Abstract
The paper deals with tdesigns that can be partitioned into sdesigns, each missing a point of the underlying set, called pointmissing sresolvable tdesigns, with emphasis on their applications in constructing tdesigns. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the \(\left( {\begin{array}{c}v +1\\ k\end{array}}\right) \) ksets chosen from a \((v+1)\)set X into \((v+1)\) mutually disjoint s\((v,k,\delta )\) designs, each missing a different point of X. Among others, it is shown that the existence of a pointmissing \((t1)\)resolvable t\((v,k,\lambda )\) design leads to the existence of a t\((v,k+1,\lambda ')\) design. As a result, we derive various infinite series of 4designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4\((3^n,5,5)\), 4\((3^n+2,5,5)\) and 4\((2^n+1,5,5)\), for \(n \ge 2\), and were unknown until now. We also include a recursive construction of pointmissing sresolvable tdesigns and its application.
1 Introduction
The paper is concerned with pointmissing sresolutions of tdesigns 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 sresolution. If \(w=v\), then \((X, \mathcal{B})\) is called sresolvable; in particular, if \((X, \mathcal{B})\) is the complete k(v, k, 1) design, then an sresolution of \((X, \mathcal{B})\) is called a large set of sdesigns. If \(w=v1\), then \((X, \mathcal{B})\) is called pointmissing sresolvable. A pointmissing sresolution of the complete k(v, k, 1) design is called an overlarge set of sdesigns. Pointmissing sresolvability 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]. Pointmissing sresolvability is complementarily related to what we call pencillike sresolvability for tdesigns, and vice versa. As far as we know the first example of infinite series of nontrivial pointmissing sresolvable tdesigns for \(t \ge 4\) can be found in a paper of Alltop in 1972 [2], in which the author constructed a series of 4\((2^n+1, 2^{n1}, (2^{n1}3)(2^{n2}1))\) designs for \(n \ge 4\) as the union of \(2^n+1\) mutually disjoint 3\((2^n, 2^{n1}, 2^{n2}1)\) designs. We prove theorems for constructing new tdesigns from pointmissing and pencillike sresolvable tdesigns. By using these theorems for overlarge sets of disjoint Steiner quadruple systems with \(v=3 ^n1\) 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 2designs for \(k=4\) with \(v=13, 16\) points are known to exist [10, 12, 14]. We also show a recursive construction of pointmissing sresolvable tdesigns and its application.
For the sake of clarity we include a few basic definitions. A tdesign, denoted by t\((v,k,\lambda )\), is a pair \((X, \mathcal{B})\), where X is a vset of points and \(\mathcal{B}\) is a collection of ksubsets of X, called blocks, such that every tsubset of X is a subset of exactly \(\lambda \) blocks, and \(\lambda \) is called the index of the design. A tdesign is called simple if no two blocks are identical, otherwise, it is called nonsimple. A t(v, k, 1) design is called a Steiner tdesign. 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 \((t1)\)\((v1,k1,\lambda )\) 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 \(\lambda _s = \lambda {vs \atopwithdelims ()ts}/{ks \atopwithdelims ()ts}.\) Since \(\lambda _s\) is an integer, necessary conditions for the parameters of a tdesign are \({ks \atopwithdelims ()ts}\vert \lambda {vs \atopwithdelims ()ts}\) 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 ksubsets of X, then \((X, \mathcal{B})\) is a t\((v,k, \lambda _\textrm{max})\) design, called the complete design, where \(\lambda _\textrm{max}={vt \atopwithdelims ()kt}\). If we take \(\delta \) copies of the complete design, we obtain a t\((v,k, \delta {vt \atopwithdelims ()kt})\) design, which is referred to as a trivial tdesign; otherwise, it is called a nontrivial tdesign.
2 Pointmissing sresolvable tdesigns
A t\((v,k,\lambda )\) design \((X, \mathcal{B})\) is said to be sresolvable, 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 sresolution of \((X, \mathcal{B})\) and each \(\mathcal{B}_i\) is called an sresolution 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 \(N={\left( {\begin{array}{c}vt\\ kt\end{array}}\right) }/{\lambda }\), then we say that there exists a large set of tdesigns 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 pointmissing sresolvability 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 \(1\le s \le t1\). \((X,\mathcal{B})\) is called pointmissing sresolvable, if the block set \(\mathcal{B}\) can be partitioned into mutually disjoint s\((v1,k, \delta )\) designs, each missing a point of X.
However, Definition 2.1 is equivalent to a definition that describes pointmissing 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\((v1,k, \delta )\) 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 sdesign \((X_i,\mathcal{B}_i)\). 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.
Consider two given points \(x_i \in Y\) and \(x_j \in X\setminus Y\). Since \(x_i \in Y\), there is no sdesign \((X_i,\mathcal{B}_i)\) 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 \(\delta _1 =\delta \frac{ \left( {\begin{array}{c}v2\\ s1\end{array}}\right) }{\left( {\begin{array}{c}k1\\ s1\end{array}}\right) }.\) 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 pointmissing sresolvable, if there is an integer \(m \ge 1\) such that the following hold.

