a new infinite class of s(3, {4, 5, 7}, v)

A New Infinite Class of S(3, {4,5,7},v)

Lijun Ji,1 Shaopu Zhang2

1Department of Mathematics, Soochow University, Suzhou 215006, P. R. China,E-mail: [email protected]

2Department of Mathematics and Physics, Shijiazhuang Tiedao University,Shijiazhuang 050043, P. R. China, E-mail: [email protected]

Received January 26, 2011; revised July 23, 2011

Published online 14 October 2011 in Wiley Online Library (wileyonlinelibrary.com).DOI 10.1002/jcd.20303

Abstract: A t-wise balanced design (tBD) of order v and block sizes from K, denoted byS(t,K,v), is a pair (X,B), where X is a v-element set and B is a set of subsets of X, calledblocks, with the property that |B|∈K for any B∈B and every t-element subset of X iscontained in a unique block. In this article, we shall show that there is an S(3,{4,5,7},v) forany positive integer v≡7(mod12) with v �=19. q 2011 Wiley Periodicals, Inc. J Combin Designs20: 68–80, 2012

Keywords: t-wise balanced design; s-fan design; candelabra t-system

1. INTRODUCTION

A t-wise balanced design (tBD) is a pair (X,B), where X is a finite set of points andB is a set of subsets of X, called blocks, with the property that every t-element subsetof X is contained in a unique block. If |X |=v and block sizes of B are all from K, wedenote the tBD by S(t,K ,v). When K ={k}, we simply write k for K. An S(t,k,v) iscalled a Steiner system.For t=2, a 2BD is usually called a pairwise balanced design (PBD), and much work

has been done on PBD (see [1]). However, for t>2 not much is known for tBD. In 1960,Hanani [2] showed that an S(3,4,v) exists if and only if v≡2 or 4(mod6). In 1963,Hanani [3] showed that an S(3,{4,6},v) exists if and only if v≡0(mod2). Recently, thefirst author showed that an S(3,{4,5},v) exists if and only if v≡1,2,4,5,8,10(mod12)with v �=13 [8], and that an S(3,{4,5,6},v) exists if and only if v �≡3(mod4) withv �=9,13 [9]. In this article, we are interested in the existence of an S(3,{4,5,7},v).

Contract grant sponsor: NSFC; Contract grant numbers: 10701060; 10831002; Contract grant sponsors:Qing Lan Project of Jiangsu Province (to L. Ji); NSFC; Contract grant number: 11001182 (to S. Zhang).

q 2011 Wiley Periodicals, Inc. 68

A NEW INFINITE CLASS OF S(3,{4,5,7}, v) 69

Let �(K ,r)=gcd{(k−r)(k−r−1) . . . (k− t+1) :k∈K }. Kramer [10] showed that anecessary condition for the existence of an S(t,K ,v) is

(v−r)(v−r−1) . . . (v− t+1)≡0 (mod�(K ,r))

for any 0≤r<t . It follows that a necessary condition for the existence of anS(3, {4,5,7},v) is v≡1,2,4,5,7,8,10,11(mod12).By the existence of an S(3,{4,5},v), two infinite classes of S(3,{4,5,7},v) for v≡

7,11(mod12) need to be constructed. In this article, we shall focus on the infinite classof S(3, {4,5,7},v), v≡7(mod12).In Section 2, a recursive construction for S(3,{4,5,7},v) is given in Lemma 2.6.

From this lemma we arrange the other parts of this article as follows. In Section 3,we shall prove that there is a CS(3,{4,5,7},12m+7) of type (12m :7) for m∈M={4,5, . . . ,13,19}. In Section 4, we construct an S(3,{4,5,7},12g+7) for g∈Q={2,3, . . . ,14}. In the last section, we prove the following main theorem of this article.

Theorem 1.1. There exists an S(3,{4,5,7},v) for any positive integer v≡7(mod12)with v �=19.

2. A RECURSIVE CONSTRUCTION FOR S(3, {4,5,7},v)

In this section, we shall give a recursive construction for S(3,{4,5,7},v) in Lemma 2.6.Let v be a non-negative integer, let t be a positive integer and let K be a set of positive

integers. A candelabra t-system (or t-CS) of order v, and block sizes from K denotedby CS(t,K ,v) is a quadruple (X, S,�,A) that satisfies the following properties:

(1) X is a set of v elements (called points).(2) S is a subset (called the stem of the candelabra) of X of size s.(3) �={G1,G2, . . .} is a set of non-empty subsets (called groups or branches) of

X\S, which partition X\S.(4) A is a family of subsets (called blocks) of X, each of cardinality from K.(5) Every t-subset T of X with |T ∩(S∪Gi )|<t for all i, is contained in a unique

block and no t-subsets of S∪Gi for all i, are contained in any block.

By the group type (or type) of a t-CS (X, S,�,A) we mean the list (|G||G∈� : |S|) ofgroup sizes and stem size. The stem size is separated from the group sizes by a colon.If a t-CS has ni groups of size gi , 1≤ i≤r and stem size s, then we use the notation(gn11 gn22 . . .gnrr :s) to denote group type. A candelabra system with t=3 and K ={4} iscalled a candelabra quadruple system and denoted by CQS(gn11 gn22 . . .gnrr :s).The following is a construction of S(3,{4,5,7},v) from a 3-CS with stem size 7.

Lemma 2.1. Suppose that there exists a CS(3,{4,5,7},v) of type (ga11 ga22 . . .garr :7).If there exists an S(3,{4,5,7},gi +7) with a block of size 7 for any 1≤ i≤r, then therealso exists an S(3,{4,5,7},v).

