Point-missing s-resolvable t-designs: infinite series of 4-designs with constant index

The paper deals with t-designs that can be partitioned into s-designs, each missing a point of the underlying set, called point-missing s-resolvable t-designs, with emphasis on their applications in constructing t-designs. The problem considered may be viewed as a generalization of overlarge sets which are defined as a partition of all the v+1k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\begin{array}{c}v +1\\ k\end{array}}\right) $$\end{document}k-sets chosen from a (v+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v+1)$$\end{document}-set X into (v+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v+1)$$\end{document} mutually disjoint s-(v,k,δ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,k,\delta )$$\end{document} designs, each missing a different point of X. Among others, it is shown that the existence of a point-missing (t-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(t-1)$$\end{document}-resolvable t-(v,k,λ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,k,\lambda )$$\end{document} design leads to the existence of a t-(v,k+1,λ′)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(v,k+1,\lambda ')$$\end{document} design. As a result, we derive various infinite series of 4-designs with constant index using overlarge sets of disjoint Steiner quadruple systems. These have parameters 4-(3n,5,5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3^n,5,5)$$\end{document}, 4-(3n+2,5,5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3^n+2,5,5)$$\end{document} and 4-(2n+1,5,5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2^n+1,5,5)$$\end{document}, for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, and were unknown until now. We also include a recursive construction of point-missing s-resolvable t-designs and its application.


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, λ) design (X , B) into mutually disjoint s-(w, k, δ) designs, w ≤ v, s < t, is termed an s-resolution. If w = v, then (X , B) is called s-resolvable; in particular, if (X , B) is the complete k-(v, k, 1) design, then an s-resolution of (X , B) is called a large set of s-designs. If w = v − 1, then (X , 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 Communicated by G. Ge.
B Tran van Trung trung@iem.uni-due.de 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 ≥ 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 ≥ 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 pointmissing 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, λ), is a pair (X , B), where X is a v-set of points and B is a collection of k-subsets of X , called blocks, such that every t-subset of X is a subset of exactly λ blocks, and λ 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 The smallest positive integer λ for which these necessary conditions are satisfied is denoted by λ min (t, k, v) or simply λ min . If B is the set of all k-subsets of X , then (X , B) is a t-(v, k, λ max ) design, called the complete design, where λ max = v−t k−t . If we take δ copies of the complete design, we obtain a t-(v, k, δ v−t k−t ) design, which is referred to as a trivial t-design; otherwise, it is called a non-trivial t-design.

Point-missing s-resolvable t-designs
Such a partition is called an s-resolution of (X , B) and each B i is called an s-resolution class or simply a resolution class, see e.g. [25,26].
In the most general form, the concept of point-missing s-resolvability of a t-(v, k, λ) design can be defined as follows. B) is called pointmissing s-resolvable, if the block set B can be partitioned into mutually disjoint s-(v −1, k, δ) 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.
Then the blocks of B can be written as follows.

Consider two given points
.
The discussion above suggests an equivalent formulation of Definition 2.1 as follows.

Definition 2.2
Let (X , B) be a t-(v, k, λ) design and let 1 ≤ s < t be an integer. (X , B) is said to be point-missing s-resolvable, if there is an integer m ≥ 1 such that the following hold.
. . , m, and m is called the multiplicity of the point x.
If m = 1, (X , B) is simply called point-missing s-resolvable. Moreover, if m > 1, then (X \{x}, B x ) is an s-(v − 1, k, mδ) design. Therefore, (X , B) again is a union of v mutually disjoint s-(v − 1, k, mδ) 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.

Lemma 2.1 Let (X , B) be a point-missing s-resolvable t-(v, k, λ) design and assume that each point in the resolution has multiplicity m. Then
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 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 , . . . , x v } and let is not an s-design, but rather a "pencil". Hence, the decomposition of (X , D) suggests the following definition.

Definition 2.3 Let
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 pencillike 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.  B) forms an overlarge set.

Remark 2.1 1. The proof of Theorem 2.3 shows that the constructed
The following corollary is an immediate consequence of Theorem 2.3.
The general case k ≥ t + 2 is interesting, since Theorem 2.3 provides a point-missing 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 ≥ 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.
then the simplicity of (X , B * ) depends on the structure of the resolution.
The next theorem may be viewed as the reverse of Theorem 3.1.

Theorem 3.2 Let (X , B) be a pencil-like
Then T is contained in λ blocks of (X , B), which are distributed in v classes of the pencil-like (t − 1)-resolution. Note that T is contained in δ blocks of ({i j } ∪ B i j ), for i j ∈ T , so T is contained in tδ blocks of ( Thus the block set D = x∈X B x is the union of derived designs of (X , B) at all points of X = {0, 1, 2, 3, 4, 5, 6, 7}. Here B 0 , . . . , 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 , 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]. The examples indicate an involved problem of deciding the simplicity of (X , B * ), when (X , B) has two blocks B and B with |B ∩ 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 ≥ 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.  for v = 2 n + 1, and n ≡ 3 (mod 4), for v = 3 n , and n ≡ 2 (mod 4), for v = 3 n + 2, and n ≡ 3 (mod 4).
Note that Alltop [1] has constructed infinite series of simple 4-(2 n + 1, 5, 5) designs for n odd and n ≥ 5; thus the first series extends the point number to all possible values of n.
For the ease of the reader, we include a table of known infinite series of t-designs with constant index for t ≥ 4 (Table 1).

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-
) be a copy of (Y , D) defined on X j . If vλ 0 (λ 0 − λ 1 ) < v k , then by Theorem A in [24] there are (v + 1) mutually disjoint B (1) , . . . , B (v+1) and they form a t- We prove that (X , B) is point-missing s-resolvable. Denote the partition of (X j , B ( j) ) into point-missing s-resolution classes by . . , j s } ⊆ X j . Then S will not appear in the blocks of C j , for i = j 1 , . . . , j s . In other words, S is contained in the blocks of C j exactly (v − s)δ times, which proves the claim. Further, since B = C 1 ∪ · · · ∪ C v+1 , (X , B) is point-missing s-resolvable with C 1 , . . . , C v+1 as resolution classes. Note that the value of δ can be computed in terms of t, v, k, λ by using Lemma 2.1.

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 ≥ 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.
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