) ∪ ( B∈B DB ) ∪ A ∪ A . It is routine to check that 1 2 3 ′ 4 (X × Z4 , {G × Z4 : G ∈ G }, D , D , D , D ) is the desired 3-FG(3, (3, 3, 4, 4), 48) of type 12 . □ Lemma 4.4. There is a 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m for m ∈ {5, 8, 13}. Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Proof. For the given m ∈ {5, 8}, deleting two points from an S(3, 5, 3 m + 2) yields a 1-FG(3, (4, {4, 5}), 3 m) of type 3m , where an S(3, 5, 17) exists by Lemma 3.4 and an S(3, 5, 26) exists in Table III in [3]. Apply Theorem 2.4 with the known H(k, 4, 4, 3) for k ∈ {4, 5} in Theorem 2.3 and 3-FG(3, (3, 3, 4, 4), 16) of type 44 in Lemma 4.1. Then we obtain a 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m . For m = 13, start with a 1-FG(3, (4, {4, 5, 13}), 39) of type 313 , which exists from the proof of [6, Lemma 4.14]. Apply Theorem 2.4 with the known H(r , 4, 4, 3) for r ∈ {4, 5, 13} and 3-FG(3, (3, 3, 4, 4), 16) of type 44 . The desired design is obtained. □ Lemma 4.5. There exists a 3-FG(3, (3, 3, 4, 4), 144) of type 1212 . Proof. Let (X , G , T ) be an H(12, 3, 4, 3) containing a subdesign H(12, 3, 4, 2)(X , G , B) where G = {G1 , . . . , G12 }. Such a design exists from [7, Lemma 4.12]. We shall construct the desired design on X × Z4 with groups G × Z4 , G ∈ G . For each block of B ∈ T \ B, by Theorem 2.3 we can construct an H(4, 4, 4, 3) on B × Z4 having groups {x} × Z4 , x ∈ B. Denote its block set by CB . For each block B ∈ B, by Lemma 4.1 we can construct a 3-FG(3, (3, 3, 4, 4), 16) of type 44 on B × Z4 having groups {x} × Z4 , x ∈ B. Denote its block sets of two GDD(2, 3, 16)’s of type 44 by DB1 and DB2 , respectively. Denote the block set of a GDD(2, 4, 16) by DB3 and the set of the other blocks by DB′ . Similar to the proof of Lemma 4.3, for each 1 ≤ i ≤ 12, we consider the multipartite graph ΓGi on Gi × Z4 that contains all edges {(x, s), (x′ , s′ )} with x ̸ = x′ . G G Let FGi = {F1 i , . . . , F8 i } be a one-factorization of ΓGi . Let G

Gj

A = {{a, b, c , d}: {a, b} ∈ Fk i , {c , d} ∈ Fk , 1 ≤ k ≤ 8, 1 ≤ i < j ≤ 12}.

For 1 ≤ i ≤ 3, let Di = ∪B∈G DBi . Denote D′ = (∪B∈B DB′ ) ∪ (∪B∈T \B CB ). It is routine to check that (X × Z4 , {G × Z4 : G ∈ G }, D1 , D2 , D3 , D′ ∪ A) is the desired design.

□

When we give another construction for 3-FGs, a special 2-FG(3, (4, 4, K ), v ) and a special GDD are needed. Let (X , G , B1 , B2 , T ) be a 2-FG(3, (4, 4, K ), v ). For any two points from two distinct groups, link them with a red or a blue edge, and for any two points from the same group, link them with a blue edge. If such a coloring satisfies the following properties, then the 2-FG is said to be a good 2-FG(3, (4, 4, K ), v ): (P1) each block of B1 contains exactly two disjoint red edges; |A|(|A|−2) red (P2) there is a subset A ⊂ T satisfying that each block A ∈ A has exactly |A|/2 disjoint blue edges and 2 edges, each red edge is contained in exactly one block A ∈ A, and all blue edges contained in A ∈ A are from groups. If the block sizes of A and T \ A are from K1 and K2 , respectively, then the good 2-FG is denoted by 2-FG(3, (4, 4, (K1 , K2 )), v ). A 3-FG(3, (3, 3, 4, 4), v ) of type r m is called compatible to a good 2-FG if the 3-FG and the good 2-FG have the common block set B1 ∪ A. Lemma 4.6. There exists a GDD(3, {3, 4}, 12) of type 34 such that there are three disjoint blocks of size 4 and the blocks of size 3 can be partitioned into two parts, each together with the three disjoint blocks, form the block set of a GDD(2, {3, 4}, 12) of the same group set. Proof. Take the point set Z12 and the group set G = {{4i + j: 0 ≤ i ≤ 2}: 0 ≤ j ≤ 3}. Construct blocks Hb = {3a + b: 0 ≤ a ≤ 3}, 0 ≤ b ≤ 2, and {i, 1 + i, 6 + i, 7 + i}, 0 ≤ i ≤ 5. The other blocks are generated by the following three base blocks modulo 12, where the first two base blocks of size 3 generate B1 under (+2 mod 12) and B2 = {B + 1: B ∈ B1 }. 012

