1 Introduction

The paper is concerned with point-missing s-resolutions of t-designs and applications thereof. In general, a partition of a t-\((v,k,\lambda )\) design \((X, \mathcal{B})\) into mutually disjoint s-\((w,k,\delta )\) designs, \(w \le v\), \(s < t\), is termed an s-resolution. If \(w=v\), then \((X, \mathcal{B})\) is called s-resolvable; in particular, if \((X, \mathcal{B})\) is the complete k-(v, k, 1) design, then an s-resolution of \((X, \mathcal{B})\) is called a large set of s-designs. If \(w=v-1\), then \((X, \mathcal{B})\) is called point-missing s-resolvable. A point-missing s-resolution of the complete k-(v, k, 1) design is called an overlarge set of s-designs. Point-missing s-resolvability remains still sparsely investigated; however, several computational and theoretical works on the subject can be found in the literature [9, 13, 15, 16, 19, 20, 23]. Point-missing s-resolvability is complementarily related to what we call pencil-like s-resolvability for t-designs, and vice versa. As far as we know the first example of infinite series of non-trivial point-missing s-resolvable t-designs for \(t \ge 4\) can be found in a paper of Alltop in 1972 [2], in which the author constructed a series of 4-\((2^n+1, 2^{n-1}, (2^{n-1}-3)(2^{n-2}-1))\) designs for \(n \ge 4\) as the union of \(2^n+1\) mutually disjoint 3-\((2^n, 2^{n-1}, 2^{n-2}-1)\) designs. We prove theorems for constructing new t-designs from point-missing and pencil-like s-resolvable t-designs. By using these theorems for overlarge sets of disjoint Steiner quadruple systems with \(v=3 ^n-1\) and \(v=3^n+1\) points constructed by Teirlinck [23], including the already known case with \(v=2^n\), we derive various infinite series of 4-\((v+1,5,5)\) designs, which were unknown until now. It is worthy of note that no large sets of Steiner quadruple systems are constructed to date; however, large sets of Steiner 2-designs for \(k=4\) with \(v=13, 16\) points are known to exist [10, 12, 14]. We also show a recursive construction of point-missing s-resolvable t-designs and its application.

For the sake of clarity we include a few basic definitions. A t-design, denoted by t-\((v,k,\lambda )\), is a pair \((X, \mathcal{B})\), where X is a v-set of points and \(\mathcal{B}\) is a collection of k-subsets of X, called blocks, such that every t-subset of X is a subset of exactly \(\lambda \) blocks, and \(\lambda \) is called the index of the design. A t-design is called simple if no two blocks are identical, otherwise, it is called non-simple. A t-(v, k, 1) design is called a Steiner t-design. For any point \(x\in X\), let \(\mathcal{B}_x= \{ B {\setminus } \{x\}: \, x \in B \in \mathcal{B} \}\). Then \((X{\setminus } \{x\}, \mathcal{B}_x)\) is a \((t-1)\)-\((v-1,k-1,\lambda )\) design, called a derived design of \((X, \mathcal{B})\). It can be shown by simple counting that a t-\((v,k,\lambda )\) design is an s-\((v,k,\lambda _s)\) design for \(0 \le s \le t\), where \(\lambda _s = \lambda {v-s \atopwithdelims ()t-s}/{k-s \atopwithdelims ()t-s}.\) Since \(\lambda _s\) is an integer, necessary conditions for the parameters of a t-design are \({k-s \atopwithdelims ()t-s}\vert \lambda {v-s \atopwithdelims ()t-s}\) for \(0 \le s \le t\). The smallest positive integer \(\lambda \) for which these necessary conditions are satisfied is denoted by \(\lambda _\textrm{min}(t,k,v)\) or simply \(\lambda _\textrm{min}\). If \(\mathcal{B}\) is the set of all k-subsets of X, then \((X, \mathcal{B})\) is a t-\((v,k, \lambda _\textrm{max})\) design, called the complete design, where \(\lambda _\textrm{max}={v-t \atopwithdelims ()k-t}\). If we take \(\delta \) copies of the complete design, we obtain a t-\((v,k, \delta {v-t \atopwithdelims ()k-t})\) design, which is referred to as a trivial t-design; otherwise, it is called a non-trivial t-design.

2 Point-missing s-resolvable t-designs

A t-\((v,k,\lambda )\) design \((X, \mathcal{B})\) is said to be s-resolvable, for \(0< s < t\), if its block set \(\mathcal{B}\) can be partitioned into \(N \ge 2\) classes \(\mathcal{B}_1, \ldots , \mathcal{B}_N\) such that each \((X, \mathcal{B}_i)\) is an s-\((v,k,\delta )\) design for \(i=1, \ldots , N\). Such a partition is called an s-resolution of \((X, \mathcal{B})\) and each \(\mathcal{B}_i\) is called an s-resolution class or simply a resolution class, see e.g. [25, 26].

If the complete k-(v, k, 1) design can be partitioned into N disjoint t-\((v,k,\lambda )\) designs, where \(N={\left( {\begin{array}{c}v-t\\ k-t\end{array}}\right) }/{\lambda }\), then we say that there exists a large set of t-designs denoted by LS[N](t, k, v) or by \(LS_{\lambda }(t,k,v)\) to emphasize the value \(\lambda \).

In the most general form, the concept of point-missing s-resolvability of a t-\((v,k,\lambda )\) design can be defined as follows.

Definition 2.1

Let \((X,\mathcal{B})\) be a t-\((v,k,\lambda )\) design and let \(1\le s \le t-1\). \((X,\mathcal{B})\) is called point-missing s-resolvable, if the block set \(\mathcal{B}\) can be partitioned into mutually disjoint s-\((v-1,k, \delta )\) designs, each missing a point of X.

However, Definition 2.1 is equivalent to a definition that describes point-missing resolutions with more exact details. We now give an explanation.