Proof. Let (X, S,G,T ) be a CS(3,{4,5,7},v) of type (ga11 ga22 . . .garr :7). For eachG∈G, construct an S(3,{4,5,7}, |G|+7) on G∪S having S as a block. Such a designexists by assumption. Denote its block set by FG . Let F =(∪G∈GFG \{S})∪{S}∪T .Then (X,F) is the desired design. �

Journal of Combinatorial Designs DOI 10.1002/jcd

70 JI AND ZHANG

This lemma shows that the CS(3,{4,5,7},v) is useful in the construction ofS(3, {4,5,7},v). To obtain such CSs, we state a fundamental construction for 3-CSswhich is a special case of Hartmam’s fundamental construction [5].Let v be a non-negative integer, t be a positive integer and K be a set of positive

integers. A group divisible t-design (or t-GDD) of order v and block sizes from Kdenoted by GDD(t,K ,v) is a triple (X,G,B) such that

(1) X is a set of v elements (called points),(2) G={G1,G2, . . .} isasetofnon-emptysubsets (calledgroups)ofXwhichpartitionX,(3) B is a family of subsets of X (called blocks) each of cardinality from K such that

each block intersects any given group in at most one point, and(4) each t-set of points from t distinct groups is contained in exactly one block.

The type of the GDD is defined to be the list (|G||G∈G) of group sizes.Let (X, S,G,A) be a CS(3,K ,v) of type (ga11 ga22 . . .garr :s) with s>0 and let

S={∞1, . . . ,∞s}. For 1≤ i≤s, let Ai ={A\{∞i } : A∈A,∞i ∈ A} and AT ={A∈A : A∩S=∅}. Then (s+3)-tuple (X,G,A1,A2, . . . ,As,AT ) is called an s-fan design(as in [5]). If block sizes of Ai and AT are from Ki (1≤ i≤s) and KT , respec-tively, then the s-fan design is denoted by s-FG(3, (K1,K2, . . . ,Ks,KT ),

∑ri=1 ai gi )

of type ga11 ga22 . . .garr . Conversely, for each subdesign GDD(2,Ki ,∑r

i=1 ai gi ) of ans-FG(3, (K1,K2, . . . ,Ks,KT ),

∑ri=1 ai gi ) of type ga11 ga22 . . .garr , adjoining a new point

∞i to each block, and taking S={∞1, . . . ,∞s} as a stem, we then obtain a 3-CS oftype (ga11 ga22 . . .garr :s).Hanani [4, Theorem 5.1] gave some 1-FGs which we state below.

Theorem 2.2. Let q be a prime power. Then there exists an S(3,q+1,q2+1) and a1-FG(3, (q,q+1),q2) of type qq .

Theorem 2.3 (Hartman [5]). Suppose there is an e-FG(3, (K1, . . . ,Ke,KT ),v) oftype ga11 ga22 . . .garr . If there exists a CS(3, L ,bk1+a1) of type (bk1 :a1) for any k1∈K1,

a GDD(3, L ,bki +ai ) of type bki a1i for any ki ∈Ki and 2≤ i≤e, and a GDD(3, L ,bk)of type bk for any k∈KT , then there exists a CS(3, L ,bv+∑

1≤i≤e ai ) of type((bg1)a1(bg2)a2 . . . (bgr )ar :∑1≤i≤e ai ).

By deleting some points of the 1-FG of type qq in Theorem 2.2, an infinite class of1-FGs was given as follows [8]. In the sequel, we denote the set {k :a≤k≤b and k isan integer} by [a,b].Lemma 2.4 (Ji [8]). Let U ={l≥15 : l is an integer}\{17,26,27,29,31,33}. Thenfor u∈U there is a 1-FG(3, (M,N ),u) of type ga11 . . .garr , where M=[4,13]∪{19},N ={k≥4 :k is an integer}, and gi ∈Q=[2,14] for 1≤ i≤r .

A GDD(3,4,v) of type rm is called an H design (as in [12]) and denoted byH(m,r,4,3).

Theorem 2.5 (Ji [6] and Mills [12]). For m>3 and m �=5, an H(m,r,4,3) exists ifand only if rm is even and r(m−1)(m−2) is divisible by 3. For m=5, H(5,r,4,3)exists if r is even, r �=2 and r �≡10,26(mod48).

Starting with a 1-FG in Lemma 2.4 and taking special input designs in Theorem 2.3we can construct a CS(3, ({4,5,7}),v) so as to obtain an S(3,{4,5,7},v).



Lemma 2.6. Suppose that there exists a CS(3,{4,5,7},12m+7) of type (12m :7) forany m∈M. Suppose that there also exists an S(3,{4,5,7},12g+7) for any g∈Q. Thenthere exists an S(3,{4,5,7},12u+7) for any u∈U.

Proof. For u∈U , by Lemma 2.4 there is a 1-FG(3, (M,N ),u) of type ga11 ga22 . . .garr ,where gi ∈Q for 1≤ i≤r . Apply Theorem 2.3 with b=12. Since there exists aCS(3, {4,5,7},12m+7) of type (12m :7) for any m∈M by assumption and anH(n,12,4,3) for any n≥4 by Theorem 2.5, we obtain a CS(3,{4,5,7},12u+7) oftype ((12g1)a1(12g2)a2 . . . (12gr )ar :7). By assumption, for any g∈Q there exists anS(3, {4,5,7},12g+7) which must have a block of size 7. We apply Lemma 2.1 toobtain an S(3,{4,5,7},12u+7). �

3. SOME SMALL CS(3, {4,5,7},12m+7)’s

In this section, we show that there exists a CS(3,{4,5,7},12m+7) of type (12m :7) forany m∈M .