138

0235

It is easy to see that (X , G , Bi ∪ {H0 , H1 , H2 }) is a GDD(2, {3, 4}, 12). All blocks form the block set of the desired design. □ Lemma 4.7. Suppose that there exists a good 2-FG(3, (4, 4, (K1 , 4)), rs) of type r s and a 3-FG(3, (3, 3, 4, 4), rs) of type r s compatible to the good 2-FG. If there is a G(k1 /2, 6, 4, 3) for any k1 ∈ K1 , then there exists a 3-FG(3, (3, 3, 4, 4), 3rs) of type (3r)s . Proof. Let (X , G , B1 , B2 , T ) be a given good 2-FG(3, 4, 4, (K1 , 4), v ) and let (X , G , C1 , C2 , B1 , T ′ ) be a given 3-FG(3, (3, 3, 4, 4), v ) compatible to the good 2-FG. Then there is a subset A ⊂ T ∩ T ′ such that B1 ∪ A have the properties (P1) and (P2). For each block B ∈ B2 , by Lemma 4.6 we can construct a special GDD(3, {3, 4}, 12) on B × Z3 , x ∈ B, such that B × {i}, 0 ≤ i ≤ 2, are blocks. Then the blocks of size 3 can be partitioned into DB1 and DB2 with the property that j (B × Z3 , {{x} × Z3 : x ∈ B}, {B × {i}: i ∈ Z3 } ∪ DB ) is a GDD(2, {3, 4}, 12) for j = 1, 2. Denote the set of blocks of size 4 ′ except B × {i} (0 ≤ i ≤ 2) by DB . For each block B ∈ B1 , construct an S∗ (3, {4, 6}, 12) on B × Z3 such that {x, y} × Z3 is a block, where {x, y} is colored red in B. Such a 3-BD exists from the proof of [7, Lemma 4.10]. Denote its block set of the subdesign GDD(2, 4, 12) by εB and the set of the other blocks of size 4 by εB′ . For each B ∈ T \ A, construct an H(4, 3, 4, 3) on B × Z3 having groups {x} × Z3 , x ∈ B, such that B × {i}, 0 ≤ i ≤ 2, are blocks. Such a design exists by [15, Theorem Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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12.4]. Denote its set of blocks except B × {i}(i ∈ Z3 ) by FB . For each block A ∈ A, construct a G(|A|/2, 6, 4, 3) on A × Z3 with groups {x, y} × Z3 , where {x, y} is a blue edge in A. Such a design exists by assumption. Denote its block set by HA . For 0 ≤ i ≤ 2, construct a 3-FG(3, (3, 3, 4, 4), v ) compatible to the good 2-FG on X × {i} having {G × {i}: G ∈ G } as its group set and {B × {i}: B ∈ C1 ∪ C2 ∪ B1 ∪ T ′ } as its block set. Let G ′ = {G × Z3 : G ∈ G } and C i = (∪B∈B2 DBi ) ∪ (∪0≤j≤2 {C × {j}: C ∈ Ci }) for i = 1, 2. Denote C 3 = ∪B∈B1 εB and 4 C = (∪B∈B1 εB′ ) ∪ (∪B∈B2 DB′ ) ∪ (∪B∈T \A FB ) ∪ (∪0≤j≤2 {C × {j}: C ∈ T ′ }) ∪ (∪A∈A HA ). Then (X × Z3 , G ′ , C 1 , C 2 , C 3 , C 4 ) is the desired design, where (X × Z3 , G ′ , C i ) is a subdesign GDD(2, 3, 3rs) for i = 1, 2 and a subdesign GDD(2, 4, 3rs) for i = 3. This completes the proof. □ a

a

A GDD(3, K , v ) of type g1 1 g2 2 · · · gr r is called s-fan if its block set B can be partitioned into disjoint subsets B1 , · · · , Bs and T such that for each i, 1 ≤ i ≤ s, Bi is the block set of a GDD(2, Ki , v ) of the same group set. If block sizes of T are all from KT , then it is denoted by s-fan GDD(3, (K1 , K2 , . . . , Ks , KT ), v ). In the sequel, to shorten the list of base blocks we use multipliers whenever it is possible. An element m ∈ Zv is called a multiplier of a design if for any block B = {x1 , x2 , . . . , xr }, mB = {mx1 , mx2 , . . . , mxr } is also a block of the design. It is clear that the set of all multipliers form a group, denoted by P. When multipliers are used, we list all the multipliers and those base blocks with which all base blocks can be generated by using the multipliers. In other words, all blocks of the design can be generated from the shortened list of base blocks under the automorphism group {x ↦ → mx + b : m ∈ P , b ∈ Zv } of the design. In some cases, to avoid repetition we list corresponding multipliers for each base block in the shortened list. a

Lemma 4.8. There exists a 3-FG(3, (3, 3, 4, 4), 228) of type 1219 . Proof. We shall construct a good 2-FG(3, (4, 4, (4, 4)), 76) and a 3-FG(3, (3, 3, 4, 4), 76) compatible to the good 2-FG on Z76 having groups Gj = {19i + j: 0 ≤ i ≤ 3}, 0 ≤ j ≤ 18. From the proof of [6, Lemma 4.9], a good 2-FG(3, (4, 4, 4), 76) of type 419 exists. The base blocks of a subdesign GDD(2, 4, 76) are listed below, and multiplying each of them by 5 gives the base blocks of the other subdesign GDD(2, 4, 76). Denote the block set of the first GDD(2, 4, 76) by B. 0138

0 4 29 60

0 6 18 42

0 9 48 63

0 10 33 59

0 11 32 46

Let S = {2, 8, 9, 15, 17, 18}. For any two distinct points x, y, if |x − y| ∈ S ∪ (38 − S), then we link x and y with a red edge, otherwise a blue edge. Then B has the property (P1). Also there is a subset A = {{k, d + k, 38 + k, 38 + d + k}: d ∈ S , 0 ≤ k ≤ 37} having the property (P2). Now, we construct a 3-fan GDD(3, (3, 3, 4, 4), 76) with the multiplier group Q = {5i : 0 ≤ i ≤ 8} The list of base blocks is as follows, where the underlined base blocks generated the block set of a subdesign GDD(2, 4, 76) and Q ′ = {1, 5, 25}. The blocks with a star (or two stars) under the action of the multiplier group generate the set of base blocks of a GDD(2, 3, 76). Denote the set of all blocks of this 3-fan GDD by C1 m ∈ Q ′: m ∈ Q:

0 0 0 0 0

4 32∗ 1 11∗ 10 33 59 1 14 35 1 36 71

0 0 0 0 0

4 48∗∗ 1 66∗∗ 11 32 46 1 16 42 1 40 74

0 0 0 0

1 1 1 1

38 25 21 53 41 64

0 0 0 0

4 1 1 2

29 60 9 29 22 48 6 16

0 0 0 0

6 1 1 2

18 10 23 13

42 50 55 36

0 0 0 0

9 1 1 2

48 13 33 18

63 18 44 62

From such a 3-fan GDD, for 0 ≤ i ≤ 18, let Fi1 = {{i, 38 + i}, {19 + i, 57 + i}}, Fi2 = {{i, 19 + i}, {38 + i, 57 + i}}, = {{i, 57 + i}, {19 + i, 38 + i}}. Let C2 = {{a, b, c , d}: {a, b} ∈ Fmj , {c , d} ∈ Fnj , 0 ≤ m < n ≤ 18, 1 ≤ j ≤ 3}. Then C1 ∪ C2 form the block set of a 3-FG(3, (3, 3, 4, 4), 76). It is easy to see that the 2-FG has the common block subset B ∪ A with the above 3-FG. So, the 3-FG is indeed compatible to the good 2-FG. Since there is a G(2, 6, 4, 3) by Lemma 2.1, the conclusion follows by Lemma 4.7 . □