1.
\(\mathcal{B}= \mathcal{B}_{x_1} \cup \cdots \cup \mathcal{B}_{x_v}\), where \(X=\{x_1, \ldots , x_v\},\)

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\((v1,k,\delta )\) design, \(j=1, \ldots , m,\) and m is called the multiplicity of the point x.
If \(m=1\), \((X,\mathcal{B})\) is simply called pointmissing sresolvable. Moreover, if \(m > 1\), then \((X{\setminus } \{x\}, \mathcal{B}_x)\) is an s\((v1,k,m\delta )\) design. Therefore, \((X,\mathcal{B})\) again is a union of v mutually disjoint s\((v1,k,m\delta )\) design, each missing a different point of X. Hence, in general, when we speak of pointmissing sresolvable tdesigns we mean \(m=1\).
If the complete k(v, k, 1) design can be partitioned into v mutually disjoint s\((v1, k, \delta )\) designs (i.e. pointmissing sresolvable), then we have an overlarge set of s\((v1, k, \delta )\) designs.
Lemma 2.1
Let \((X, \mathcal{B})\) be a pointmissing sresolvable t\((v,k, \lambda )\) design and assume that each point in the resolution has multiplicity m. Then
In particular, if the complete t(v, t, 1) design is pointmissing \((t1)\)resolvable, then the designs in the resolution are Steiner \((t1)\)\((v1,t,1)\) designs.
Proof
By assumption, we have
where \((X \setminus \{x\}, \mathcal{B}^i_x)\) is an s\((v1, k, \delta )\) 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 \(m(vs)\delta \) times in the blocks of \(\mathcal{B}\). On the other hand, the number of blocks in \(\mathcal{B}\) containing S is \(\lambda _s=\frac{\left( {\begin{array}{c}vs\\ ts\end{array}}\right) }{\left( {\begin{array}{c}ks\\ ts\end{array}}\right) }\lambda .\) Therefore \(\lambda _s=m(vs)\delta \) and thus \(\delta = \frac{\lambda _s}{m(vs)}\), as desired. \(\square \)
Recall that the complement of an sresolvable tdesign is again sresolvable. However, it is not true with a pointmissing sresolvable tdesign. 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 pointmissing sresolvable tdesign with parameters t\((v,k,\lambda )\) and let \((X,\overline{\mathcal{D}})\) be its complement which has parameters t\((v,vk,\overline{\lambda })\), where \(\overline{\lambda }= \lambda \left( {\begin{array}{c}vk\\ t\end{array}}\right) /\left( {\begin{array}{c}k\\ t\end{array}}\right) \). Let \(\mathcal{D} =\mathcal{D}_1\cup \cdots \cup \mathcal{D}_v\) be a partition of \(\mathcal{D}\) into v pointmissing sresolution classes, where \((X_i, \mathcal{D}_i)\) is an s\((v1,k,\delta )\) design, for \(i=1, \ldots , v\). The complement of \((X_i, \mathcal{D}_i)\) (within \(X_i\)) is an s\((v1, v1k, \widetilde{\delta })\) design \((X_i, \widetilde{\mathcal{D}}_i)\) with \(\widetilde{\delta }=\delta \left( {\begin{array}{c}v1k\\ s\end{array}}\right) /\left( {\begin{array}{c}k\\ s\end{array}}\right) \). 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 sdesign, 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\((v1,k1,\delta )\) 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 spencil 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\((v1,k1,\delta )\) design, then \((X, \mathcal{B})\) is said to be pencillike sresolvable.
As observed above, the complement of a pointmissing sresolvable tdesign is a pencillike sresolvable tdesign. Conversely, it is straightforward to check that the complement of a pencillike sresolvable tdesign is a pointmissing sresolvable tdesign. Hence the notion of pointmissing sresolvability and that of pencillike sresolvability are complementary equivalent. We record this fact in the following lemma.
Lemma 2.2
A tdesign is pointmissing sresolvable if and only if its complement is pencillike sresolvable.
The next theorem shows a relation between certain classes of tdesigns and pointmissing \((t1)\)resolvable tdesigns, in terms of derived designs.
Theorem 2.3
Let \((X, \mathcal{B})\) be a simple t\((v,k, \lambda )\) design with \(B\cap B'\le k2\) for any two distinct blocks \(B, B' \in \mathcal{B}\). Then there exists a simple pointmissing \((t1)\)resolvable t\((v, k1, (kt)\lambda )\) 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 \((t1)\)\((v1,t,1)\) designs.
Proof
For a given point \(x \in X\) consider the derived design \((X{\setminus } \{x\}, \mathcal{B}_x)\) at x with parameters \((t1)\)\((v1, k1, \lambda )\). 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\((v, k1, (kt)\lambda )\) 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 \((kt)\) blocks D containing T. Thus there are \((kt)\lambda \) blocks \(D \in \mathcal{D}\) containing T in total, as desired. The simplicity of \((X, \mathcal{D})\) is a consequence of the property: \(B\cap B'\le k2\), \(B, B' \in \mathcal{B}\), \(B \not = B'\), 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 \(B\cap B'=k1\), 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 \((t1)\)\((v1,t,1)\) derived designs of \((X, \mathcal{B})\) forms an overlarge set. \(\square \)
Remark 2.1