Lemma 3.1. There exists a CS(3,{4,5,7},12m+7) of type (12m :7) for m∈{4,7,11,19}.Proof. For m=4, there is a 7-FG(3, (4, . . . ,4,4),48) of type 124 by [8, Lemmas 3.6and 3.7], which leads to a CS(3,{4,5},55) of type (124 :7) according to the relationshipbetween s-FG and 3-CS.For m∈{7,11,19}, the construction of a 5-FG(3, (3,3,3,3,4,4),12m) of type 12m

[7, Lemmas 4.13–4.15] implies the existence of a 7-FG(3, (3,3,3,3,4,4,4,4),12m).So, there is a CS(3,{4,5,7},12m+7) of type (12m :7). �

Lemma 3.2. There is a GDD(3,4,23) of type 4471 and a CS(3,{4,5},23) of type(44 :7).Proof. Let (X,G,B) be a GDD(3,4,22) of type 4461, such a design exists in [11].Let G={G0, . . . ,G4} and |G4|=6. It is easy to see that all blocks disjoint from G4form the block set B′ of a GDD(2,4,16) of type 44 on X \G4 with group set G\{G4}.Let C={B∪{∞}: B∈B′}. Then (X∪{∞},{G0,G1,G2,G3,G4∪{∞}}, (B\B′)∪C) isa GDD(3, {4,5},23) of type 4471.For 0≤ i≤3, let Fi ={F1

i ,F2i ,F3

i } be a one-factorization on Gi . Let

D={{x, y, x ′, y′} : {x, y}∈Fki ,{x ′, y′}∈Fk

j ,0≤ i< j ≤3,1≤k≤3}.Then (X∪{∞},G4∪{∞},G\{G4}, (B\B′)∪C∪D) is a CS(3,{4,5},23) of type (44 :7).

�

Lemma 3.3. There exists a CS(3,{4,5,7},12m+7) of type (12m :7) for m∈{5,8}.Proof. For m∈{5,8}, an S(3,5,3m+2) exists in [4]. Deleting two points gives a2-FG(3, (4,4,5),3m) of type 3m , which is also a 1-FG(3, (4,{4,5}),3m) of type 3m .Apply Theorem 2.3 with b=4, the known CS(3,{4,5},23) of type (44 :7) by Lemma 3.2and H(r,4,4,3) for r =4,5 by Theorem 2.5. We then obtain a CS(3,{4,5,7},12m+7)of type (12m :7). �


72 JI AND ZHANG

Lemma 3.4. There exists a CS(3,{4,5,7},12m+7) of type (12m :7) for m∈{9,13}.Proof. For each given m, there is a 1-FG(3, (4,4),3m+1) of type 13m+1 by [7,Theorem 1.3]. Deleting one point from this 1-FG gives a 2-FG(3, (4,3,4),3m) oftype 3m . Apply Theorem 2.3 with b=4, the known H(4,4,4,3) by Theorem 2.5and CS(3, {4,5},19) of type (44 :3). The last one can be obtained from a 3-FG(3, (4,4,4,4),16) of type 44, which exists by [8, Lemmas 3.6 and 3.7]. So, we obtain aCS(3, {4,5,7},12m+7) of type (12m :7). �

Lemma 3.5. There exists a CS(3,{4,5,7},79) of type (126 :7).Proof. Deleting two points from an S(3,6,26) gives a 2-FG(3, (5,5,6),24) oftype 46. Apply Theorem 2.3 with b=3, the known CS(3,{4,5},15) of type (35 :0) (i.e.,2-FG(3, (4,4,5),15) of type 35 in the proof of Lemma 3.3), GDD(3,4,22) of type3571 in [13] and H(6,3,4,3) by Theorem 2.5. We then obtain a CS(3,{4,5,7},79) oftype (126 :7). �

A GDD(3,K ,v) of type ga11 ga22 . . .garr is called s-fan if its block set B can be parti-tioned into disjoint subsets B1, . . . ,Bs and T such that for each i, 1≤ i≤s, Bi is theblock set of a GDD(2,Ki ,v) of the same type. If the block sizes of T are all from KT ,then it is denoted by s-fan GDD(3, (K1,K2, . . . ,Ks,KT ),v). When the s-fan GDD hasuniform type gn and g is even, we can modify it to obtain an s-FG of the same type by[8, Lemma 3.7].The following construction for GDD is also a special case of Hartmam’s fundamental

construction [5].Theorem 3.6 (Hartman [5]). Suppose there is an e-fan GDD(3, (K1, . . . ,Ke,KT ),v)

of type ga11 ga22 . . .garr . If there exists a GDD(3, L ,bki +ai ) of type bki a1i for any ki ∈Ki

and 1≤ i≤e, and a GDD(3, L ,bk) of type bk for any k∈KT , then there exists aGDD(3, L ,bv+∑

1≤i≤e ai ) of type (bg1)a1(bg2)a2 . . . (bgr )ar (∑

1≤i≤e ai )1.

Lemma 3.7. There exists a CS(3,{4,5,7},151) of type (1212 :7).Proof. We start with a 1-fan GDD(3, (4,4),36) of type 312, which exists by [7, Lemma4.11]. Apply Theorem 3.6 with b=4, the known GDD(3,{4,5},23) of type 4471 byLemma 3.2. We then obtain a GDD(3,{4,5},151) of type 121271. Further by [8, Lemma3.7], there is a CS(3,{4,5},151) of type (1212 :7). �

Before we give a construction for CS(3,{4,5,7},127) of type (1210 :7), we need aCQS with an additional property as a starting design and a special S(3,{4,6},12) as aninput design.Let (X, S,G,T ) be a CQS(mk :s). If it contains a subdesign GDD(2,4,mk)(X \

S,G,B), then it shortly denoted by CQS∗(mk :s). For any two points from two distinctgroups, link them with a red or 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, thenwe call the CQS∗ a good CQS∗:(P1) each block of B contains exactly two disjoint red edges; and(P2) there is A⊂T \B satisfying that each block of A meets in exact two groups and

has four red edges, and each red edge is contained in exact one block A∈A.



Let (Y,G) be a 1-design of type 34, where G={G0,G1,G2,G3}. An S(3,{4,6},12)(Y,C) is denoted by S∗(3,{4,6},12) if it satisfies that (1) {Gi ∪Gi+1}, i=0,2 are theonly blocks of size six; (2) there is D⊂C such that (Y,G,D) is a GDD(2,4,12).From the proof of [7, Lemma 4.10], we have the following.