Fi3

Lemma 4.9. There exists a 3-FG(3, (3, 3, 4, 4), 120) of type 1210 . Proof. First, we construct a good 2-FG(3, 4, 4, ({4, 10}, 4), 40) and a 3-FG(3, (3, 3, 4, {4, 10}), 40), compatible to the good 2-FG on Z40 having groups Gj = {10i + j: 0 ≤ i ≤ 3}, 0 ≤ j ≤ 9. Their common block set contains the blocks in A = {{4i + j: 0 ≤ i ≤ 9}: 0 ≤ j ≤ 3}∪{{i, i + 5, i + 20, i + 25}: 0 ≤ i ≤ 19} and blocks {2i, 2i + 5, 2i + 10, 2i + 15}, {2i, 2i + 10, 2i + 25, 2i + 35}, 0 ≤ i ≤ 19. The other common blocks are generated by the following 26 base blocks modulo 40, where the underlined blocks generate the block set B1 of a GDD(2, 4, 40) of type 410 and the base blocks with a star generate the block set B2 of the other GDD(2, 4, 40) of type 410 . 0 0 0 0

1 4 13 1 2 21 1 10 11 10 18 28

0 0 0 0

2 7 24 2 4 22 2 10 12 10 19 29

0 6 21 32 0 3 6 23 0 3 10 13

0 1 28 37∗ 0 6 12 26 0 4 10 14

0 2 18 35∗ 0 7 14 27 0 10 16 26

0 6 14 25∗ 0 9 18 29 0 10 17 27

Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Then we construct the other blocks of the 2-FG(3, (4, 4, (10, 4)), 40), which are generated by the following base blocks. 0 0 0 0 0 0

1 1 2 3 5 7

37 17 18 16 19 18 25 16 21 19 28

0 0 0 0 0 0

1 1 2 4 5 8

34 38 19 22 23 26 9 35 18 27 17 31

0 0 0 0 0

1 2 2 4 5

56 5 11 17 25 11 33 19 26

0 0 0 0 0

1 2 3 4 6

89 31 37 8 35 15 19 15 31

0 0 0 0 0

1 2 3 4 6

12 14 8 15 11 14 17 21 17 29

0 0 0 0 0

1 2 3 4 6

27 27 15 18 18

29 34 28 26 24

0 0 0 0 0

1 2 3 5 6

15 16 9 33 16 27 12 17 19 27

Combining these blocks and the common blocks, we obtain a 2-FG(3, (4, 4, 4), 40), where B1 and B2 are two block sets of GDD(2, 4, 40)’s. In fact, the above 2-FG is also a good 2-FG(3, (4, 4, ({4, 10}, 4)), 40). For any two points x and y with x > y, if x − y ∈ {4, 5, 8, 12, 15, 16, 24, 25, 28, 32, 35, 36}, then we link x and y with a red edge, otherwise a blue edge. Further B1 has property (P1) and A has property (P2). Thus it is indeed a good 2-FG. Then we construct the other blocks of the required 3-FG compatible to the 2-FG, which are generated by the following base blocks, where the first (resp. second) six base blocks generate the block set of a GDD(2, 3, 40) of type 410 . 0 0 0 0 0 0 0

1 1 1 2 3 4 5

3 38 78 5 19 7 32 11 17 18 27

0 0 0 0 0 0 0

4 4 1 2 3 4 6

9 35 9 14 23 37 11 36 27 33 15 27

0 0 0 0 0 0 0

6 6 1 2 3 4 6

18 28 27 32 68 8 27 18 26 19 31

0 0 0 0 0 0 0

7 7 1 2 3 4 7

26 21 12 15 9 33 16 35 19 23 18 25

0 0 0 0 0 0 0

8 8 1 2 3 5 8

23 25 26 29 11 13 9 25 11 16 17 31

0 0 0 0 0 0 0

11 27 11 24 1 16 17 2 14 16 3 18 34 5 12 33 156

0 0 0 0

1 2 3 5

18 15 19 17

19 17 22 28

Since this 3-FG and the 2-FG have A ∪ B1 in common, this 3-FG is compatible to the 2-FG. Further, since a G(5, 6, 4, 3) exists by Lemma 2.1, the conclusion follows by Lemma 4.7. This completes the proof. □ Lemma 4.10. There exists a 3-FG(3, (3, 3, 4, 4), 72) of type 126 . Proof. We shall construct a 3-FG(3, (3, 3, 4, 4), 72) on Z72 having groups Gj = {6i + j: 0 ≤ i ≤ 11}, 0 ≤ j ≤ 5. The block set contains three kinds of blocks described below. For each i ∈ S = {1, 2, . . . , 17} \ {3, 6, 9, 12, 15}, construct blocks {j, j + i, j + 2i, j + 36 + i}, j ∈ Z72 . When i runs through S, all these blocks form the first part of the block set. j j For 0 ≤ j ≤ 5, let F ′ = {F1 , . . . , F11 } be a one-factorization of the complete graph on Gj such that F11 = {{j + 6i, j + 6i + 36}: 0 ≤ i ≤ 5}. It is clear that such a one-factorization exists. For any {c , d} ∈ Fkj and any {c ′ , d′ } ∈ Fkl , j j+3 construct blocks {c , d, c ′ , d′ }, where 0 ≤ j < l ≤ 5 and 1 ≤ k ≤ 10. For 0 ≤ j ≤ 2, {c , d} ∈ F11 and any {c ′ , d′ } ∈ F11 , ′ ′ construct blocks {c , d, c , d }. All these blocks form the second part of the block set. The blocks in the third part of the block set are generated by the following base blocks under the multiplier group Q = {1, −1}, where the ten blocks of size 3 form the set B of base blocks of a GDD(2, 3, 72) with the same group and −B is the set of base blocks of the other GDD(2, 3, 72). Note that the underlined blocks form the set of base blocks of a GDD(2, 4, 72) with the same groups. m = 1:

m ∈ Q:

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 9 10 1 28 29 3 35 38 7 16 63 7 34 45 9 23 32 13 32 45 16 35 51 13 14 39 5 21 44 2 17 21 3 16 20 5 25 46 8 19 28 14 29 51

0 0 0 0 0 0 0

1 14 15 1 32 33 5 14 19 7 21 28 8 21 59 9 34 43 13 33 46

0 0 0 0 0 0 0

1 16 17 1 34 35 5 15 20 7 22 29 8 23 57 10 21 61 14 33 53

0 0 0 0 0 0 0

1 20 21 257 5 16 61 7 26 53 8 25 33 11 26 57 15 31 56

0 0 0 0 0 0 0

1 22 23 2 9 11 5 27 50 7 27 52 8 31 39 11 28 55 15 32 55

0 0 0 0 0 0 0

1 26 27 2 35 39 5 32 37 7 32 39 8 35 43 13 29 56 16 33 55

0 0 0 0 0 0 0

49 16 38 8 22 53 2 22 25 3 22 26 5 26 34 9 19 44

0 0 0 0 0 0 0

7 15 19 40 9 29 46 2 23 27 3 28 32 5 28 39 9 26 40

0 0 0 0 0 0 0

10 27 20 43 158 2 28 31 4 13 35 7 17 38 10 23 37

0 0 0 0 0 0 0

11 37 1 4 11 2 10 13 2 29 33 4 14 61 7 20 35 10 29 44

0 0 0 0 0 0 0

13 41 2 15 40 2 16 19 3 14 34 5 13 22 7 23 33 10 32 43 □

Lemma 4.11. There exists a 3-FG(3, (3, 3, 4, 4), 108) of type 129 . Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Proof. We shall construct a 3-FG(3, (3, 3, 4, 4), 108) on Z108 having groups Gj = {9i + j: 0 ≤ i ≤ 11}, 0 ≤ j ≤ 8. The block set will contain three kinds of blocks described below. First, construct two GDD(2, 3, 108)’s. Developing the following base blocks under (+2, mod 108) gives the block set B of a GDD(2, 3, 108) and B + 1 is the block set of the other GDD(2, 3, 108). All these blocks form the first part of the block set. 0 j 2j, 0 ≤ j ≤ 27, j ̸ ≡ 0 (mod 9), j ≡ 1 (mod 2) 1 1 + j 1 + 2j, 27 ≤ j ≤ 54, j ̸ ≡ 0 (mod 9), j ≡ 1 (mod 2) 0 4 12 1 5 101 0 16 44 1 17 81 0 20 60 1 21 69

0 24 56

1 25 77

The blocks in the second part of the block set are generated by the following base blocks with the multiplier group Q = {±13i : 0 ≤ i ≤ 8}, where the underlined blocks generate the block set of a GDD(2, 4, 108) and Q ′ = {13i : 0 ≤ i ≤ 8}, Q ′′ = {13i : 0 ≤ i ≤ 2}. Note that the base blocks with a star generate blocks twice, i.e., {0, 3, 54, 57} = {0, 3, 54, 57} + 54, each should be taken only once. m=1

m ∈ Q ′′

m ∈ Q′ m∈Q

0 0 0 0 0 0 0 0 0 0 0

4 16 20 8 24 92 12 32 44 3 9 12 3 51 87 6 63 87 1 17 92 2 33 77 1 3 94 11 30 71 1 34 51

0 0 0 0 0 0 0 0 0 0 0

4 24 28 8 32 60 12 52 68 3 15 24 3 60 96 6 12 60 1 40 41 2 4 56 4 38 62 13 44 69 1 54 55∗

0 0 0 0 0 0 0

4 32 52 8 56 84 16 48 64 3 18 21 6 15 21 3 54 57∗ 1 43 44

0 5 12 73 0157

0 0 0 0 0

4 60 80 8 40 48 20 44 84 3 27 30 6 18 24

0 4 44 48 0 8 52 64

0 8 20 28 0 12 28 40

0 3 33 36 0 6 27 51

0 3 45 48 0 6 30 36

0 2 16 94

0 2 21 89

0 2 31 79

0 6 22 88 0 1 6 47

0 8 29 57 0 1 11 25

0 10 33 65 0 1 14 52

j

j

Now, we shall give the third part of the block set. For 0 ≤ j ≤ 8, let F j = {F1 , . . . , F11 } be a one-factorization of the j complete graph on Gj such thatF11 = {{j + 9i, j + 9i + 54}: 0 ≤ i ≤ 5}. It is clear that such a one-factorization exists. For j ′ ′ any {c , d} ∈ Fk and any {c , d } ∈ Fkl , construct blocks {c , d, c ′ , d′ }, where 0 ≤ j < l ≤ 8, j ̸ ≡ l (mod 3) and 1 ≤ k ≤ 10. All these blocks form the third part of blocks. □ Lemma 4.12. There exists a 3-FG(3, (3, 3, 4, 4), 84) of type 127 . Proof. A 3-FG(3, (3, 3, 4, 4), 84) is constructed on Z84 with groups Gj = {7i + j: 0 ≤ i ≤ 11}, 0 ≤ j ≤ 6, and the multiplier group Q = {47i : 0 ≤ i ≤ 5}. The shortened list of base blocks is as follows, where the underlined blocks generate the block set of a GDD(2, 4, 84) with the same groups. The blocks of size 3 under the multiplier group Q ′ = {25i : 0 ≤ i ≤ 2} generate the set B of base blocks of a GDD(2, 3, 84), −B is the set of base blocks of the other GDD(2, 3, 84). m ∈ {1, −1} m ∈ Q′

m∈Q

0 0 0 0 0 0 0 0 0 0 0 0

4 20 1 10 11 2 15 17 3 20 23 2 4 44 15 21 36 13 11 30 61 1 25 40 3 11 33 1 14 22 2 23 28