1.
The proof of Theorem 2.3 shows that the constructed t\((v, k1, (kt)\lambda )\) design is not simple, if there are two blocks \(B,B'\in \mathcal{B}\) with \(B\cap B'= k1\).

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 \((t1)\)\((v1, k1, 1)\) designs, but rather a pointmissing \((t1)\)resolution of a t\((v, k1, (kt))\) 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 pointmissing \((t1)\)resolvable t\((v,k1, kt)\) 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 pointmissing \((t1)\)resolvable t(v, t, 1) design, i.e. an overlarge set of Steiner \((t1)\)\((v1,t,1)\) designs. Note that the converse of this statement is not true, i.e. if there exists an overlarge set of Steiner \((t1)\)\((v1,t,1)\) 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 \(v=3^n1\), \(n \ge 2\) 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 pointmissing \((t1)\)resolvable t\((v,k1, kt)\) 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 pointmissing 4resolvable 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 pointmissing 3resolvable 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 pointmissing 3resolvable 4\((q+1,5,20)\) design having a 3(q, 5, 10) design in the resolution.
Corollary 3.3 shows an interesting example of 4designs that are 3resolvable, and pointmissing 3resolvable as well.
3 Constructions of tdesigns from pointmissing \((t1)\)resolvable tdesigns
Recall that Lemma 2.2 shows a natural connection between pointmissing and pencillike sresolvability via the complement action. However, we observe that pointmissing \((t1)\)resolvable tdesigns may be used to construct pencillike \((t1)\)resolvable tdesigns which are not related to the complementary connection, as shown in the following theorem.
Theorem 3.1
Let \((X, \mathcal{B})\) be a pointmissing \((t1)\)resolvable t\((v,k, \lambda )\) design with \((t1)\)\((v1, k, \delta )\) designs in the resolution. Then there is a pencillike \((t1)\)resolvable t\((v,k+1,t\delta + \lambda )\) design \((X, \mathcal{B}^*)\). If \(B\cap B'\le k2\) 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 \(B\cap B'= k1\), 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 \((t1)\)\((v1, k, \delta )\) design in the pointmissing \((t1)\)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 pencillike \((t1)\)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 \((t1)\)\((v1, k, \delta )\) 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 \((vt)\) resolution classes \(\mathcal{B}_j\) with \(j \not \in T \) have \(\lambda \) blocks containing T. Therefore the \((vt)\) 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 \(B\cap B'\le k2\) 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 \(B\cap B'\le k2\), 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 pencillike \((t1)\)resolvable t\((v,k, \lambda )\) design with \((t1)\)\((v1, k1, \delta )\) designs in the resolution. Then there is a pointmissing \((t1)\)resolvable t\((v,k1,\lambda  t\delta )\) design \((X, \mathcal{B}^*)\). If \(B\cap B'\le k2\) 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 \(B\cap B'= k1\), then the simplicity of \((X, \mathcal{B}^*)\) depends on the structure of the pencillike \((t1)\)resolution.
Proof
Let \(X=\{1, \ldots , v\}\). For \(i \in X\) denote \((X{\setminus } \{i\}, \mathcal{B}_i)\) the \((t1)\)\((v1, k1, \delta )\) design in the pencillike \((t1)\)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\((v,k1,\lambda  t\delta )\) design, which is pointmissing \((t1)\)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 pencillike \((t1)\)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 \((vt)\) 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 pointmissing \((t1)\)resolvable. Assume that \(B\cap B'\le k2\) 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 \(B\cap B'= k1\) remains a main open question. In fact, examples for simple as well as nonsimple \((X, \mathcal{B}^*)\) 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