Let \(X=\{x_1, \ldots , x_v\}\) and let \(X_i= X\setminus \{x_i\}\), \(i=1, \ldots , v\). Let \(m_i\) denote the number of s-\((v-1,k, \delta )\) designs \((X_i,\mathcal{B}_i)\) missing \(x_i\) in the resolution. First we show that any \(x_i \in X\) is a missing point of an s-design \((X_i,\mathcal{B}_i)\). More precisely, let \(Y \subseteq X\) be the subset of X such that there is no design \((X_i,\mathcal{B}_i)\) missing point \(x_i\), when \(x_i \in Y\). Assume that \(Y \not = \emptyset \). Then the blocks of \(\mathcal{B}\) can be written as follows.

$$\begin{aligned} \mathcal{B}= \bigcup _{x_h\in X\setminus Y} m_h\mathcal{B}_h, \mathrm{\ where} \quad m_h\mathcal{B}_h:=\underbrace{ \mathcal{B}_h \cup \cdots \cup \mathcal{B}_h }_{{m_h} \mathrm \ times}. \end{aligned}$$

Consider two given points \(x_i \in Y\) and \(x_j \in X\setminus Y\). Since \(x_i \in Y\), there is no s-design \((X_i,\mathcal{B}_i)\) missing \(x_i\). Thus \(x_i\) appears in each design \((X_h,\mathcal{B}_h)\), where \(x_h \in X\setminus Y\), therefore \(x_i\) appears in \(\sum _{x_h \in X{\setminus } Y}m_h\delta _1\) times in the blocks of \(\mathcal{B},\) where \(\delta _1 =\delta \frac{ \left( {\begin{array}{c}v-2\\ s-1\end{array}}\right) }{\left( {\begin{array}{c}k-1\\ s-1\end{array}}\right) }.\) Whereas the point \(x_j \in X\setminus Y\) appears in \(\sum _{x_h \in X{\setminus } \{Y \cup \{x_j\}\} } m_h\delta _1\) times in the blocks of \(\mathcal{B}\), which is a contradiction if \(Y \not = \emptyset .\) Further, we show that \(m_1= \cdots = m_v\). W.l.o.g., assume by contradiction that \(m_1 \not = m_2\). Then the number of blocks containing \(x_1\) (resp. \(x_2\)) is then \(\sum _{x\in X {\setminus } \{x_1\}} m_x\delta _1=m_2\delta _1+ \sum _{i=3}^v m_i\delta _1\) (resp. \(\sum _{x\in X {\setminus } \{x_2\}} m_x\delta _1= m_1\delta _1+ \sum _{i=3}^v m_i\delta _1\)). Since \(m_2\delta _1+ \sum _{i=3}^v m_i\delta _1 = m_1\delta _1+ \sum _{i=3}^v m_i\delta _1\), we have \(m_2\delta _1=m_1\delta _1\), or equivalently \(m_2=m_1\), contradicting the assumption. Thus we must have \(m_1= \cdots = m_v\).

The discussion above suggests an equivalent formulation of Definition 2.1 as follows.

Definition 2.2

Let \((X,\mathcal{B})\) be a t-\((v,k, \lambda )\) design and let \(1 \le s < t\) be an integer. \((X,\mathcal{B})\) is said to be point-missing s-resolvable, if there is an integer \(m \ge 1\) such that the following hold.

  1. 1.

    \(\mathcal{B}= \mathcal{B}_{x_1} \cup \cdots \cup \mathcal{B}_{x_v}\), where \(X=\{x_1, \ldots , x_v\},\)

  2. 2.

    \(\mathcal{B}_x= \mathcal{B}_x^1 \cup \cdots \cup \mathcal{B}_x^m\), each \((X\setminus \{x\}, \mathcal{B}_x^j)\) is an s-\((v-1,k,\delta )\) design, \(j=1, \ldots , m,\) and m is called the multiplicity of the point x.

If \(m=1\), \((X,\mathcal{B})\) is simply called point-missing s-resolvable. Moreover, if \(m > 1\), then \((X{\setminus } \{x\}, \mathcal{B}_x)\) is an s-\((v-1,k,m\delta )\) design. Therefore, \((X,\mathcal{B})\) again is a union of v mutually disjoint s-\((v-1,k,m\delta )\) design, each missing a different point of X. Hence, in general, when we speak of point-missing s-resolvable t-designs we mean \(m=1\).

If the complete k-(v, k, 1) design can be partitioned into v mutually disjoint s-\((v-1, k, \delta )\) designs (i.e. point-missing s-resolvable), then we have an overlarge set of s-\((v-1, k, \delta )\) designs.

Lemma 2.1

Let \((X, \mathcal{B})\) be a point-missing s-resolvable t-\((v,k, \lambda )\) design and assume that each point in the resolution has multiplicity m. Then

$$\begin{aligned} \delta =\lambda {\left( {\begin{array}{c}v-s\\ t-s\end{array}}\right) }/ {\left( {\begin{array}{c}k-s\\ t-s\end{array}}\right) }{m(v-s)}. \end{aligned}$$

In particular, if the complete t-(v, t, 1) design is point-missing \((t-1)\)-resolvable, then the designs in the resolution are Steiner \((t-1)\)-\((v-1,t,1)\) designs.