0 0 0 0 0

8 1 2 4 3

40 16 17 32 54 15 19 6 45

0 0 0 0 0

12 36 1 19 20 2 41 45 4 40 48 4 8 46

0 0 0 0

2 3 5 6

8 10 9 12 24 65 12 48

0 0 0 0

2 11 75 3 18 69 5 29 60 11 22 53

0 0 0 0

2 3 1 3

0 0 0 0 0 0

5 11 12 25 57 1 30 34 3 15 41 1 15 28 3 7 38

0 0 0 0 0 0

15 46 15 37 55 1 32 41 4 17 22 1 21 29 3 17 56

0 0 0 0 0 0

1 1 2 4 2 4

46 5 13 18 24 23 36 7 51 11 49

0 0 0 0 0 0

8 1 2 1 2 5

0 0 0 0 0 0

10 26 46 1 23 26 2 33 46 1 8 35 2 16 21 6 14 62 □

17 41 12 18 19 57 7 36 14 30 12 35

12 74 19 22 2 43 21 24

Lemma 4.13. There exists a 3-FG(3, (3, 3, 4, 4), 132) of type 1211 . Proof. A 3-FG(3, (3, 3, 4, 4), 132) is constructed on Z32 with groups Gj = {11i + j: 0 ≤ i ≤ 11}, 0 ≤ j ≤ 10, and the multiplier group Q = {±25i : 0 ≤ i ≤ 4}. The shortened list of base blocks is as follows, where the underlined base blocks generate the block set of a GDD(2, 4, 132) with the same groups. The first four base blocks of size 3 under the action of the multiplier group Q ′ = {25i : 0 ≤ i ≤ 4} generate the set B of base blocks of a GDD(2, 3, 13), −B is the set of base blocks of the other GDD(2, 3, 132). Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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m ∈ Q′

m∈Q

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

156 1 19 20 1 65 68 2 26 28 4 9 13 2 4 68 23 33 56 13 4 57 84 6 34 65 1 52 59 2 34 74 3 9 63 4 23 90 1 12 55 2 22 46 6 17 77

0 0 0 0 0 0

1 1 2 2 4 3

78 24 25 57 47 87 12 124 6 69

0 0 0 0 0 0

1 1 2 2 5 5

9 10 28 29 68 59 75 19 24 10 71

0 0 0 0 0 0

1 1 2 3 6 6

13 31 10 13 24 12

14 32 12 16 30 72

0 0 0 0 0 0

1 15 16 1 42 43 2 14 16 3 26 29 9 38 47 10 20 76

0 0 0 0 0 0

1 1 2 3 1 5

17 18 57 58 17 19 48 51 2 67 33 38

0 0 0 0 0 0 0 0 0

4 10 3 46 93 13 36 71 1 61 63 2 40 69 3 15 41 4 30 47 1 22 34 2 33 90

0 0 0 0 0 0 0 0 0

5 17 10 25 70 1 21 26 2 9 15 2 41 53 3 39 56 4 43 73 1 23 44 3 14 55

0 0 0 0 0 0 0 0 0

9 1 1 2 2 3 5 1 4

56 30 36 20 42 45 41 33 22

50 40 23 48 76 65 45 103

0 0 0 0 0 0 0 0 0

14 40 78 2 18 113 1 39 46 2 29 32 3 7 20 4 18 85 6 26 86 2 11 79 4 26 44

0 0 0 0 0 0 0 0 0

8 17 108 5 12 81 1 47 60 2 30 43 3 8 31 4 20 56 1 11 56 2 13 77 5 22 93 □

Combining Lemmas 4.3–4.5, Lemmas 4.8–4.13, we have the following. Lemma 4.14. There is a 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m for m ∈ M = {4, 5, . . . , 13} ∪ {19}. 5. Small orders for 1-AFSQSs In this section, we show that there exists a 1-AFSQS(12k + 2) for any k ∈ Q where Q = {2, 3, . . . , 14} \ {7}. Lemma 5.1 ([11]). There is a 1-AFSQS(v ) for v ∈ {26, 38}. Lemma 5.2. There exists a 1-AFSQS(170). Proof. Start with a 1-FG(3, (4, 4), 42) of type 67 from [11, Lemma 5.1]. We can obtain a 3-FG(3, (3, 3, 4, 4), 168) of type 247 by applying Theorem 2.4 with the known 3-FG(3, (3, 3, 4, 4), 16) of type 44 from Lemma 4.1 and H(4, 4, 4, 3) from Theorem 2.3. Further apply Lemma 2.2 with the known 1-AFSQS(26) from Lemma 5.1. The conclusion follows. □ A lattice t-design, denoted by LD(m, n, K , t), is a quadruple (X , G , H, B) where X is a set of mn elements, G is a partition of X into m subsets of size n, H is a partition of X into n subsets of size m, B is a family of subsets with cardinality from K , called blocks, satisfying the following properties: (1) |G ∩ H | = 1 for each G ∈ G and each H ∈ H, (2) each block intersects each G ∈ G in at most one point, (3) each block intersects each H ∈ H in at most one point, and (4) each t-set of points, which meets each G ∈ G in at most one point and meets each H ∈ H in at most one point, is contained in exactly one block. G is called the group set and H is called the hole set. Mohácsy and Ray-Chaudhuri pointed out an equivalence between ordered designs and lattice designs [15]. Theorem 5.3 ([15,17]). There is an LD(4, m, 4, 3) for any integer m ≥ 4 with m ̸ = 7. Lemma 5.4. There is an H(4, 5, {3, 4}, 3) containing a subdesign H(4, 5, 4, 2) and there are five disjoint blocks in the subdesign H(4, 5, 4, 2) such that the blocks of size 3 can be partitioned into two parts, each, together with the five disjoint blocks, forming the block set of an H(4, 5, {3, 4}, 2). Proof. Take a point set Z20 with groups {4i + j: 0 ≤ i ≤ 4}, 0 ≤ j ≤ 3. Its block set contains 5 special blocks {i, i + 5, i + 10, i + 15}, 0 ≤ i ≤ 4. It also contains blocks {i, i + 1, i + 10, i + 11}, {i, i + 3, i + 10, i + 13}, 0 ≤ i ≤ 9, and the blocks generated by the following base blocks modulo 20. Note that the blocks generated by the underlined blocks, together with the five special blocks, form the block set of a GDD(2, 4, 20). Furthermore, developing the base blocks of size 3 through (+2 mod 20), together with the five special blocks, form the block set of a GDD(2, {3, 4}, 20). The other blocks of size 3, together with the 5 five special blocks, also form the block set of the other GDD(2, {3, 4}, 20). 012

036

1 10 19

1 8 15

0257

0 1 3 14

0 1 7 18

0 5 6 11

□

Lemma 5.5. There exists a 1-AFSQS(98). Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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11