Lemma 3.8. There exists an S∗(3,{4,6},12).

Lemma 3.9. If there exists a good CQS∗(mk :s), then there exists a CQS∗((3m)k :3s).Proof. Let (X, S,G,T ) be the given good CQS∗(mk :s) and (X \S,G,B) be its subsde-sign GDD(2,4,mk). Then there isA⊂T \B having (P2). We shall construct the desireddesign on X ′ = X×Z3 having G′ ={G×Z3 :G∈G} as its group set and S′ = S×Z3 asits stem.For each block B∈T \(A∪B), construct an H(4,3,4,3) on B×Z3 having groups

{x}×Z3, x ∈ B. Denote its block set by CB . For each block B∈B∪A, construct anS∗(3, {4,6},12) on B×Z3 such that {x, y}×Z3 is a block, where {x, y} is one of thetwo red or blue edges in B according to B∈B or B∈A. Such an S∗(3,{4,6},12) existsfrom Lemma 3.8. Denote the block set of the GDD(2,4,12) of type 34 by DB and theset of the other blocks of size 4 by D′

B .Let E=∪B∈BDB and E ′ =(∪B∈B∪AD′

B)∪(∪B∈ADB)∪(∪B∈T \(A∪B)CB). Then(X ′, S′,G′,E∪E ′) is a CQS∗((3m)k :3s) with a subdesign GDD(2,4,3mk) (X ′ \S′,G′,E). �

Lemma 3.10. There is a CS(3,{4,5,7},127) of type (1210 :7).Proof. We first construct a 3-fan GDD(3, (3,3,4,4),40) of type 410 on Z40 withgroup set G={{10i+ j :0≤ i≤3} :0≤ j ≤9}. The required blocks are generated by thefollowing base blocks modulo 40, where the first six blocks of size 3 generate theblock set of a GDD(2,3,40) of the same group set, so do the second six blocks andthe last three underlined blocks generate the blocks of a GDD(2,4,40). Denote theblock sets of these subdesigns by B1,B2, and B3, respectively, and the set of the otherblocks by T .

0 1 3 0 4 9 0 6 18 0 7 21 0 8 23 0 11 240 1 5 0 2 8 0 3 17 0 7 22 0 9 21 0 11 270 1 2 9 0 1 6 7 0 1 12 14 0 1 15 16 0 1 17 22 0 1 18 290 1 19 27 0 1 23 32 0 1 24 37 0 1 28 33 0 1 36 38 0 2 6 350 2 11 17 0 2 13 37 0 2 14 27 0 2 15 23 0 2 16 19 0 2 18 360 2 21 25 0 2 26 33 0 2 31 34 0 3 6 28 0 3 7 29 0 3 8 320 3 14 36 0 3 15 22 0 3 18 34 0 3 19 31 0 3 21 35 0 4 8 150 4 12 17 0 4 16 31 0 4 23 28 0 4 25 32 0 4 29 35 0 5 14 310 5 18 27 0 5 19 33 0 6 12 23 0 6 13 19 0 6 21 32 0 7 23 310 1 4 13 0 2 7 24 0 6 14 25

For 0≤ i≤9, let F1i ={{i,20+i},{10+i,30+i}}, F2

i ={{i,10+i},{20+i,30+i}},F3i ={{i,30+i},{10+i,20+i}}, and Fi ={F1

i ,F2i ,F3

i }. Let C={{a,b,c,d} : {a,b}∈F jm, {c,d}∈F j

n ,0≤m<n≤9 and 1≤ j ≤3}. Then (Z40,G,B1,B2,B3,T ∪C) is a3-FG(3, (3,3,4,4),40) of type 410. Adjoining two new points to blocks of two


74 JI AND ZHANG

GDD(2,3,40)’s, respectively, and considering the set of these two points as a stem, weobtain a CQS∗(410 :2).In fact, this CQS∗ is also a good CQS∗(410 :2). For any two points x, ywith 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, other-wise a blue edge. Further, there are B3 having (P1) and A={{{k,4+k,20+k,24+k},{k,8+k,20+k,28+k},{k,5+k,20+k,25+k}} :0≤k≤19}⊂C having (P2). Thus, itis indeed a good CQS∗.Further, applying Lemma 3.9, we have a CQS∗(1210 :6). Adjoin a new point to each

block of the subdesign GDD(2,4,120) and add the new point to the stem. We thenobtain a CS(3,{4,5,7},127) of type (1210 :7). �

Combining Lemmas 3.1, 3.3–3.5, 3.7, and 3.10, we obtain the main result of thissection.

Lemma 3.11. There exists a CS(3,{4,5,7},12k+7) of type (12k :7) for k∈M=[4,13]∪{19}.

4. SMALL ORDERS FOR S(3, {4,5,7},v)

In this section, we shall construct an S(3,{4,5,7},12k+7) for k∈Q=[2,14].Lemma 4.1. There exists an S(3,{4,5,7},12k+7) for k∈{2,3}.Proof. By [8, Lemma 3.5], there is a 1-FG(3, (4,4),30) of type 65. From the proof of[7, Lemma 5.4] there is a 1-FG(3, (4,4),42) of type 67. By adjoining a new point to eachblock and each group of the subdesign GDD(2,4,12k+6) of the 1-FG(3, (4,4),12k+6)of type 62k+1 for k∈{2,3}, we then obtain an S(3,{4,5,7},12k+7). �

Lemma 4.2. There exists an S(3,{4,5,7},12k+7) for k∈{8,10,12,14}.Proof. For k∈{8,12}, by [8, Lemmas 3.6 and 3.7], there is a 7-FG(3, (4, . . . ,4,4),12k)of type (3k)4, so, there is a CS(3,{4,5},12k+7) of type ((3k)4 :7). Further applyingLemma 2.1 with the known S(3,{4,5,7},3k+7) by Lemma 4.1, we have the desireddesign.For k∈{10,14}, there is a 1-FG(3, (4,4),3k) of type 6k/2 as stated in the proof

of Lemma 4.1. Apply Theorem 2.3 with known CS(3,{4,5,7},23) of type (44 :7) byLemma 3.2 and H(4,4,4,3) by Theorem 2.5. We then obtain a CS(3,{4,5,7},12k+7)of type ((24)k/2 :7). Further applying Lemma 2.1 with the known S(3,{4,5,7},31) byLemma 4.1, we have the desired design. �

Lemma 4.3. There exists a 1-FG(3, (4,4),114) of type 619.