Proof

By assumption, we have

$$\begin{aligned} \mathcal{B} = \bigcup _{x \in X}\{ \mathcal{B}_x^1 \cup \cdots \cup \mathcal{B}_x^m\}, \end{aligned}$$

where \((X \setminus \{x\}, \mathcal{B}^i_x)\) is an s-\((v-1, k, \delta )\) design. Let \(S=\{x_1, \ldots , x_s\}\subseteq X\). Then S does not appear in any block of \(\mathcal{B}^i_{x_j}\), for \(j=1, \ldots , s\) and \(i=1, \ldots , m\). Further, S appears in each \(\mathcal{B}^i_{x_j}\) with \(j \not = 1, \ldots , s\), exactly \(\delta \) times. Thus S appears \(m(v-s)\delta \) times in the blocks of \(\mathcal{B}\). On the other hand, the number of blocks in \(\mathcal{B}\) containing S is \(\lambda _s=\frac{\left( {\begin{array}{c}v-s\\ t-s\end{array}}\right) }{\left( {\begin{array}{c}k-s\\ t-s\end{array}}\right) }\lambda .\) Therefore \(\lambda _s=m(v-s)\delta \) and thus \(\delta = \frac{\lambda _s}{m(v-s)}\), as desired. \(\square \)

Recall that the complement of an s-resolvable t-design is again s-resolvable. However, it is not true with a point-missing s-resolvable t-design. Let \(X:=\{x_1, \ldots , x_v\}\) and let \(X_i:= X\setminus \{x_i\}\), \(i=1, \ldots , v\). To simplify the typing we write: if \(Y \subseteq X\), then \(\overline{Y}:= X \setminus Y\), whereas if \(Y \subseteq X_i\), then \(\widetilde{Y}:= X_i {\setminus } Y\). Let \((X, \mathcal{D})\) be a point-missing s-resolvable t-design with parameters t-\((v,k,\lambda )\) and let \((X,\overline{\mathcal{D}})\) be its complement which has parameters t-\((v,v-k,\overline{\lambda })\), where \(\overline{\lambda }= \lambda \left( {\begin{array}{c}v-k\\ t\end{array}}\right) /\left( {\begin{array}{c}k\\ t\end{array}}\right) \). Let \(\mathcal{D} =\mathcal{D}_1\cup \cdots \cup \mathcal{D}_v\) be a partition of \(\mathcal{D}\) into v point-missing s-resolution classes, where \((X_i, \mathcal{D}_i)\) is an s-\((v-1,k,\delta )\) design, for \(i=1, \ldots , v\). The complement of \((X_i, \mathcal{D}_i)\) (within \(X_i\)) is an s-\((v-1, v-1-k, \widetilde{\delta })\) design \((X_i, \widetilde{\mathcal{D}}_i)\) with \(\widetilde{\delta }=\delta \left( {\begin{array}{c}v-1-k\\ s\end{array}}\right) /\left( {\begin{array}{c}k\\ s\end{array}}\right) \). So, we have \(\overline{\mathcal{D}} = \overline{\mathcal{D}}_1 \cup \cdots \cup \overline{\mathcal{D}}_v = (\{x_1\}\cup \widetilde{\mathcal{D}}_1)\cup \cdots \cup (\{x_v\}\cup \widetilde{\mathcal{D}}_v)\), where \(\{x_i\}\cup \widetilde{\mathcal{D}}_i = \{ \{x_i\}\cup \widetilde{D}\; | \; \widetilde{D} \in \widetilde{\mathcal{D}}_i \}\). Thus, \(\overline{\mathcal{D}}_i= (\{x_i\}\cup \widetilde{\mathcal{D}}_i)\) is not an s-design, but rather a “pencil”. Hence, the decomposition of \((X,\overline{\mathcal{D}})\) suggests the following definition.

Definition 2.3

Let \(X=\{x_1, \ldots , x_v\}\) and denote \(X_i:= X\setminus \{x_i\}\), \(i=1, \ldots , v\). Let \((X, \mathcal{B})\) be a t-\((v,k,\lambda )\) design. If for some \(x_i \in X\) there exists an s-\((v-1,k-1,\delta )\) design \((X_i, \mathcal{B}_i)\) for \(1 \le s < t\), then we call \(\{x_i\}\cup \mathcal{B}_i= \{ \{x_i\}\cup \widetilde{B}\; | \; \widetilde{B} \in \widetilde{\mathcal{B}}_i \} \subseteq \widetilde{\mathcal{B}}\) an s-pencil of \((X, \mathcal{B})\). If \(\mathcal{B}=(\{x_1\}\cup \mathcal{B}_1) \cup \cdots \cup (\{x_v\}\cup \mathcal{B}_v)\), where \((X_i, \mathcal{B}_i)\) is an s-\((v-1,k-1,\delta )\) design, then \((X, \mathcal{B})\) is said to be pencil-like s-resolvable.

As observed above, the complement of a point-missing s-resolvable t-design is a pencil-like s-resolvable t-design. Conversely, it is straightforward to check that the complement of a pencil-like s-resolvable t-design is a point-missing s-resolvable t-design. Hence the notion of point-missing s-resolvability and that of pencil-like s-resolvability are complementary equivalent. We record this fact in the following lemma.

Lemma 2.2

A t-design is point-missing s-resolvable if and only if its complement is pencil-like s-resolvable.

The next theorem shows a relation between certain classes of t-designs and point-missing \((t-1)\)-resolvable t-designs, in terms of derived designs.

Theorem 2.3

Let \((X, \mathcal{B})\) be a simple t-\((v,k, \lambda )\) design with \(|B\cap B'|\le k-2\) for any two distinct blocks \(B, B' \in \mathcal{B}\). Then there exists a simple point-missing \((t-1)\)-resolvable t-\((v, k-1, (k-t)\lambda )\) design \((X, \mathcal{D})\). In particular, if \((X, \mathcal{B})\) is a Steiner t-\((v,t+1,1)\) design, then there exists an overlarge set of Steiner \((t-1)\)-\((v-1,t,1)\) designs.