Proof. Let (X , G , B, T ) be a 1-FG(3, (5, {5, 6}), 24) of type 46 , which can be obtained by deleting two points from an S(3, 6, 26) in Lemma 3.4. First, we construct a 3-FG(3, (3, 3, 4, 4), 96) on X × Z4 with groups X × {i}, i ∈ Z4 . For each block B ∈ T , by Theorem 5.3 we can construct an LD(4, |B|, 4, 3) on B × Z4 with groups B × {i}, i ∈ Z4 , and holes {x} × Z4 , x ∈ B. Denote its block set by DB . For each block B ∈ B, by Lemma 5.4, we can construct an H(4, 5, {3, 4}, 3) containing a subdesign H(4, 5, 4, 2) on B × Z4 with groups B × {i}, i ∈ Z4 such that {x} × Z4 , x ∈ B, are the five special blocks in the subdesign H(4, 5, 4, 2) and that the blocks of size 3 can be partitioned into two parts, each, together with the five disjoint blocks, forming the block set of an H(4, 5, {3, 4}, 2). Denote the set of blocks in the subdesign H(4, 5, 4, 2) except the five special blocks by A3B . Denote two parts of blocks of size 3 by A1B and A2B , respectively. Denote by A4B the set of the other blocks of size 4 except the five special blocks. For each group G, by Lemma 4.1 we can construct a 3-FG(3, (3, 3, 4, 4), 16) on G × Z4 having groups G × {i}, i ∈ Z4 . Denote its block set of a subdesign GDD(2, 4, 16) by DG3 . Denote its two block sets of subdesigns GDD(2, 3, 16) by DG1 and DG2 , and the set of the other blocks by DG4 . For each i ∈ Z4 , consider a graph Γi on X × {i} containing all edges of the form {(x, i), (y, i)} where x, y are from distinct groups of the 1-FG(3, (5, {5, 6}), 24). Such a graph can be thought as a graph on Z24 having all edges {a, b} i with |a − b| ∈ L = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11} . By Lemma 4.2 there is a one-factorization. Let {F1i , . . . , F20 } be a onefactorization of the graph Γi . For any {c , d} ∈ Fki and {c ′ , d′ } ∈ Fkl , construct blocks {c , d, c ′ , d′ }, where 0 ≤ i < l ≤ 3 and 1 ≤ k ≤ 20. Denote the set of all these blocks by ε . For G, G′ ∈ G , G ̸ = G′ , 0 ≤ i < j ≤ 3, by Lemma 2.1 we can construct a G(2, 4, 4, 3) with groups G × {i} and G′ × {j}. Denote the set of all these blocks by ε ′ . For 1 ≤ i ≤ 4, let F i = (∪B∈B AiB ) ∪ (∪G∈G DGi ). Then (X × Z4 , {X × {i}: i ∈ Z4 }, F 1 , F 2 , F 3 , F 4 ∪ ε ∪ ε ′ ) is a 3-FG(3, (3, 3, 4, 4), 96). From such a 3-FG, we can obtain the desired design by applying Lemma 2.2 with the known 1-AFSQS(26) in Lemma 5.1. □ Lemma 5.6. There exists a 1-AFSQS(122). Proof. We start with a 2-FG(3, (4, 4, 5), 15)(X , G , B1 , B2 , T ) of type 35 , which can be obtained by deleting two points from an S(3, 5, 17) in Lemma 3.4. For each block B ∈ T , by Theorem 2.3 we can construct an H(5, 8, 4, 3) on B × Z8 having group {x} × Z8 , x ∈ B. Denote its block set by AB . For each block B ∈ B1 , construct a 2-FG(3, (3, 3, 4), 32) on B × Z8 having group {x} × Z8 . Such a design exists from [8, Theorem 5.3]. Denote its block sets of the two subdesigns GDD(2, 3, 32) by CB1 and CB2 . Denote the set of the other blocks by CB3 . For each block B ∈ B2 , construct an H(4, 8, 4, 3) containing a subdesign H(4, 8, 4, 2) on B × Z8 having group {x} × Z8 , x ∈ B. Such a design exists by [11, Lemma 2.2]. Denote the block set of a subdesign H(4, 8, 4, 2) by DB1 and the set of the other blocks by DB2 . For i ∈ {1, 2}, let F i = ∪B∈B1 CBi , F 3 = ∪B∈B2 DB1 , and F ′ = (∪B∈B1 CB3 ) ∪ (∪B∈B2 DB2 ) ∪ (∪B∈T AB ). Then (X × Z8 , {G × Z8 : G ∈ G }, F 1 , F 2 , F 3 , F ′ ) is a 3-FG(3, (3, 3, 4, 4), 120) of type 245 . From such a 3-FG, we can obtain the desired design by applying Lemma 2.2 with the known 1-AFSQS(26) in Lemma 5.1. □ Lemma 5.7. There exists a 1-AFSQS(146). Proof. Similar to the proof of Lemma 4.3, we first construct a 3-FG(3, (3, 3, 4, 4), 144) of type 364 . Let (X , G ) be a 1-design of type 34 where G = {G0 , G1 , G2 , G3 }. Let (X , T ) be an S∗ (3, {4, 6}, 12), which exists from the proof of [7, Lemma 4.10]. Let its two disjoint blocks of size six be Gi ∪ Gi+1 , i = 0, 2. Also, there is a subset B ⊂ T such that (X , G , B) is a GDD(2, 4, 12). We shall construct the desired design on X × Z12 with groups Gi × Z12 , 0 ≤ i ≤ 3. For each block B ∈ T with |B| = 4, by Theorem 2.3 we can construct an H(4, 12, 4, 3) on B × Z12 having groups {x} × Z12 , x ∈ B. Denote the block set by CB . For each block B ∈ B, construct a 3-FG(3, (3, 3, 4, 4), 48) on B × Z12 having groups {x} × Z12 , x ∈ B. Such a 3-FG exists from Lemma 4.3. Denote its block sets of the two subdesigns GDD(2, 3, 48)’s by DB1 and DB2 . Denote the block set of the subdesign GDD(2, 4, 48) by DB3 and the set of other blocks by DB′ . For 0 ≤ i ≤ 3, we consider the multipartite graph Γi on Gi × Z12 that contains all edges {(x, s), (x′ , s′ )} with x ̸ = x′ . Such a graph can be thought as a graph on Z36 having all edges {a, b} with |a − b| ∈ L = {i: 1 ≤ i ≤ 18, i ̸ ≡ 0 (mod 3)}. i By Lemma 4.2 there is a one-factorization of the graph Γi . Let F i = {F1i , . . . , F24 } be a one-factorization of the graph Γi . j j+1 j For j = 0, 2, let A = {{a, b, c , ⋃ d}: {a, b} ∈ Fk , {c , d} ∈⋃Fk , 1 ≤ k ≤ 24}. ⋃ i ′ ′ 0 2 For 1 ≤ i ≤ 3, let Di = B∈B DB . Let D = ( B∈T \B,|B|=4 CB ) ∪ ( B∈B DB ) ∪ A ∪ A . It is routine to check that 1 2 3 ′ (X × Z12 , {G × Z12 : G ∈ G }, D , D , D , D ) is the desired 3-FG(3, (3, 3, 4, 4), 144) of type 364 . From such a 3-FG, we can obtain the desired design by applying Lemma 2.2 with the known 1-AFSQS(38) in Lemma 5.1. □ Lemma 5.8. There exists a 1-AFSQS(50). Proof. Let G = (GF (49), +), where the primitive polynomial f (x) = x2 + x + 3 over Z7 is used to generate the field. Let ξ be a primitive element of the field. Denote the field elements 0, ξ 1 , ξ 2 , . . . , ξ 48 by 0, 1, 2, . . . , 48. We construct a 1-AFSQS(50) on G ∪ {∞} with the multiplier group Q = {16, 32, 48}. The shortened list of base blocks is as follows, where the underlined base blocks generate the block set of an S(2, 4, 49). Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