Proof. We first construct a 1-fan GDD(3, (4,4),38) on Z38 having groups { j,19+ j},0≤ j ≤18. For i ∈ Z3 and b∈ Z19, define �(i,b) : Z38→ Z38 by the rule �(i,b)(x)=7i x+2b for all x ∈ Z38. The first six base blocks under the action of the permutation(0 1 2 3 . . . 37), together with other base blocks under the action of the automorphismgroup {�(i,b) :b∈ Z19, i ∈ Z3} generate the required blocks, where the underlined baseblocks generate the block set B of a subdesign GDD(2,4,38) with the same groups.



Let T be the block set generated by the other base blocks.

0 2 16 25 0 4 10 11 0 8 20 22 0 13 22 36 0 27 28 34 0 16 18 300 1 10 22 1 2 11 23 0 1 2 3 1 2 4 6 0 1 4 5 1 2 5 130 1 6 23 1 2 7 31 0 1 8 18 1 2 9 17 0 1 9 30 1 2 10 340 1 12 26 1 2 14 27 0 1 14 15 1 2 15 16 0 1 16 33 0 1 17 291 2 18 30 1 2 19 22 1 2 35 37 0 2 6 30 1 3 7 29 0 2 10 231 3 18 31 0 3 6 32 1 4 7 13 0 3 8 28 1 4 10 33 0 3 13 33

Let C={{i, i+19, j, j+19}:0≤i< j≤18}. Then (Z38,∅,G,B,T ∪C) is a CQS∗(219:0).In fact, the above CQS∗ is also a good CQS∗(219 :0). Let S={1,7,8,11,12,18}.

For any two distinct points x, y with x>y, if x− y∈ S∪(−S), then we link x andy with a red edge, otherwise a blue edge. Further, there are B having (P1) and A={{k,d+k,19+k,19+d+k} :d∈ S and 0≤k≤18}⊂C having (P2). Thus, it is indeed agood CQS∗.Applying Lemma 3.9 yields a CQS∗(619 :0), i.e., a 1-FG(3, (4,4),114) of type 619.

�


Proof. The desired design is constructed on Z54 with groups G j ={9i+ j : i ∈ Z6},j ∈ Z9. its block set B is described below.For j ∈ Z3, construct a CQS(63 :0) on {3i+ j : i ∈ Z18} with groups G j ,G3+ j ,G6+ j .

The blocks of these CQS’s form one part of B.For i ∈ Z3 and b∈ Z54, define �(i,b) : Z54→ Z54 by the rule �(i,b)(x)=19i x+b for all

x ∈ Z54. The other part of B is generated by the following blocks under the action of theautomorphism group {�(i,b) :b∈ Z54, i ∈ Z3}, where the underlined blocks generate theblock set of a GDD(2,4,54) of type 69 modulo 54 and the length of the orbit generatedby the last block is 81.

0 1 3 13 0 1 42 52 0 4 19 26 0 4 32 39 0 5 21 29 0 5 30 380 6 17 40 0 6 20 43 0 1 2 9 0 1 4 33 0 1 6 46 0 1 10 350 1 12 43 0 1 14 18 0 1 16 39 0 1 17 27 0 1 19 23 0 1 21 260 1 22 51 0 1 24 25 0 1 29 34 0 1 45 49 0 2 4 20 0 2 5 480 2 8 51 0 2 10 28 0 2 14 24 0 2 25 27 0 2 32 42 0 2 34 450 4 8 49 0 4 11 47 0 5 10 41 0 5 15 44 0 4 27 31

�


Proof. The desired design is constructed on Z66 with groups G j ={11i+ j : i ∈ Z6},j ∈ Z11. Its block set B is described below.For j ∈ Z3, construct an SQS(22) on {3i+ j : i ∈ Z22}. The blocks of these three SQS’s

form one part of B.For i ∈ Z5 and b∈ Z66, define �(i,b) : Z66→ Z66 by the rule �(i,b)(x)=25i x+b for all

x ∈ Z66. The other part of B is generated by the following blocks under the action ofthe automorphism group {�(i,b) :b∈ Z66, i ∈ Z5}, where the underlined blocks generatethe block set of a GDD(2,4,66) of type 611 modulo 66.


76 JI AND ZHANG

0 1 3 7 0 5 20 39 0 8 17 45 0 10 24 40 0 12 35 53 0 1 2 90 1 4 5 0 1 6 11 0 1 12 13 0 1 14 15 0 1 16 20 0 1 17 180 1 21 43 0 1 22 23 0 1 25 39 0 1 26 59 0 1 28 41 0 1 30 580 1 33 56 0 1 46 48 0 2 4 54 0 2 7 61 0 2 11 15 0 2 13 400 2 14 46 0 2 18 32 0 2 22 48 0 2 42 47 0 2 44 63 0 3 13 560 3 14 26

�


Proof. The desired design is constructed on Z90 with groups G j ={15i+ j : i ∈ Z6},j ∈ Z15. Its block set B is described below.For j ∈ Z3, construct a CQS(65 :0) on {3i+ j : i ∈ Z30} with groups G j ,G j+3,G j+6,

G j+9, and G j+12. The blocks of these three CQS’s form the first part of B.For j ∈ Z5, construct a CQS(63 :0) on {5i+ j : i ∈ Z30} with groups G j ,G j+5, and

G j+10. The blocks of these five CQS’s form the second part of B.For j ∈ Z15, let Fj ={F1

j ,F2j ,F

3j ,F

4j }, where Fk

j are defined as follows:

F1j = {{ j,30+ j},{15+ j,45+ j},{60+ j,75+ j}},

F2j = {{ j,15+ j},{30+ j,60+ j},{45+ j,75+ j}},

F3j = {{ j,60+ j},{15+ j,75+ j},{30+ j,45+ j}}, and

F4j = {{ j,75+ j},{15+ j,30+ j},{45+ j,60+ j}}.