Proof

For a given point \(x \in X\) consider the derived design \((X{\setminus } \{x\}, \mathcal{B}_x)\) at x with parameters \((t-1)\)-\((v-1, k-1, \lambda )\). Here \(\mathcal{B}_x =\{ B{\setminus } \{x\} \mid x \in B, \; B\in \mathcal{B} \}\). Define \(\mathcal{D}= \bigcup _{x \in X} \mathcal{B}_x.\) We claim that \((X, \mathcal{D})\) is a t-\((v, k-1, (k-t)\lambda )\) design. Let \(T=\{x_1, \ldots , x_t\} \subseteq X\). Then there are \(\lambda \) blocks of \(\mathcal{B}\), say, \(B_1, \ldots , B_{\lambda }\) containing T. Each \(B_i\), \(i=1, \ldots , \lambda \), gives rise to a set \(\mathbb {D}_i = \{D=B_i {\setminus } \{x\}\; \mid x \in B_i {\setminus } T\} \subseteq \mathcal{D}\) having \((k-t)\) blocks D containing T. Thus there are \((k-t)\lambda \) blocks \(D \in \mathcal{D}\) containing T in total, as desired. The simplicity of \((X, \mathcal{D})\) is a consequence of the property: \(|B\cap B'|\le k-2\), \(B, B' \in \mathcal{B}\), \(B \not = B'\), which can be seen as follows. Let \(D, D'\) be two blocks of \(\mathcal{D}\). If \(D, D' \in \mathcal{B}_x\) for some \(x \in X\), then \(D \not = D'\), since \((X {\setminus } \{x\}, \mathcal{B}_x)\) is the derived design at x. If \(D \in \mathcal{B}_x\) and \(D' \in \mathcal{B}_y\) with \(x \not =y\), then again \(D \not =D'\). This is because if \(D=D'\), then the two blocks \(B=D\cup \{x\}\) and \(B'=D'\cup \{y\}\) of \(\mathcal{B}\) would have \(|B\cap B'|=k-1\), a contradiction. In addition, if \((X, \mathcal{B})\) is a Steiner t-\((v,t+1,1)\) design, then \((X, \mathcal{D})\) becomes the complete t-(v, t, 1) design. In other words, the set of v distinct \((t-1)\)-\((v-1,t,1)\) derived designs of \((X, \mathcal{B})\) forms an overlarge set. \(\square \)

Remark 2.1

  1. 1.

    The proof of Theorem 2.3 shows that the constructed t-\((v, k-1, (k-t)\lambda )\) design is not simple, if there are two blocks \(B,B'\in \mathcal{B}\) with \(|B\cap B'|= k-1\).

  2. 2.

    It should be stressed that the set of v distinct derived designs of a Steiner t-(v, k, 1) design with \(k > t+1\) in Theorem 2.3 will not form an overlarge set of \((t-1)\)-\((v-1, k-1, 1)\) designs, but rather a point-missing \((t-1)\)-resolution of a t-\((v, k-1, (k-t))\) design.

The following corollary is an immediate consequence of Theorem 2.3.

Corollary 2.4

Assume that there exists a Steiner t-(v, k, 1) design. Then there exists a point-missing \((t-1)\)-resolvable t-\((v,k-1, k-t)\) design.

The case \(k=t+1\) of Corollary 2.4 is known as examples of overlarge sets of Steiner designs, see [23]. Thus, if there exists a Steiner t-\((v,t+1,1)\) design, then there exists a point-missing \((t-1)\)-resolvable t-(v, t, 1) design, i.e. an overlarge set of Steiner \((t-1)\)-\((v-1,t,1)\) designs. Note that the converse of this statement is not true, i.e. if there exists an overlarge set of Steiner \((t-1)\)-\((v-1,t,1)\) designs, it is not necessarily true that a Steiner t-\((v, t+1,1)\) design exists. For example, Östergård and Pottonen [17] have shown that a Steiner 4-(17, 5, 1) design does not exist. Nevertheless, there exists an overlarge set of Steiner 3-(16, 4, 1) designs, see [23]. And crucially, Teirlinck [23] has shown that there are overlarge sets of Steiner 3-(v, 4, 1) designs for \(v=3^n-1\), \(n \ge 2\) and \( v= 3^n+1\), \(n\ge 1\), despite the fact that only a finite number of Steiner 4-(v, 5, 1) designs are hitherto known.

The general case \(k \ge t+2\) is interesting, since Theorem 2.3 provides a point-missing \((t-1)\)-resolvable t-\((v,k-1, k-t)\) design, which is not a complete design. Examples about this case can be seen, for instance, from Steiner 5-(24, 8, 1) and 5-(28, 7, 1) designs. Here we obtain point-missing 4-resolvable 5-(24, 7, 3) and 5-(28, 6, 2) designs, where designs in the resolution are Steiner 4-(23, 7, 1) and 4-(27, 6, 1) designs, respectively. Similarly, there are point-missing 3-resolvable 4-(23, 6, 3) and 4-(27, 5, 2) designs having Steiner 3-(22, 6, 1) and 3-(26, 5, 1) designs in the resolution, respectively.

As a further application of Theorem 2.3, we consider the infinite series of 4-\((q+1,6,10)\) designs with \(q=2^n\), \(n \ge 5\) and \(\gcd (n,6)=1\), [8], having the property that any two blocks of the designs intersect in at most 4 points. Thus we have the following result.

Corollary 2.5

Let \(q=2^n\), \(n \ge 5\) and \(\gcd (n,6)=1\). Then there exists a point-missing 3-resolvable 4-\((q+1,5,20)\) design having a 3-(q, 5, 10) design in the resolution.

Corollary 3.3 shows an interesting example of 4-designs that are 3-resolvable, and point-missing 3-resolvable as well.

3 Constructions of t-designs from point-missing \((t-1)\)-resolvable t-designs

Recall that Lemma 2.2 shows a natural connection between point-missing and pencil-like s-resolvability via the complement action. However, we observe that point-missing \((t-1)\)-resolvable t-designs may be used to construct pencil-like \((t-1)\)-resolvable t-designs which are not related to the complementary connection, as shown in the following theorem.

Theorem 3.1

Let \((X, \mathcal{B})\) be a point-missing \((t-1)\)-resolvable t-\((v,k, \lambda )\) design with \((t-1)\)-\((v-1, k, \delta )\) designs in the resolution. Then there is a pencil-like \((t-1)\)-resolvable t-\((v,k+1,t\delta + \lambda )\) design \((X, \mathcal{B}^*)\). If \(|B\cap B'|\le k-2\) for any two distinct blocks \(B, B' \in \mathcal{B}\), then \((X, \mathcal{B}^*)\) is simple. Further, if there are two blocks \(B, B' \in \mathcal{B}\) with \(|B\cap B'|= k-1\), then the simplicity of \((X, \mathcal{B}^*)\) depends on the structure of the resolution.

