Doubly resolvable Steiner quadruple systems of orders $$2^{2n+1}$$ 2 2 n + 1
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Doubly resolvable Steiner quadruple systems of orders 22n+1 Juanjuan Xu1 · Jingjun Bao2 · Lijun Ji1 Received: 27 April 2020 / Revised: 6 August 2020 / Accepted: 8 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A t-(v, k, λ) design is a pair (X , B), where X is a v-element set and B is a set of k-subsets of X , called blocks, with the property that every t-subset of X is contained in exactly λ blocks. A t-(v, k, λ) design (X , B) is said to be (s, μ)-resolvable if B can be partitioned into B1 | · · · |Bc such that each (X , Bi ) is an s-(v, k, μ) design, further, if each (X , Bi ) is also (r , ν)-resolvable, then such an (s, μ)-resolvable t-design is called (s, μ)(r , ν)-doubly resolvable. In 1980, Hartman constructed a (2, 3)(1, 1)-doubly resolvable 3-(v, 4, 1) design for v ∈ {20, 32, 44, 68, 80, 104} and a (2, 3)-resolvable 3-(27 , 4, 1) design. In this paper, we construct (2, 3)(1, 1)-doubly resolvable 3-(22n+1 , 4, 1) designs for all positive integers n. Keywords Steiner quadruple system · H design · Resolvable Mathematics Subject Classification 05B05
1 Introduction A t-(v, k, λ) design is a pair (X , B), where X is a v-element set and B is a set of k-subsets of X , called blocks, with the property that every t-subset of X is contained in exactly λ blocks. A t-(v, k, λ) design is also denoted by Sλ (t, k, v). If λ = 1, then it is usually called a Steiner system and denoted by S(t, k, v). An S(1, k, v) is simply a partition of X into k-subsets and is called a parallel class. An S(2, 3, v) is called a Steiner triple system. An S(3, 4, v) is called
Communicated by L. Teirlinck. Research is supported by NSFC Grants 11701303 (J. Bao), 11871363 (L. Ji).
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Lijun Ji [email protected] Juanjuan Xu [email protected] Jingjun Bao [email protected]
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Department of Mathematics, Soochow University, Suzhou 215006, China
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Department of Mathematics, Ningbo University, Ningbo 315211, China
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J. Xu et al.
a Steiner quadruple system and denoted by SQS(v). It is well-known that there is an SQS(v) if and only if v ≡ 2, 4 (mod 6) [7]. An Sλ (t, k, v) (X , B) is said to be (s, μ)-resolvable if its block set can be partitioned into B1 | · · · |Bc such that each (X , Bi ) is an Sμ (s, k, v). The Bi ’s are called resolution classes. Further, if each (X , Bi ) is also (r , ν)-resolvable, then such an (s, μ)-resolvable t-design is called (s, μ)(r , ν)-doubly resolvable. Much work has been done for (1, μ)-resolvable Sλ (2, k, v), see [1,6] for detail results. An (s, 1)-resolvable S(k, k, v) is a partition of the set of all k-subsets of a v-element set X into S(s, k, v)’s and is usually called a large set of Steiner systems. A large set of Steiner triple systems of order v exists if and only if v ≡ 1, 3 (mod 6) and v = 7 [16,17,19]. A large set of S(2, 4, 13) exists [5]. A (2, 1)(1, 1)-doubly resolvable S(3, 3, v) is a large set of v − 2 disjoint Kirkman triple systems of order v, the interested reader can refer to [4]. In 1987, Hartman almost determined the existence of (1, 1)-
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