Construct the third part of B : {{a,b,c,d} : {a,b}∈Fkj ,{c,d}∈Fk

l ,0≤ j<l≤14, i �≡j (mod3), i �≡ j (mod5),1≤k≤4}.For i ∈ Z3 and b∈ Z90, define �(i,b) : Z90→ Z90 by the rule �(i,b)(x)=31i x+b for all

x ∈ Z90. The other part of B is generated by the following blocks under the action ofthe automorphism group {�(i,b) :b∈ Z90, i ∈ Z3}, where the underlined blocks generatethe block set of a GDD(2,4,90) of type 615 modulo 90.

0 1 3 7 0 17 38 63 0 5 13 24 0 5 71 82 0 9 40 58 0 9 41 590 10 39 53 0 10 47 61 0 12 34 67 0 12 35 68 0 16 36 64 0 16 42 700 1 84 88 0 17 44 69 0 1 4 5 0 1 6 8 0 1 83 85 0 1 9 230 1 68 82 0 1 11 12 0 1 13 14 0 1 18 22 0 1 69 73 0 1 20 210 1 24 25 0 1 26 27 0 1 28 29 0 1 33 34 0 1 35 36 0 1 37 380 1 39 41 0 1 50 52 0 1 42 43 0 1 44 47 0 2 5 16 0 2 76 870 2 9 13 0 2 79 83 0 2 10 12 0 2 14 18 0 2 74 78 0 2 19 210 2 22 35 0 2 57 70 0 2 23 28 0 2 64 69 0 2 24 26 0 2 25 380 2 54 67 0 2 27 53 0 2 39 65 0 2 43 48 0 2 44 49 0 3 8 200 3 73 85 0 3 10 82 0 3 11 83 0 4 9 85 0 4 10 77 0 4 17 840 4 14 68 0 4 26 80 0 4 20 24 0 4 23 50 0 4 44 71 0 4 41 530 4 42 52 0 4 43 51 0 5 11 51 0 5 44 84 0 5 18 23 0 5 19 420 5 53 76 0 5 41 54 0 6 14 47 0 6 49 82 0 7 17 24 0 7 21 400 7 57 76 0 8 17 27 0 8 71 81 0 8 24 74 0 1 2 46 0 2 4 470 4 8 49 0 7 14 52

�




Proof. The desired design is constructed on Z138 with groups G j ={23i+ j : i ∈ Z6},j ∈ Z23. its block set B is described below.For j ∈ Z3, construct an SQS(46) on {3i+ j : i ∈ Z46}. The blocks of these three SQS’s

form one part of B.For i ∈ Z11 and b∈ Z138, define �(i,b) : Z138→ Z138 by the rule �(i,b)(x)=13i x+b for

all x ∈ Z138. The other part of B is generated by the following blocks under the action ofthe automorphism group {�(i,b) :b∈ Z138, i ∈ Z11}, where the underlined blocks generatethe block set of a GDD(2,4,138) of type 623.

0 1 3 7 0 1 132 136 0 1 4 30 0 1 109 135 0 2 5 460 2 94 135 0 1 5 6 0 1 8 9 0 1 10 11 0 1 12 130 1 14 15 0 1 16 17 0 1 18 19 0 1 20 21 0 1 22 230 1 24 31 0 1 108 115 0 1 27 29 0 1 110 112 0 1 32 330 1 34 35 0 1 36 37 0 1 38 39 0 1 40 41 0 1 42 430 1 44 45 0 1 46 48 0 1 91 93 0 1 51 52 0 1 57 580 1 59 60 0 1 61 62 0 1 63 64 0 1 68 71 0 2 8 120 2 128 132 0 2 9 23 0 2 117 131 0 2 14 16 0 2 18 200 2 22 24 0 2 26 58 0 2 82 114 0 2 35 54 0 2 86 1050 2 37 39 0 2 41 43 0 2 42 44 0 2 45 95 0 2 60 800 2 61 63 0 3 19 122 0 3 38 74 0 3 67 103 0 3 41 1000 3 46 64 0 3 77 95 0 3 59 82 0 4 20 24 0 4 42 1000 6 35 41 0 1 2 70 0 2 4 71

�Lemma 4.8. There exists a 1-FG(3, (4,4),78) of type 613.

Proof. The desired design is constructed on Z78 with groups G j ={13i+ j : i ∈ Z6},j ∈ Z13.For j ∈ Z13, let Fj ={F1

j ,F2j ,F

3j ,F

4j }, where Fk

j are defined as follows:

F1j = {{ j,26+ j},{13+ j,39+ j},{52+ j,65+ j}},

F2j = {{ j,13+ j},{26+ j,52+ j},{39+ j,65+ j}},

F3j = {{ j,52+ j},{13+ j,65+ j},{26+ j,39+ j}}, and

F4j = {{ j,65+ j},{13+ j,26+ j},{39+ j,52+ j}}.

Define block set B={{a,b,c,d} : {a,b}∈Fkj ,{c,d}∈Fk

l ,0≤ j<l≤12,1≤k≤4}. Theother blocks are generated by the following base blocks modulo 78, where the underlinedblocks generated the block set of the required subdesign.