Proof

Let \(X=\{1, \ldots , v\}\). For \(i \in X\) denote \((X{\setminus } \{i\}, \mathcal{B}_i)\) the \((t-1)\)-\((v-1, k, \delta )\) design in the point-missing \((t-1)\)-resolution. Define \(\mathcal{B}_i^* = \{i\}\cup \mathcal{B}_i =\{ \{i\} \cup B \; | \; B \in \mathcal{B}_i \}\), for \(i=1, \ldots , v\), and \(\mathcal{B}^* = \bigcup _{i\in X} \mathcal{B}_i^*. \) We claim that \((X, \mathcal{B}^*)\) is a pencil-like \((t-1)\)-resolvable t-\((v,k+1,t\delta + \lambda )\) design. Let \(T=\{i_1, \ldots , i_t\} \subseteq X\). Consider a resolution class \(\mathcal{B}_j\) with \(j\in T\). Since \((X{\setminus } \{j\}, \mathcal{B}_j)\) is a \((t-1)\)-\((v-1, k, \delta )\) design, it follows that \(\{i_1, \ldots , i_t\}{\setminus } \{j\}\) is contained in \(\delta \) blocks of \(\mathcal{B}_j\). Therefore \(\{j\}\cup \{i_1, \ldots , i_t\}{\setminus } \{j\} = \{i_1, \ldots , i_t\}\) is contained in \(\delta \) blocks of \(\mathcal{B}_j^*\). Thus \(\mathcal{B}_{i_1}^*, \ldots , \mathcal{B}_{i_t}^*\) together have \(t\delta \) blocks containing T. Further, the \((v-t)\) resolution classes \(\mathcal{B}_j\) with \(j \not \in T \) have \(\lambda \) blocks containing T. Therefore the \((v-t)\) classes \(\mathcal{B}^*_j\) with \(j \not \in T \) together have \(\lambda \) blocks containing T. It follows that \((X, \mathcal{B}^*)\) is a t-\((v,k+1,t\delta + \lambda )\) design. Assume that \(|B\cap B'|\le k-2\) for any two distinct blocks \(B, B' \in \mathcal{B}\). Let \(B^*, B'^* \in \mathcal{B}^*\) be the two corresponding blocks of B and \(B'\). If \(B^*, B'^* \in \mathcal{B}_i^*\), then \(B^*=\{i\} \cup B\) and \(B'^*=\{i\} \cup B'\), so \(B^* \not = B'^*\), since \(B \not =B'\). The other case is that \(B^* \in \mathcal{B}_i^*\) and \(B'^* \in \mathcal{B}_j^*\) for \(i\not =j\), thus \(B^*=\{i\} \cup B\), \(B'^*=\{j\} \cup B'\), where \(B\in \mathcal{B}_i\) and \(B'\in \mathcal{B'}_j\). Since \(|B\cap B'|\le k-2\), we have \(B^* \not = B'^*\). Thus \((X, \mathcal{B}^*)\) is simple. \(\square \)

The next theorem may be viewed as the reverse of Theorem 3.1.

Theorem 3.2

Let \((X, \mathcal{B})\) be a pencil-like \((t-1)\)-resolvable t-\((v,k, \lambda )\) design with \((t-1)\)-\((v-1, k-1, \delta )\) designs in the resolution. Then there is a point-missing \((t-1)\)-resolvable t-\((v,k-1,\lambda - t\delta )\) design \((X, \mathcal{B}^*)\). If \(|B\cap B'|\le k-2\) for any two distinct blocks \(B, B' \in \mathcal{B}\), then \((X, \mathcal{B}^*)\) is simple. Further, if there are two blocks \(B, B' \in \mathcal{B}\) with \(|B\cap B'|= k-1\), then the simplicity of \((X, \mathcal{B}^*)\) depends on the structure of the pencil-like \((t-1)\)-resolution.

Proof

Let \(X=\{1, \ldots , v\}\). For \(i \in X\) denote \((X{\setminus } \{i\}, \mathcal{B}_i)\) the \((t-1)\)-\((v-1, k-1, \delta )\) design in the pencil-like \((t-1)\)-resolution of \((X, \mathcal{B})\). We have \(\mathcal{B}=(\{1\}\cup \mathcal{B}_1)\cup \cdots \cup (\{v\}\cup \mathcal{B}_v)\) Define \(\mathcal{B}^* = \mathcal{B}_1\cup \cdots \cup \mathcal{B}_v\). We claim that \((X, \mathcal{B}^*)\) is a t-\((v,k-1,\lambda - t\delta )\) design, which is point-missing \((t-1)\)-resolvable. Let \(T=\{i_1, \ldots , i_t\} \subseteq X\). Then T is contained in \(\lambda \) blocks of \((X, \mathcal{B})\), which are distributed in v classes of the pencil-like \((t-1)\)-resolution. Note that T is contained in \(\delta \) blocks of \((\{i_j\}\cup \mathcal{B}_{i_j})\), for \(i_j \in T\), so T is contained in \(t\delta \) blocks of \((\{i_1\}\cup \mathcal{B}_{i_1}) \cup \cdots \cup (\{i_t\}\cup \mathcal{B}_{i_t})\) (i.e., T is not contained in any block of \(\mathcal{B}_{i_1} \cup \cdots \cup \mathcal{B}_{i_t}\)). The remaining \((v-t)\) classes \(\{(\{1\}\cup \mathcal{B}_{1}) \cup \cdots \cup (\{v\}\cup \mathcal{B}_{v}) \}{\setminus } \{ (\{i_1\}\cup \mathcal{B}_{i_1}) \cup \cdots \cup (\{i_t\}\cup \mathcal{B}_{i_t}) \}\) of \((X, \mathcal{B})\) will have \(\lambda -t\delta \) blocks containing T. Moreover, if T is contained in a block \(\{j\}\cup B \in (\{j\}\cup \mathcal{B}_{j})\), \(j \in \{1, \ldots , v \} {\setminus } T\), then T is contained in \(B\in \mathcal{B}_{j}\). Hence, \(\mathcal{B}_1\cup \cdots \cup \mathcal{B}_v\) will have \(\lambda -t\delta \) blocks containing T and \((X, \mathcal{B}^*)\) is point-missing \((t-1)\)-resolvable. Assume that \(|B\cap B'|\le k-2\) for any two distinct blocks \(B, B' \in \mathcal{B}\). Obviously, the two corresponding blocks \(B^*, B'^* \in \mathcal{B}^*\) are distinct. Thus \((X, \mathcal{B}^*)\) is simple. \(\square \)