12

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m = 48 m∈Q

0 0 0 0 0 0

i 8 + i ∞ 0 i 16 + i 40 + i 0 4 5 14 08 123 1 8 15 0 1 10 11 0 1 1 21 44 0 1 22 24 0 1 1 31 42 0 1 34 46 0 2 2 24 39 0 3 5 15 03

10 21 12 35 23 38 58 6 44

1 0 0 0 0 0

≤i≤8 11 12 18 1 14 47 1 26 48 2 7 37 4 21 32

0 0 0 0

146 1 16 37 1 27 30 2 11 21 □

0 0 0 0

1 1 1 2

5 19 18 20 28 43 19 38

Lemma 5.9. There exists a 1-AFSQS(62). Proof. We construct a 1-AFSQS(62) on Z61 ∪ {∞} with the multiplier group Q = {9i : 0 ≤ i ≤ 4}. The shortened list of base blocks is as follows, where the underlined base blocks generate the block set of an S(2, 4, 61). 0 0 0 0 0

1 1 1 2 2

5∞ 12 16 29 57 7 15 35 56

0 0 0 0 0

2 1 1 2 4

10 14 30 14 15

0 0 0 0 0

∞ 15 36 49 46

1 1 1 2 5

0 0 0 0

6 50 17 18 31 39 17 47 15 28

1 1 1 2

24 19 23 32 46 19 33

0 0 0 0

1 1 1 2

78 20 22 37 59 21 44

0 0 0 0

1 1 1 2

9 10 24 26 38 40 25 30

0 0 0 0

1 11 1 25 1 43 2 31 □

13 33 51 48

Lemma 5.10. There exists a 1-AFSQS(74). Proof. We construct a 1-AFSQS(74) on Z73 ∪ {∞} with the multiplier group Q = {2i : 0 ≤ i ≤ 8}. The shortened list of base blocks is as follows, where the underlined base blocks under the multiplier group {8i : 0 ≤ i ≤ 2} generate the block set of an S(2, 4, 73) and Q ′ = {2i : 0 ≤ i ≤ 2}. m ∈ Q′ m∈Q

0 0 0 0 0 0

1 3 1 1 1 3

9 52 49 ∞ 3 20 13 17 28 56 14 56

0 0 0 0 0 0

5 41 45 11 58 ∞ 4 30 35 1 21 26 1 29 51 3 17 48

0 3 10 27

0 11 26 61

0 1 65 ∞

0 5 33 ∞

0 0 0 0

0 0 0 0

0 1 10 14 0 1 24 43 0 1 47 59

0 1 11 12 0 1 25 57 0 1 49 61 □

1 1 1 3

25 22 60 30 50 28 62

1 1 1 3

67 23 58 31 40 42 47

Lemma 5.11. There exists a 1-AFSQS(110). Proof. We construct a 1-AFSQS(110) on Z109 ∪ {∞} with the multiplier group Q = {3i : 0 ≤ i ≤ 26}. The shortened list of base blocks is as follows, where the underlined base blocks under the multiplier group {27i : 0 ≤ i ≤ 8} generate the block set of an S(2, 4, 119) and Q ′ = {3i : 0 ≤ i ≤ 8}. m ∈ Q′ m∈Q

0 0 0 0

1 1 1 1

64 ∞ 3 60 16 17 37 89

0 0 0 0

2 1 1 1

19 ∞ 25 18 86 39 58

0 0 0 0

1 1 1 1

46 52 67 19 27 41 71

0 0 0 0

2 1 1 1

92 104 89 21 53 55 57

0 1 10 11 0 1 24 66 0 1 59 82

0 1 14 15 0 1 25 54 □

Lemma 5.12. There exists a 1-AFSQS(134). Proof. We construct a 1-AFSQS(134) on Z133 ∪ {∞} with the multiplier group Q = {4i : 0 ≤ i ≤ 8}. The shortened list of base blocks is as follows, where the blocks {0, 7 m, 56 m, 59 m}, m ∈ {1, 4, 16} and the underlined blocks generate the block set of an S(2, 4, 133) modulo 133 and Q ′ = {1, 4, 16}. m=1 m ∈ Q′ m∈Q

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 95 ∞ 7 84 ∞ 13∞ 18 50 97 1 10 14 1 28 34 1 48 50 1 60 61 1 87 99 1 116 125 2 19 47 2 58 60 2 99 121 3 22 127 3 51 114 7 51 97

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

19 38 76 7 56 59 6 45 ∞ 19 41 80 1 12 13 1 30 35 1 49 54 1 62 63 1 91 105 1 119 126 2 20 62 2 59 70 2 105 116 3 24 75 3 65 107 1 20 21

