A method of constructing 2-resolvable t-designs for t=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=3,4$$\end{document}

The paper introduces a method for constructing 2-resolvable t-designs for t=3,4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=3,4$$\end{document}. The main idea is based on the assumption that there exists a partition of a t-design into Steiner 2-designs. A remarkable property of the method is that it enables the construction of 2-resolvable t-designs with a large variety of block sizes. For t=4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=4$$\end{document}, it is required that the Steiner 2-designs of the partition are projective planes and this case would also lead to a construction of 3-resolvable 5-designs. For instance, we show the existence of an infinite series of 3-resolvable 5-designs having N=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=5$$\end{document} resolution classes with parameters 5-(14+8m,7,10(9+8m)(1+m))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(14+8m,7, 10(9+8m)(1+m))$$\end{document} for any m≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m \ge 0$$\end{document} as a byproduct. Moreover, it turns out that the method is very effective, as it yields infinitely many 2-resolvable 3-designs. However, the question of simplicity of the constructed designs has not been yet investigated.


Introduction
A t-(v, k, λ) design is called s-resolvable if it can be partitioned into s-(v, k, δ) designs with s < t. The interesting case is s ≥ 2. Especially, the s-resolvability of the complete k-(v, k, 1) design is known in the literature as a large set of an s-(v, k, δ) design. Large sets are an essential element in proving the existence of simple t-designs for arbitrarily large t which have been intensively studied over three decades, see for instance [1, [10][11][12][13][14]. By contrast, very little is known about s-resolvability of non-trivial t-designs, when s > 1, see [4,15,[17][18][19]. We are interested in non-trivial t-designs having s-resolutions. By focussing on s = 2 we introduce a method of constructing 2-resolvable t-designs, for t = 3, 4. In essence, the method is based on the assumption that there exists a t-design which can be partitioned Communicated by L. Teirlinck. B Tran van Trung trung@iem.uni-due.de into Steiner 2-designs, and for t = 4 it is further required that the Steiner 2-designs must be projective planes. Some examples among others satisfying the assumption can be found in large sets of 2-(v, 3, 1) Steiner triple systems for v ≡ 1, 3 mod 6, v = 7, in partition of certain infinite classes of 3-(v, 4, 1) Steiner quadruple systems into 2-(v, 4, 1) designs, for v = 2 2m , m ≥ 2, [4], and v = 2 p n + 2 , p ∈ {7, 31, 127} [15], or in large sets of the projective planes of order 3, i.e. a symmetric 2-(13, 4, 1) design, [6,8]. It appears that the method is very effective, actually, when starting with examples above, it will provide a huge number of 2-resolvable 3-designs for a large variety of block sizes. Moreover, with suitable parameters for t = 4, we can also construct 4-(2k + 1, k, ) designs having 2-resolutions and therefore they can be extended to 3-resolvable 5-(2k + 2, k + 1, ) designs. For instance, the case corresponding to the projective plane of order 3 yields a 3-resolvable 5- (14,7,90) design, which in turn leads to the existence of an infinite series of 3-resolvable 5-designs having N = 5 resolution classes with parameters 5-(14 + 8m, 7, 10(9 + 8m)(1 + m)) for any m ≥ 0 as a byproduct.
We recall 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 of B. 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. It can be shown by simple counting that a t-(v, k, λ) design is an s-(v, k, λ s ) design for 0 ≤ s ≤ t, where λ s = λ v−s t−s / k−s t−s . Since λ s is an integer, necessary conditions for the parameters of a t-design are k−s t−s |λ v−s t−s for 0 ≤ s ≤ t. 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, to which we refer as a trivial t-design. Again a t-(v, k, λ) design (X , B) is said to be s-resolvable, for 0 < s < t, if its block set B can be partitioned into N ≥ 2 classes A 1 , . . . , A N such that each (X , A i ) is an s-(v, k, δ) design for i = 1, . . . , N . Each A i is called an s-resolution class or simply a resolution class and the set of N classes is called an s-resolution of (X , B). If the complete k-(v, k, 1) design is t-resolvable, i.e. it can be partitioned into N disjoint t-(v, k, λ) designs, where k > t, then we say that there exists a large set of size N of t-designs denoted by L S[N ](t, k, v) or by L S λ (t, k, v) to emphasize the value λ.
For more information about s-resolvable t-designs with 1 < s < t, see for instance [16][17][18][19]. It should be remarked that s-resolvable t-designs have been used in the construction of t-designs [16].

Description of the method
The details of the method are described in this section. Here, two elements are required.
resolution classes. Let B 1 , . . . , B N denote the resolution classes of (X , B), so each (X , B i ) is a 2-(v, k, 1) design. We call (X , B) the outer design.
Consider a fixed resolution class (X , That is, block D C i,x is formed by the union of blocks in Y i,x indexed by C, and D i,x is the set of μ 0 such blocks D C i,x . Further, define Similarly, define If (X , D) or (X , D * ) forms a t-design, we call it the constructed design. For t = 3, we show that (X , D) and (X , D * ) are 3-designs. For t = 4, if each resolution class of the outer design is a symmetric 2-(v, k, 1) design, i.e. a projective plane of order (k−1) with v = q 2 +q+1, k = q+1, we prove that (X , D) and (X , D * ) will form 4-designs. Further, it is shown that (X , D i ) and (X , D * i ) are 2-designs. Obviously, the construction method makes clear that the constructed designs (X , D) and (X , D * ) are 2-resolvable, as they are the union of designs (X , D i ) and (X , D * i ), respectively. In case t = 4 and for suitable parameters of the outer design, the constructed design can be extended to a 3-resolvable 5-design, as shown in the subsequent section. A further investigation shows that if the inner design is also 2-resolvable with L resolution classes, then the constructed design is 2-resolvable with N L resolution classes. A major advantage of the method is the fact that it enables us to construct 2-resolvable t-designs with a large variety of block sizes, because there is no restriction on the parameters of the inner designs.