The simplicity of \((X, \mathcal{B}^*)\) in Theorem 3.1 in the case that there are two blocks \(B, B' \in \mathcal{B}\) with \(|B\cap B'|= k-1\) remains a main open question. In fact, examples for simple as well as non-simple \((X, \mathcal{B}^*)\) do exist in this case. We illustrate the situation with two explicit examples. First, consider the unique Steiner 3-(8, 4, 1) design \((X, \mathcal{B})\). By applying Lemma 2.2 we have

$$\begin{aligned} \mathcal{B}_0= & {} 123 \;\; 345 \;\; 256 \;\; 136 \;\; 467 \;\; 157 \;\; 237 \\ \mathcal{B}_1= & {} 024 \;\; 235 \;\; 456 \;\; 036 \;\; 057 \;\; 267 \;\; 347 \\ \mathcal{B}_2= & {} 014 \;\; 135 \;\; 346 \;\; 056 \;\; 167 \;\; 037 \;\; 457 \\ \mathcal{B}_3= & {} 125 \;\; 246 \;\; 045 \;\; 016 \;\; 567 \;\; 027 \;\; 147 \\ \mathcal{B}_4= & {} 012 \;\; 236 \;\; 035 \;\; 156 \;\; 067 \;\; 137 \;\; 257 \\ \mathcal{B}_5= & {} 123 \;\; 034 \;\; 146 \;\; 026 \;\; 367 \;\; 017 \;\; 247 \\ \mathcal{B}_6= & {} 234 \;\; 145 \;\; 025 \;\; 013 \;\; 357 \;\; 047 \;\; 127 \\ \mathcal{B}_7= & {} 356 \;\; 046 \;\; 015 \;\; 126 \;\; 023 \;\; 134 \;\; 245 \end{aligned}$$

Thus the block set \(\mathcal{D}=\bigcup _{x\in X}\mathcal{B}_x\) is the union of derived designs of \((X, \mathcal{B})\) at all points of \(X=\{0,1,2,3,4,5,6,7\}\). Here \(\mathcal{B}_0, \ldots , \mathcal{B}_7\) form an overlarge set of Steiner 2-(7, 3, 1) designs. It is easy to check that the resulting 3-(8, 4, 4) design \((X, \mathcal{B}^*)\) is not simple, more precisely each block is repeated 4 times. The second example is chosen from the set of 11 non-isomorphic of overlarge sets for 2-(7, 3, 1) designs [18]. The following representation is taken from [15].

$$\begin{aligned} \mathcal{B}'_0= & {} 123 \;\; 145 \;\; 167 \;\; 247 \;\; 256 \;\; 346 \;\; 357 \\ \mathcal{B}'_1= & {} 026 \;\; 035 \;\; 047 \;\; 234 \;\; 257 \;\; 367 \;\; 456 \\ \mathcal{B}'_2= & {} 015 \;\; 037 \;\; 046 \;\; 136 \;\; 147 \;\; 345 \;\; 567 \\ \mathcal{B}'_3= & {} 014 \;\; 025 \;\; 067 \;\; 127 \;\; 156 \;\; 246 \;\; 457 \\ \mathcal{B}'_4= & {} 016 \;\; 023 \;\; 057 \;\; 125 \;\; 137 \;\; 267 \;\; 356 \\ \mathcal{B}'_5= & {} 017 \;\; 024 \;\; 036 \;\; 126 \;\; 134 \;\; 237 \;\; 467 \\ \mathcal{B}'_6= & {} 013 \;\; 027 \;\; 045 \;\; 124 \;\; 157 \;\; 235 \;\; 347 \\ \mathcal{B}'_7= & {} 012 \;\; 034 \;\; 056 \;\; 135 \;\; 146 \;\; 236 \;\; 245 \end{aligned}$$

It is straightforward to check that \((X, \mathcal{B}'^*)\) forms a simple 3-(8, 4, 4) design.

The examples indicate an involved problem of deciding the simplicity of \((X, \mathcal{B}^*)\), when \((X, \mathcal{B})\) has two blocks B and \(B'\) with \(|B\cap B'|=k-1.\) The most interesting case for this situation, as mentioned in Theorem 2.3, is overlarge sets of disjoint Steiner \((t-1)\)-(v, t, 1) designs, i.e. the complete t-\((v+1, t,1)\) design is point-missing \((t-1)\)-resolvable having Steiner \((t-1)\)-(v, t, 1) designs in the resolution classes. Teirlinck [23] has shown that overlarge sets for Steiner 3-\((3^n -1,4,1)\) and 3-\((3^n +1,4,1)\) designs for \(n \ge 2\) exist, including the known overlarge sets of Steiner 3-\((2^n,4,1)\) designs. By using these results we obtain the following infinite series of 4-designs with constant index as a corollary of Theorem 3.1.

Corollary 3.3

There exist infinite series of pencil-like 3-resolvable 4-designs with the following parameters:

  1. 1.

    4-\((2^n+1,5,5)\) for \(n\ge 2\),

  2. 2.

    4-\((3^n,5,5)\) for \(n\ge 2\),

  3. 3.

    4-\((3^n+2,5,5)\) for \(n\ge 2\).