0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 5 11 20 51 88 1 15 17 1 36 39 1 51 52 1 66 80 1 92 101 1 120 131 2 43 63 2 61 84 3 9 46 3 26 52 6 15 76 3 60 63

0 0 0 0 0 0 0 0 0 0 0 0 0

2 17 25 26 55 95 1 22 24 1 40 43 1 53 55 1 67 79 1 93 96 2 11 26 2 44 74 2 65 88 3 14 89 3 31 121 6 17 78

0 0 0 0 0 0 0 0 0 0 0 0 0

9 1 1 1 1 1 1 2 2 2 3 3 6

66 109 28 23 26 42 44 56 57 68 81 110 112 13 21 51 127 72 109 18 55 39 62 49 63

0 0 0 0 0 0 0 0 0 0 0 0 0

16 60 87 149 1 27 29 1 45 46 1 58 59 1 85 90 1 115 127 2 15 28 2 57 94 2 76 87 3 21 91 3 49 119 6 62 98 □

Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Lemma 5.13. There exists a 1-AFSQS(158). Proof. We construct a 1-AFSQS(157) on Z157 ∪ {∞} with the multiplier group Q = {16i : 0 ≤ i ≤ 12}. The shortened list of base blocks is as follows, where the underlined base blocks generate the block set of an S(2, 4, 157). 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 2 2 2 2 3 3

3∞ 11 12 23 24 36 37 48 51 64 66 78 81 98 107 9 15 36 62 63 122 89 137 40 60 117 123

0 0 0 0 0 0 0 0 0 0 0 0 0

5 1 1 1 1 1 1 1 2 2 2 2 3

18 ∞ 13 14 25 27 38 41 53 54 67 77 80 89 110 117 12 19 37 70 68 138 96 134 50 135

0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 2 2 2 2 3

7 26 15 16 28 29 39 47 55 61 70 71 83 92 125 150 24 33 42 148 71 142 106 119 53 139

0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 2 2 2 2 3

25 17 18 30 32 42 43 59 68 72 73 84 95 152 155 26 32 43 52 72 107 109 115 69 107

0 0 0 0 0 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 2 2 2 2 3 3

68 19 20 31 34 44 45 60 62 74 79 96 120 7 10 28 30 50 83 73 76 18 105 74 151

0 0 0 0 0 0 0 0 0 0 0 0 0

1 9 10 1 21 22 1 33 35 1 46 52 1 63 69 1 76 106 1 97 154 2 8 13 2 35 55 2 57 104 2 74 85 3 36 73 3 91 142 □

Combining Lemmas 5.1–5.2 and Lemmas 5.5–5.13, we have the following. Lemma 5.14. There exists a 1-AFSQS(12g + 2) for any g ∈ {2, 3, . . . , 14} \ {7}. 6. Conclusion Proof of Theorem 1.1. It is easy to see that the necessary condition for the existence of a 1-AFSQS(v ) is v ≡ 2 (mod 12). Write v as v = 12u + 2, where u is a positive integer. For u = 1, exhaustive search by computer shows that there does not exist a 1-AFSQS(12u + 2). For u ∈ {2, 3, . . . , 14} \ {7}, by Lemma 5.14 there is a 1-AFSQS(12u + 2). For u ≥ 15 and u ̸∈ {17, 26, 27, 29, 31, 33}, we have u ∈ U by Theorem 3.10. Since there is a 3-FG(3, (3, 3, 4, 4), 12 m) of type 12m for m ∈ {4, 5, . . . , 13, 19} by Lemma 4.14, applying Lemma 2.6 gives a 1-AFSQS(12u + 2). For u = 26, start with a 1-FG(3, (4, 4), 78) of type 613 , which exists by [10, Lemma 4.8]. Applying Theorem 2.4 with the known 3-FG(3, (3, 3, 4, 4), 16) of type 44 in Lemma 4.1 and H(4, 4, 4, 3) in Theorem 2.3 gives a 3-FG(3, (3, 3, 4, 4), 312) of type 2413 . Since there is a 1-AFSQS(26), applying Lemma 2.2 yields a 1-AFSQS(12u + 2). For u = 27, start with a 2-FG(3, (3, 4, 4), 27) of type 39 , which can be obtained by deleting one point from an SQS(28) containing a subdesign S(2, 4, 28) in [7]. For each block of size 3, input a 2-FG(3, (3, 3, 4), 36) of type 123 , which exists in [12]. For each block in the subdesign GDD(2, 4, 27) of type 39 , input an H(4, 12, 4, 3) containing an H(4, 12, 4, 2), which exists by [11, Lemma 2.2]. For other blocks of size 4, input an H(4, 12, 4, 3), which exists by Theorem 2.3. Then by applying Hartman’s fundamental construction in [4], we obtain a 3-FG(3, (3, 3, 4, 4), 324) of type 369 . Since there is a 1-AFSQS(38), applying Lemma 2.2 yields a 1-AFSQS(12u + 2). For u = 29, start with a 1-FG(3, (4, 4), 30) of type 65 . By deleting one point and considering groups as blocks, we obtain a 2-FG(3, (3, {4, 6}, 4), 29) of type 38 51 . Similar to the case u = 27, For each block of size 3, input a 2-FG(3, (3, 3, 4), 36) of type 123 . For each block B in the subdesign GDD(2, {4, 6}, 29) of type 38 51 , input a 1-fan GDD(3, (4, 4), 12|B|) of type 12|B| , which exists in the proof of [6, Lemma 4.10]. For other blocks of size 4, input an H(4, 12, 4, 3). Then by applying Hartman’s fundamental construction in [4], we obtain a 3-FG(3, (3, 3, 4, 4), 12u) of type 368 601 . Since there is a 1-AFSQS(38) and a 1-AFSQS(62), applying Lemma 2.2 yields a 1-AFSQS(12u + 2). This completes the proof. □ Note that if there is a 1-AFSQS(86), then we can use the known 1-FG(3, (M , N), u) with group sizes from [2, 14] in [6, Lemma 3.15] instead of 1-FGs in Theorem 3.10 as initial designs, and Section 3 can be deleted. Although I have spent much time constructing a 1-AFSQS(86), I failed. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The author would like to thank referees for helpful comments. Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.

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Please cite this article as: L. Ji, Existence of Steiner quadruple systems with an almost spanning block design, Discrete Mathematics (2019) 111708, https://doi.org/10.1016/j.disc.2019.111708.