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 nonisomorphic of overlarge sets for 2(7, 3, 1) designs [18]. The following representation is taken from [15].
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 \(B\cap B'=k1.\) The most interesting case for this situation, as mentioned in Theorem 2.3, is overlarge sets of disjoint Steiner \((t1)\)(v, t, 1) designs, i.e. the complete t\((v+1, t,1)\) design is pointmissing \((t1)\)resolvable having Steiner \((t1)\)(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 4designs with constant index as a corollary of Theorem 3.1.
Corollary 3.3
There exist infinite series of pencillike 3resolvable 4designs with the following parameters:

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

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

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,
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 nonisomorphic series of 4designs with parameters given in Corollary 3.3 will exist, which are simple as well as nonsimple, due to the fact that the number of nonisomorphic overlarge sets of 3(v, 4, 1) will strongly increase as v is getting large. In particular, it is important to decide whether the 4designs 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 nonsimple 4(9, 4, 1) design, which is impossible). In fact, this can also be verified directly by checking the two nonisomorphic 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 nonisomorphic overlarge sets of 3(10, 4, 1) designs as shown in [20] and found that they all yield nonsimple 4(11, 5, 5) designs.
For the ease of the reader, we include a table of known infinite series of tdesigns with constant index for \(t\ge 4\) (Table ).
Theorem 3.4
There exists a pencillike 3resolvable 4\((2^n+1,7,\frac{70}{3}(2^n5))\) design for \(n \ge 5\) and \(\gcd (n,6)=1\).
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\((2^n+1,2^n5,\frac{2}{3}\left( {\begin{array}{c}2^n5\\ 4\end{array}}\right) )\) design \((X, \bar{\mathcal{B}})\) having block intersections at most \((2^n3)\). By Theorem 2.3 there is a pointmissing 3resolvable 4\((2^n+1,2^n6,(2^n9)\frac{2}{3}\left( {\begin{array}{c}2^n5\\ 4\end{array}}\right) )\) design \((X, \bar{\mathcal{D}})\). Again, the complement of \((X, \bar{\mathcal{D}})\) is pencillike 3resolvable 4\((2^n+1,7,\frac{70}{3}(2^n5))\) design, as desired. \(\square \)
By applying Theorem 3.2 to the pointmissing 3resolvable 4\((2^n+1, 2^{n1}, (2^{n1}3)(2^{n2}1))\) design \((X, \mathcal{B})\) of Alltop [2], we obtain an interesting result. Namely, we prove that there is a pointmissing 3resolvable 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 \(B\cap B'\le 2^{n1}2\)). Let \(\mathcal{B}= \mathcal{B}_1 \cup \cdots \cup \mathcal{B}_v\) be a partition of \(\mathcal{B}\) into pointmissing 3resolution classes, i.e. each \((X_i, \mathcal{B}_i)\) is a 3\((2^n, 2^{n1}, 2^{n2}1)\) 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\((2^n+1, 2^{n1}+1, (2^{n1}+1)(2^{n2}1))\) and is pencillike 3resolvable. 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\((2^n, 2^{n1}, 2^{n2}1)\) 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 pointmissing 3resolvable 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 4design \((X, \mathcal{B}\cup \tilde{\mathcal{B}}^*)\) can be extended to a 5design. Thus we have the following theorem.
Theorem 3.5
Let \(n \ge 4\). Then

