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Constructions for Steiner Quadruple Systems with a Spanning Block Design L

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Constructions for Steiner Quadruple Systems with a Spanning Block Design L View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 261 (2003) 347–360 www.elsevier.com/locate/disc Constructions for Steiner quadruple systems with a spanning block design L. Ji, L. Zhu∗;1 Department of Mathematics, Suzhou University, Suzhou 215006, China Received 2 April 2001; accepted 30 September 2001 Dedicated to Alex Rosa on the occasion ofhis sixty-ÿfthbirthday Abstract A singular direct product construction is presented for Steiner quadruple systems with a span- ning block design. More constructions are also provided using Steiner systems S(3;k;v) and other designs. Small orders for v = 40 and 52 are constructed directly. Some inÿnite classes of orders are also obtained. c 2002 Elsevier Science B.V. All rights reserved. Keywords: Steiner quadruple system; Spanning block design; t-wise balanced design 1. Introduction A t-wise balanced design (t-BD) is a pair (X; B), where X is a ÿnite set of points and B is a set ofsubsets of X , called blocks with the property that every t-element subset of X is contained in a unique block. If |X | = v and K is the set ofblock sizes, we denote the t-BD by S(t; K; v). A Steiner system S(t; k; v)isat-BD with all blocks having size k, on a set of v points. A Steiner quadruple system oforder v, brie9y SQS(v), is an S(3; 4;v). It is well known (see [8]) that an SQS(v) exists ifand only if v ≡ 2 or 4 (mod 6). In a Steiner quadruple system (X; B), if B can be partitioned into disjoint subsets B1;:::;Bs and A such that each (X; Bi)isanS(2; 4;v) for 16i6s, then the SQS is called (as in [10]) s-fan, denoted by s-FSQS(v). In a 1-FSQS(v), the S(2; 4;v) is also called (as in [5]) a spanning block design. ∗ Corresponding author. E-mail address: [email protected] (L. Zhu). 1 Research supported by NSFC Grant 19831050. 0012-365X/03/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved. PII: S 0012-365X(02)00480-6 348 L. Ji, L. Zhu / Discrete Mathematics 261 (2003) 347–360 The existence problem for 1-FSQS(v)s has received some attention. Hartman and Phelps provided an open problem in [11]: Show that there exists an SQS(v) with a spanning block design for all v ≡ 4 (mod 12). It is also conjectured by Lindner (see [5]) that the answer to this open problem is yes. The best known result on 1-FSQS(v)s is the inÿnite class for v =4n (see [5, Theo- rem 2.3]). For such orders there are even (v − 2)=2-fan SQS(v)s (see [3]). A result on (v−2)=2-fan SQS(v)s for other orders was obtained by Teirlinck [18]. For convenience we state their results in one theorem. Theorem 1.1 (Baker [3] and Teirlinck [18]). There is a (v − 2)=2-fan SQS(v) when v =4n, or v =2(qn +1) for all n¿1 and q∈{7; 31}. The Lindner’s conjecture on the existence ofa Steiner quadruple system with a spanning block design is now far from settled. Fu [5] proved a quadrupling construc- tion that ifthere exists a 1-FSQS( v), then there exists a 1-FSQS(4v). Hartman and Phelps [11] pointed out that ifan S(3; 6;v) exists, then there exists a 1-FSQS(3v − 2). For small orders, Phelps gave in [17] a construction to obtain a 1-FSQS(28). In this paper, we provide a singular direct product construction for 1-FSQS(v)s in Section 2, as a consequence, the quadrupling construction becomes a special case. In Section 3, we give three more constructions using S(3;k;v)s and other designs. In Section 4,we construct 1-FSQS(v)s for small orders, particularly for v =40; 52. Some inÿnite classes oforders are also given in Section 4. 