Remark 3.1

It should be remarked that for all the designs in Corollary 3.3 we have \(\lambda _\textrm{min}=1 \text{ or } 5\). More precisely,

$$\begin{aligned} {\lambda _\textrm{min}=5 \;} {\left\{ \begin{array}{ll} \text {for } v=2^n+1, &{} \text {and } n \equiv 3 \pmod 4, \\ \text {for } v=3^n, &{} \text {and } n \equiv 2 \pmod 4, \\ \text {for } v=3^n+2, &{} \text {and } n \equiv 3 \pmod 4. \end{array}\right. } \end{aligned}$$

Note that Alltop [1] has constructed infinite series of simple 4-\((2^n+1, 5,5)\) designs for n odd and \(n \ge 5\); thus the first series extends the point number to all possible values of n.

It is very likely that many non-isomorphic series of 4-designs with parameters given in Corollary 3.3 will exist, which are simple as well as non-simple, due to the fact that the number of non-isomorphic overlarge sets of 3-(v, 4, 1) will strongly increase as v is getting large. In particular, it is important to decide whether the 4-designs in Corollary 3.3 are simple or not. As an observation we take a close look at the first design in each of the 4-\((3^n,5,5)\) and 4-\((3^n+2,5,5)\) series. These are 4-(9, 5, 5) and 4-(11, 5, 5) designs, corresponding to \(n=2\). Note that each 4-(9, 5, 5) design is simple, since its complement is the complete 4-(9, 4, 1) design (otherwise, we would have a non-simple 4-(9, 4, 1) design, which is impossible). In fact, this can also be verified directly by checking the two non-isomorphic overlarge sets of 3-(8, 4, 1) designs given in [9], yielding 4-(9, 5, 5) designs. Note also that 4-(9, 5, 5) is the parameters of the second design in the 4-\((2^n+1,5,5)\) series. The case of 4-(11, 5, 5) designs is quite different. We have inspected the complete list of 21 non-isomorphic overlarge sets of 3-(10, 4, 1) designs as shown in [20] and found that they all yield non-simple 4-(11, 5, 5) designs.

For the ease of the reader, we include a table of known infinite series of t-designs with constant index for \(t\ge 4\) (Table ).

Theorem 3.4

There exists a pencil-like 3-resolvable 4-\((2^n+1,7,\frac{70}{3}(2^n-5))\) design for \(n \ge 5\) and \(\gcd (n,6)=1\).

Table 1 Known infinite series of t-designs with constant index for \(t \ge 4\)
Full size table

Proof

Each 4-\((2^n+1,6,10)\) design \((X, \mathcal{B})\) with \(n \ge 5\) and \(\gcd (n,6)=1\) in [8] has the property that \(|B\cap B'|\le 4\) for any two distinct blocks \(B, B' \in \mathcal{B}\). Its complement is a 4-\((2^n+1,2^n-5,\frac{2}{3}\left( {\begin{array}{c}2^n-5\\ 4\end{array}}\right) )\) design \((X, \bar{\mathcal{B}})\) having block intersections at most \((2^n-3)\). By Theorem 2.3 there is a point-missing 3-resolvable 4-\((2^n+1,2^n-6,(2^n-9)\frac{2}{3}\left( {\begin{array}{c}2^n-5\\ 4\end{array}}\right) )\) design \((X, \bar{\mathcal{D}})\). Again, the complement of \((X, \bar{\mathcal{D}})\) is pencil-like 3-resolvable 4-\((2^n+1,7,\frac{70}{3}(2^n-5))\) design, as desired. \(\square \)

By applying Theorem 3.2 to the point-missing 3-resolvable 4-\((2^n+1, 2^{n-1}, (2^{n-1}-3)(2^{n-2}-1))\) design \((X, \mathcal{B})\) of Alltop [2], we obtain an interesting result. Namely, we prove that there is a point-missing 3-resolvable design \((X, \mathcal{B}^*)\) with the same parameters as \((X, \mathcal{B})\) and disjoint from \((X, \mathcal{B})\) (recall that any two distinct blocks \(B, B'\in \mathcal{B}\) have \(|B\cap B'|\le 2^{n-1}-2\)). Let \(\mathcal{B}= \mathcal{B}_1 \cup \cdots \cup \mathcal{B}_v\) be a partition of \(\mathcal{B}\) into point-missing 3-resolution classes, i.e. each \((X_i, \mathcal{B}_i)\) is a 3-\((2^n, 2^{n-1}, 2^{n-2}-1)\) design with \(X_i= X{\setminus } \{i\}\). Consider \((X,\bar{\mathcal{B}})\) as the complement of \((X, \mathcal{B})\). So, \((X,\bar{\mathcal{B}})\) has parameters 4-\((2^n+1, 2^{n-1}+1, (2^{n-1}+1)(2^{n-2}-1))\) and is pencil-like 3-resolvable. Here, \(\bar{\mathcal{B}}= (\{1\}\cup \tilde{\mathcal{B}}_1)\cup \cdots \cup (\{v\}\cup \tilde{\mathcal{B}}_v)\), where \(\tilde{\mathcal{B}}_j\) is the complement of \(\mathcal{B}_j\) in \(X_j\), and \((X_j,\tilde{\mathcal{B}}_j)\) is a 3-\((2^n, 2^{n-1}, 2^{n-2}-1)\) design, for \(j=1, \ldots , v\). The proof of Theorem 3.2 shows that \((X, \tilde{\mathcal{B}}^*)\) with \(\tilde{\mathcal{B}}^*= \tilde{\mathcal{B}}_1 \cup \cdots \cup \tilde{\mathcal{B}}_v\), is point-missing 3-resolvable with \((X_j,\tilde{\mathcal{B}}_j)\) as the design in the resolution. Clearly, \((X, \mathcal{B})\) and \((X, \tilde{\mathcal{B}}^*)\) are disjoint and they have the same parameters. Further, the 4-design \((X, \mathcal{B}\cup \tilde{\mathcal{B}}^*)\) can be extended to a 5-design. Thus we have the following theorem.