1.
There exists a simple pointmissing 3resovable 4\((2^n+1, 2^{n1}, 2(2^{n1}3)(2^{n2}1))\) design,

2.
There exists a simple 5\((2^n+2, 2^{n1}+1, 2(2^{n1}3)(2^{n2}1))\) design.
4 A construction of pointmissing sresolvable tdesigns
In this section we show that the recursive construction of tdesigns in [24] can be extended to a construction of pointmissing sresolvable tdesigns. More precisely, we prove the following theorem.
Theorem 4.1
Assume that there exists a pointmissing sresolvable t\((v,k,\lambda )\) design having s\((v1,k,\delta )\) 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 pointmissing sresolvable t\((v+1,k,(v+1t)\lambda )\) design having s\((v,k,(vs)\delta )\) designs in its resolution.
Proof
Assume that \((Y, \mathcal{D})\) is a pointmissing sresolvable 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\((v+1,k,(v+1t)\lambda )\) design \((X, \mathcal{B})\), where
We prove that \((X, \mathcal{B})\) is pointmissing sresolvable. Denote the partition of \((X_j,\mathcal{B}^{(j)})\) into pointmissing sresolution classes by
with \((X_{i,j}, \mathcal{C}_{i}^{(j)})\) as an s\((v1, k, \delta )\) design, where \(X_{i,j}= X_j {\setminus } \{i\}\) and \(i\in X_j\). For each point \(j \in X \) define
We claim that \((X_j, \mathcal{C}_j)\) is an s\((v,k, (vs)\delta )\) 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 \((vs)\) 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 \((vs)\delta \) times, which proves the claim. Further, since
\((X, \mathcal{B})\) is pointmissing sresolvable 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 4designs \((X, \mathcal{D})\) constructed by Alltop in [2]. \((X, \mathcal{D})\) has parameters 4\((2^n+1, 2^{n1}, (2^{n1}3)(2^{n2}1))\), \(n \ge 4\), and is pointmissing 3resolvable with 3\((2^n, 2^{n1}, 2^{n2}1)\) 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 pointmissing 3resolvable 4\((2^n+2, 2^{n1}, (2^n2)(2^{n1}3)(2^{n2}1))\) designs. The parameters of the 3designs in the resolution are 3\((2^n+1, 2^{n1},(2^n2)(2^{n2}1)).\)
5 Conclusion
The paper deals with pointmissing sresolvable tdesigns with emphasis on their use in constructing tdesigns. 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 pointmissing sresolvable tdesigns including an application.
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Trung, T.v. Pointmissing sresolvable tdesigns: infinite series of 4designs with constant index. Des. Codes Cryptogr. (2023). https://doi.org/10.1007/s10623023012068
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DOI: https://doi.org/10.1007/s10623023012068
Keywords
 Pointmissing sresolvable tdesign
 Overlarge set of sdesigns
Mathematics Subject Classification
 05B05