0 i 2i i+39 where 1≤ i≤19 and i �=13

0 1 3 7 0 1 72 76 0 5 14 29 0 5 54 69 0 8 28 46 0 8 40 580 10 35 51 0 10 37 53 0 11 30 47 0 11 42 59 0 12 33 56 0 12 34 570 1 4 5 0 1 6 8 0 1 71 73 0 1 9 10 0 1 11 12 0 1 15 160 1 17 18 0 1 19 20 0 1 21 22 0 1 23 24 0 1 25 28 0 1 51 540 1 29 30 0 1 31 32 0 1 33 34 0 1 35 36 0 1 37 38 0 2 5 110 2 69 75 0 2 9 12 0 2 68 71 0 2 10 14 0 2 66 70 0 2 16 180 2 17 19 0 2 20 22 0 2 21 23 0 2 24 30 0 2 50 56 0 2 25 27


78 JI AND ZHANG

0 2 29 31 0 2 32 34 0 2 33 35 0 2 36 38 0 2 37 43 0 3 8 110 3 14 17 0 3 15 18 0 3 19 22 0 3 20 23 0 3 21 24 0 3 25 300 3 51 56 0 3 31 34 0 3 32 35 0 3 33 36 0 3 37 40 0 3 38 430 4 9 16 0 4 66 73 0 4 10 18 0 4 64 72 0 4 11 15 0 4 19 230 4 20 24 0 4 21 25 0 4 22 29 0 4 53 60 0 4 27 31 0 4 28 320 4 33 37 0 4 34 38 0 4 35 42 0 4 40 47 0 4 36 46 0 5 15 200 5 16 21 0 5 17 22 0 5 19 24 0 5 23 28 0 5 25 32 0 5 51 580 5 33 38 0 5 34 41 0 5 42 49 0 5 35 47 0 5 36 48 0 5 37 460 6 15 21 0 6 16 22 0 6 17 23 0 6 18 24 0 6 20 29 0 6 55 640 6 25 31 0 6 27 33 0 6 34 46 0 6 38 50 0 6 35 49 0 6 36 440 6 40 48 0 6 37 47 0 7 15 22 0 7 17 24 0 7 18 28 0 7 57 670 7 19 34 0 7 51 66 0 7 21 31 0 7 54 64 0 7 23 30 0 7 35 450 7 40 50 0 7 37 48 0 8 17 25 0 8 19 27 0 8 20 30 0 8 56 660 8 22 33 0 8 53 64 0 8 23 51 0 8 35 63 0 8 24 62 0 8 29 370 8 31 55 0 8 32 54 0 8 36 45 0 8 41 50 0 9 19 68 0 9 20 440 9 43 67 0 9 21 30 0 9 25 34 0 9 27 60 0 9 28 49 0 9 38 590 9 31 40 0 9 32 51 0 9 36 55 0 9 33 54 0 10 25 55 0 10 33 630 10 27 56 0 10 32 61 0 10 34 48 0 10 40 54 0 11 23 66 0 11 27 610 11 28 62 0 11 29 60 0 11 31 58 0 11 32 43 0 11 34 45 0 11 36 530 11 38 49 0 12 28 58 0 12 32 62 0 12 29 53 0 12 37 61 0 12 30 490 12 41 60 0 12 31 59 0 12 36 54 0 14 30 51 0 14 41 62 0 14 31 450 14 32 57 0 14 35 60 0 14 33 58 0 14 34 59 0 14 36 50 0 14 37 550 15 31 53 0 15 40 62 0 15 32 47 0 15 33 49 0 15 44 60 0 15 35 580 15 36 51 0 15 37 56 0 15 38 55 0 16 33 61 0 16 35 59 0 16 36 580 17 35 57 0 17 38 60 0 17 37 58

�Lemma 4.9. There is an S(3,{4,5,7},12k+7) for k∈{4,5,6,7,9,11}.Proof. For each given k, adjoin a new point to each block and each group of thesubdesign GDD(2,4,12k+6) of type 62k+1 of the known 1-FG(3, (4,4),12k+6) oftype 62k+1, which exists by Lemmas 4.3–4.8. We then obtain an S(3,{4,5,7},12k+7).

�Lemma 4.10. There is an S(3,{4,5,7},163).Proof. We construct an S(3,{4,5,7},163) on Z163. For i ∈ Z27 and b∈ Z163, define�(i,b) : Z163→ Z163 by the rule �(i,b)(x)=6i x+b for all x ∈ Z163. The block set aregenerated by the following base blocks under the action of the automorphism group{�(i,b) :b∈ Z163, i ∈ Z27}, where the lengths of orbits generated by the first six baseblocks are 9×163.

0 1 20 59 0 2 47 125 0 3 14 60 0 1 105 144 0 2 40 118 0 3 106 1520 1 2 4 7 11 17 0 1 5 14 21 0 1 9 22 0 1 15 19 0 1 18 24 0 1 23 320 1 25 29 0 1 28 35 0 1 30 44 0 1 33 95 0 1 36 98 0 1 40 720 1 41 88 0 1 45 129 0 1 46 120 0 1 50 71 0 1 51 152 0 1 54 1210 1 61 111 0 1 63 109 0 1 69 84 0 1 74 145 0 1 75 117 0 1 81 850 1 87 118 0 1 91 149 0 1 92 146 0 2 14 81 0 2 68 114

�Combining Lemmas 4.1 and 4.2 and Lemmas 4.9 and 4.10, we have the following

result.



Lemma 4.11. There exists an S(3,{4,5,7},12k+7) for any k∈Q=[2,14].

5. CONCLUSION

Lemma 5.1. There exists a CS(3,{4,5,7},6k+1) of type (6k :1) for any k∈[5,9].Proof. For k∈{5,7,9}, there is a 1-FG(3, (4,4),6k) of type 6k as stated in the proofof Lemmas 4.1 and 4.4. So, there is a CS(3,{4,5},6k+1) of type (6k :1).For k=8, taking one point of 1-FG(3, (7,8),49) of type 77 by Theorem 2.2 as a

stem and considering the blocks and groups containing this point in the subdesignGDD(2,7,49) of type 77 with it deleted as groups, we obtain a CS(3,{7,8},49) of type(68 :1). Construct an S(3,4,8) for each block of size 8. The result is a CS(3,{4,7},49)of type (68 :1).For k=6, we need only to construct a 1-FG(3, (4,4),36) of type 66 on Z36 with

groups Gi ={i, i+6, i+12, . . . , i+30}, 0≤ i≤5.Let Fi={F0

i ,F1i ,F2

i ,F3i ,F4

i } be a one-factorization of the complete graph K6 on Gi .