Theorem 3.5

Let \(n \ge 4\). Then

  1. 1.

    There exists a simple point-missing 3-resovable 4-\((2^n+1, 2^{n-1}, 2(2^{n-1}-3)(2^{n-2}-1))\) design,

  2. 2.

    There exists a simple 5-\((2^n+2, 2^{n-1}+1, 2(2^{n-1}-3)(2^{n-2}-1))\) design.

4 A construction of point-missing s-resolvable t-designs

In this section we show that the recursive construction of t-designs in [24] can be extended to a construction of point-missing s-resolvable t-designs. More precisely, we prove the following theorem.

Theorem 4.1

Assume that there exists a point-missing s-resolvable t-\((v,k,\lambda )\) design having s-\((v-1,k,\delta )\) designs in its resolution. If \(v\lambda _0(\lambda _0-\lambda _1) < \left( {\begin{array}{c}v\\ k\end{array}}\right) \), then there exists a point-missing s-resolvable t-\((v+1,k,(v+1-t)\lambda )\) design having s-\((v,k,(v-s)\delta )\) designs in its resolution.

Proof

Assume that \((Y, \mathcal{D})\) is a point-missing s-resolvable t-\((v,k,\lambda )\) design. Let \(X=\{1, \ldots , v+1\}\) and denote \(X_j= X {\setminus } \{j\}\) for \(j=1, \ldots , v+1\). Let \((X_j, \mathcal{B}^{(j)})\) be a copy of \((Y, \mathcal{D})\) defined on \(X_j\). If \(v\lambda _0(\lambda _0-\lambda _1) < \left( {\begin{array}{c}v\\ k\end{array}}\right) \), then by Theorem A in [24] there are \((v+1)\) mutually disjoint \(\mathcal{B}^{(1)}, \ldots , \mathcal{B}^{(v+1)}\) and they form a t-\((v+1,k,(v+1-t)\lambda )\) design \((X, \mathcal{B})\), where

$$\begin{aligned} \mathcal{B} = \bigcup _{j=1}^{v+1} \mathcal{B}^{(j)}. \end{aligned}$$

We prove that \((X, \mathcal{B})\) is point-missing s-resolvable. Denote the partition of \((X_j,\mathcal{B}^{(j)})\) into point-missing s-resolution classes by

$$\begin{aligned} \mathcal{B}^{(j)}=\overbrace{\mathcal{C}_{1}^{(j)} \cup \cdots \cup \mathcal{C}_{j-1}^{(j)} \cup \mathcal{C}_{j+1}^{(j)} \cup \cdots \cup \mathcal{C}_{v+1}^{(j)}}^v, \end{aligned}$$

with \((X_{i,j}, \mathcal{C}_{i}^{(j)})\) as an s-\((v-1, k, \delta )\) design, where \(X_{i,j}= X_j {\setminus } \{i\}\) and \(i\in X_j\). For each point \(j \in X \) define

$$\begin{aligned} \mathcal{C}_j= \overbrace{ \mathcal{C}_j^{(1)} \cup \mathcal{C}_j^{(2)}\cup \cdots \cup \mathcal{C}_j^{(j-1)} \cup \mathcal{C}_j^{(j+1)} \cup \cdots \cup \mathcal{C}_j^{(v+1)}}^v. \end{aligned}$$

We claim that \((X_j, \mathcal{C}_j)\) is an s-\((v,k, (v-s)\delta )\) design. Let \(S=\{j_1, \ldots , j_s \} \subseteq X_j\). Then S will not appear in the blocks of \(\mathcal{C}_j^{(j_1)}, \mathcal{C}_j^{(j_2)}, \ldots , \mathcal{C}_j^{(j_s)}.\) Hence S appears in \((v-s)\) block sets \(\mathcal{C}_j^{(i)}\), for \(i \not = j_1, \ldots , j_s\). In other words, S is contained in the blocks of \(\mathcal{C}_j\) exactly \((v-s)\delta \) times, which proves the claim. Further, since

$$\begin{aligned} \mathcal{B}= \mathcal{C}_1 \cup \cdots \cup \mathcal{C}_{v+1}, \end{aligned}$$

\((X, \mathcal{B})\) is point-missing s-resolvable with \(\mathcal{C}_1, \ldots , \mathcal{C}_{v+1} \) as resolution classes. Note that the value of \(\delta \) can be computed in terms of \(t,\; v,\; k, \; \lambda \) by using Lemma 2.1. \(\square \)

As an application of Theorem 4.1 consider the infinite series of 4-designs \((X, \mathcal{D})\) constructed by Alltop in [2]. \((X, \mathcal{D})\) has parameters 4-\((2^n+1, 2^{n-1}, (2^{n-1}-3)(2^{n-2}-1))\), \(n \ge 4\), and is point-missing 3-resolvable with 3-\((2^n, 2^{n-1}, 2^{n-2}-1)\) designs in its resolution. For \(n \ge 5\) the condition \(v\lambda _0(\lambda _0-\lambda _1) < \left( {\begin{array}{c}v\\ k\end{array}}\right) \) is satisfied, therefore Theorem 4.1 gives the following corollary.

Corollary 4.2

For \(n \ge 5\), there exists an infinite series of simple point-missing 3-resolvable 4-\((2^n+2, 2^{n-1}, (2^n-2)(2^{n-1}-3)(2^{n-2}-1))\) designs. The parameters of the 3-designs in the resolution are 3-\((2^n+1, 2^{n-1},(2^n-2)(2^{n-2}-1)).\)

5 Conclusion

The paper deals with point-missing s-resolvable t-designs with emphasis on their use in constructing t-designs. Among others, we show the existence of infinite series of 4-(v, 5, 5) designs with \(v= 2^n+1,\; 3^n,\; 3^n+2\) for \(n \ge 2\). It remains an open question about the simplicity of the designs in these series. We also present a recursive construction of point-missing s-resolvable t-designs including an application.