2. Singular direct product construction Let m be a positive integer and let X be a set of points. Let T= {T1;:::;Tm} be a partition of X into disjoint sets Ti, which we will call the groups of T.Byatransverse of T we mean a subset of X that meets each Ti in at most one point. An H design on T is deÿned to be a collection of k-element transverses of T, called blocks, such that each t-element transverse is contained in exactly one of them. If |T1| = ···= |Tm| = r, we denote it by H(m; r; k; t). In [14], Mills showed the following. Lemma 2.1. For m¿3, m =5, an H(m; r; 4; 3) exists if and only if rm is even and r(m − 1)(m − 2) is divisible by 3. For m =5, an H(5;r;4; 3) exists if r is divisible by 4 or 6. An H(m; r; 4; 3) is called s-fan ifits block set B can be partitioned into disjoint subsets B1;:::;Bs and A such that each Bi is the block set ofan H(m; r; 4; 2) for 16i6s.AnH(4;r;4; 2) is known as a transversal design TD(4;r). It is proved in [5] that a P-cube oforder v exists ifthere exist three mutually orthogonal Latin squares oforder v. Since a P-cube implies the existence ofa 1-fan H(4;v;4; 3) and also from [1, Table 2.73] there exist three mutually orthogonal Latin squares oforder v¿4 and v=6; 10. Thus, we have the following. L. Ji, L. Zhu / Discrete Mathematics 261 (2003) 347–360 349 Lemma 2.2. For v¿4 and v=6; 10, there exists a 1-fan H(4;v;4; 3). We can now state the singular direct product construction for 1-FSQS. This con- struction has been used by many authors, see [2,13] and is originally due to Hanani (see [8]), as pointed out by Granville and Hartman [6]. Construction 2.3 (SDP). If there exist a 1-FSQS(v) with a subdesign 1-FSQS(w), a (1 + w=4)-fan H(4; (v − w)=4; 4; 3) and a 1-FSQS(u), then there exists a 1-FSQS(u(v − w)+w). Proof. Let (U; B) be a 1-FSQS(u), where (U; B0)isanS(2; 4;u) such that B0 ⊂B. Take any block B in B. Let g =(v − w)=4 and s = w=4. If B=∈ B0, we construct an H(4;g;4; 3) on B × G having groups {b}×G; b∈B, where |G| = g.IfB∈B0,we construct a (1 + s)-fan H(4;g;4; 3) on B × G having groups {b}×G; b∈B. Thus, we obtain a (1 + s)-fan H(u; g; 4; 3) on U × G. Denote the block set ofthe H design by A and the block sets ofthe 1 + s disjoint H(u; g; 4; 2)s by A0; A1;:::;As. For 16i6s, adjoin a new point xi to each block B of Ai. For every such block B∪{xi} ofsize 5, construct an H(5; 4; 4; 3) on {B∪{xi}} × Z4 having groups {b}×Z4, b∈B∪{xi}. For every block B∈A and B=∈ Ai,16i6s, construct an H(4; 4; 4; 3) on B × Z4 having groups {b}×Z4, b∈B. When B∈A0 the H(4; 4; 4; 3) should be 1-fan. Such a 1-fan H(4; 4; 4; 3) exists from Lemma 2.2. We have obtained an H design with groups W = {x1;:::;xs}×Z4 and {y}×G × Z4 for y∈U. Let C denote its block set. This H design contains an H(u; 4g; 4; 2) on U × G × Z4. Denote its block set by C0. y y On each {y}×G × Z4 for y∈U, let {F1 ;:::;F4g−1} be a one-factorization of a K4g. y For any two points y; y ∈U and for any 16j64g−1, take any pair {a; b} in Fj and y any pair {c; d} in Fj to form a block {a; b; c; d}. Denote the set ofall these blocks by F. By assumption, there exists a 1-FSQS(v) with a subdesign 1-FSQS(w). For y∈U, we may construct a 1-FSQS(v)on({y}×G × Z4)∪W , having block set Dy, such that the 1-FSQS(v) contains a subdesign 1-FSQS(w)onW , having block set E.Wemay also assume that Dy contains a subset Dy0 which is the block set ofan S(2; 4;v). Let E =E∩D . Then, E is the block set ofan S(2; 4;w). Denote D= (D − E) and 0 y0 0 y∈U y D0= y∈U (Dy0 − E0). Let L=C∪F∪D ∪ E and L0=C0 ∪D0 ∪E0: It is a routine matter to see that L is the block set ofan SQS( u(v − w)+w) based on the set (U × G × Z4)∪W , which has a spanning block design with block set L0. This completes the proof. When w = 0 in Construction 2.3, we have the direct product construction. 350 L. Ji, L. Zhu / Discrete Mathematics 261 (2003) 347–360 Lemma 2.4 (DP). If there exist a 1-FSQS(u) and a 1-FSQS(v), then there exists a 1-FSQS(uv). Proof. Apply Construction 2.3 with w = 0, where a 1-fan H(4;v;4; 3) always exists from Lemma 2.2.
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