For any {x, y}∈F ji and any {x ′, y′}∈F j

i+3, construct blocks {x, y, x ′, y′}, where 0≤i≤2 and 0≤ j ≤4. The other required blocks are generated by the following baseblocks modulo 36, where the underlined base blocks generated the block set B of aGDD(3,4,36) under (+2 mod 36).

0 1 2 10 1 3 20 34 1 4 8 17 1 5 16 27 1 6 22 29 0 1 3 40 1 5 6 0 1 7 8 0 1 11 14 0 1 12 13 0 1 15 19 0 1 16 170 1 18 28 0 1 22 27 0 1 23 26 0 2 4 27 0 2 5 30 0 2 6 320 2 7 12 0 2 8 21 0 2 9 23 0 2 11 18 0 2 13 17 0 2 14 310 2 15 29 0 2 16 26 0 2 20 24 0 2 22 28 0 3 8 19 0 3 10 320 3 11 23 0 3 17 29 0 3 20 31 0 4 8 24 0 4 9 28 0 4 11 170 4 12 31 0 4 19 29 0 5 11 27 0 5 12 23 0 5 13 20 0 5 18 260 6 13 23 0 6 17 26 0 6 20 28

�

Lemma 5.2. There exists an S(3,{4,5,7},12k+7) for any k∈{17,26,27,29,31,33}.Proof. For k=17, deleting all points of two groups from a 1-FG(3, (7,8),49) oftype 77 by Theorem 2.2 and considering the truncated groups as blocks, we obtain a1-FG(3, ({5,7}, [4,8]),2k+1) of type 12k+1. For k∈{26,27,29,31}, deleting 63−2kpoints from two groups of a 1-FG(3, (8,9),64) of type 88 by Theorem 2.2 such thatthe sizes of the truncated groups are from {0,5,6,7,8} and considering the truncatedgroups as blocks, we obtain a 1-FG(3, ({5,8}, [4,9]),2k+1) of type 12k+1. For k=33,deleting 14 points from three groups of a 1-FG(3, (9,10),81) of type 99 such that thesizes of the truncated groups are from {0,5,6,7,8,9} and considering the truncatedgroups as blocks, we obtain a 1-FG(3, ({5,9}, [4,10]),2k+1) of type 12k+1.For each given k, start with 1-FG above and apply Theorem 2.3 with b=6, the known

H(l ′,6,4,3) for l ′ ∈[4,10] by Theorem 2.5 and CS(3,{4,5,7},6l+1) type (6l :1) forl∈[5,9] by Lemma 5.1. We obtain a CS(3,{4,5},12k+7) of type (62k+1 :1), whichleads to an S(3,{4,5},12k+7) trivially. �

We are now in a position to give the proof of our main theorem.


80 JI AND ZHANG

Proof of Theorem 1.1. An S(3,{4,5,7},7) exists trivially. For k∈[2,14], anS(3, {4,5,7}, 12k+7) exists by Lemma 4.11. For k∈{17,26,27,29,31,33}, anS(3, {4,5,7},12k+7) exists by Lemma 5.2. For each of other values v, applyingLemma 2.6 with the known input 3-CS’s by Lemma 3.11 and S(3,{4,5,7},12g+7)by Lemma 4.11, we obtain the result. �

Remark. By the existence of an S(3,{4,5},v) in [8] and Theorem 1.1, an infinite classof S(3, {4,5,7},v) for v≡11(mod12) need to be constructed. Since we could not findan S(3, {4,5,7},12g+11) with a hole of size 11 for g∈[2,14], the similar constructionfor S(3, {4,5,7},12k+11) from CS(3,{4,5,7},12k+11) of type (12k :11) could notwork. Although we have worked hard on it, we could not find effective constructionsfor S(3, {4,5,7},12k+11). It is our future work.

ACKNOWLEDGMENTS

The authors thank Professor L. Zhu for helpful suggestions on the article and the referees formany helpful comments.

REFERENCES

[1] R. J. R. Abel, F. E. Bennett, and M. Greig, PBD-closure, in: The CRC Handbook ofCombinatorial Designs (C. J. Colbourn and J. H. Dinitz, Eds.), CRC Press, Boca Raton, 2007,pp. 247–255.

[2] H. Hanani, On quadruple systems, Can J Math 12 (1960), 145–157.

[3] H. Hanani, On some tactical configurations, Can J Math 15 (1963), 702–722.

[4] H. Hanani, A class of three-designs, J Comb Theory Ser A 26 (1979), 1–19.

[5] A. Hartman, The fundamental construction for 3-designs, Discrete Math 124 (1994), 107–132.

[6] L. Ji, An improvement on H design, J Comb Des 17 (2009), 25–35.

[7] L. Ji, Existence of Steiner quadruple systems with a spanning block design, Discrere Math,to appear.

[8] L. Ji, On the 3BD-closed set B3({4,5}), Discrete Math 287 (2004), 55–67.

[9] L. Ji, On the 3BD-closed set B3({4,5,6}), J Comb Des 12 (2004), 92–102.

[10] E. S. Kramer, Some results on t-wise balanced designs, ARS Comb 15 (1983), 179–192.

[11] K. Lauinger, D. Kreher, R. Rees, and D. Stinson, Computing transverse t-designs, J CombMath Comb Comput 54 (2005), 33–56.

[12] W. H. Mills, On the existence of H designs, Congr Numer 79 (1990), 129–141.

[13] S. Zhang, On candelabra quadruple systems, ARS Comb 99 (2011), 335–352.


a new infinite class of s(3, {4, 5, 7}, v)

Documents