(X, D) and (X, D i ) designs
We use the notation as described in the construction method. In the first step we show that (X , D i ) is a 2-(v, (k −1)+1, δ) design, and in the next step (X , D) is a 3-(v, (k −1)+1, ) design. Step As usual μ 1 (resp. μ 2 ) denote the number of blocks of (Y , C) containing a point (resp. two points). For a given point Let B be the unique block of B i containing {a, b}. We distinguish two types of points of X , namely points x ∈ B and points Step 2 (X , D) is a 3-(v, (k − 1) + 1, ) design.
Let T = {a, b, c} ⊆ X . Note that among the N resolution classes B 1 , . . . , B N of (X , B) there are λ classes, say, B 1 , . . . , B λ having the property that each has a unique block containing T .
(i) We first focus on blocks D containing T constructed from classes B 1 , . . . , B λ . Consider Each point of type (I) gives μ 2 blocks D ⊇ T . Hence points of type (I) contribute 3(k − 1)μ 2 blocks D ⊇ T . Each point of type (II) gives μ blocks D ⊇ T . Hence points of type (II) contribute Hence, Cases (i) and (ii) together show that To compute the values of and δ in terms of v, k, λ, , μ we have to separate two cases: = 2 and = 3. = 2 : In this case the inner design is the 2-( v−1 k−1 , 2, 1) design, which is considered as a degenerated 3-design with μ = 0, μ 2 = 1 and . Replacing μ 2 and μ 1 by their values in the formulas for δ and and so simplifying we obtain

The case with 2-resolvable inner designs
We further study the resolvability of the constructed designs when the inner designs are 2-resolvable. Suppose that the inner 3-

(X, D * ) and (X, D * i ) designs
To show that (X , D * ) and (X , D * i ) are designs, a very similar proof as above is to be employed, therefore it will be omitted. The results show that (X , Putting the explicit values of μ 1 , μ 2 , both δ * and * are expressed in terms of v, k, λ, , μ as shown in the next theorem. The resolvability of (X , D * ) and (X , D * i ) is the same as that of (X , D) and (X , D i ). We summarize the results in the following theorem.

2-Resolvable 4-designs
This section deals with the case, where the designs in the resolution of the outer design are symmetric 2-(v, k, 1) designs, i.e. each resolution class is a projective plane of parameters 2-(q 2 + q + 1, q + 1, 1). Obviously, (X , D i ) and (X , D * i ) are 2-designs, as shown in the previous section. We prove that (X , D) and (X , D * ) are 4-designs. (k − 1) + 1, 3) design (X, D)

4-(v,
Again use the notation as described in the construction method. We omit the proof that (X , D i ) is a 2-(v, (k − 1) + 1, δ) design, as it is the same as that in the previous section. Here, we focus on the proof in the main step that (X , D) is a 4-(v, (k − 1) + 1, ) design. Main step (X , D) is a 4-(v, (k − 1) + 1, ) design.
To simplify the writing we temporarily keep the parameters 2-(v, k, 1) for the symmetric design of the resolution, and will replace them with 2-(q 2 + q + 1, q + 1, 1) at the end of the proof. Let In summary, cases (i), (ii), (iii) together yield Putting v = q 2 + q + 1, k = q + 1, N = λ (q 2 +q−1)(q 2 +q−2) , i = 1, 2, 3, we find that The 4-design (X , D) is 2-resolvable with N resolutions classes, because it is the union of 2-designs (X , D i )s. Further, if the inner design (Y , C) is also 2-resolvable with L resolution classes, then the same argument as above shows that (X , D) is 2-resolvable with N L resolution classes.

4-(v, (k − 1), 3 * ) design (X, D * )
Again, this case may be handled in a similar manner as that of (X , D), and therefore we will omit the proof, despite the fact that several tiresome calculations for * have to be carefully carried out.
We record the results for both cases in the following theorem.
We state the result in the following theorem.
Further, if the 4-(q + 1, q+1 2 , μ) design is 2-resolvable with L resolution classes, then the 5-(q 2 + q + 2, q(q+1) 2 + 1, * ) design is 3-resolvable with N L resolution classes and each class is a 3-(q 2 + q + 2, q(q+1) 2 + 1, δ * L ) design. In 1978, Magliveras conjectured that there will exist a large set of projective planes of order q for q ≥ 3, provided q is the order of a projective plane. This conjecture is still an unsettled problem, except for q = 3, [8]. The main assumption of Theorems 4.1 and 5.2 is the existence of a 2-resolvable 4-(q 2 + q + 1, q + 1, λ) design as the outer design, whose resolution classes are projective planes of order q. In particular, if we take the complete 4-(q 2 + q + 1, q + 1, q 2 +q−3 q−3 ) design as the outer design, then the assumption is equivalent to the existence of a large set of projective planes of order q. To further clarify Theorems 4.1 and 5.2 we focus on this special case.

Conclusion
The paper presents a method for constructing 2-resolvable t-designs for t = 3, 4 based on the assumption that there exists a partition of a t-design into Steiner 2-designs. The case t = 4 corresponds to partitioning a 4-design into projective planes. Especially, if the order of the projective planes is odd, it also enables to construct 3-resolvable 5-designs with a largest possible block size. In general, the method appears to be very effective, as it yields infinitely many 2-resolvable 3-designs with a large variety of blocks sizes. A study of simplicity of the constructed designs remains a challenging problem.
Funding Open Access funding enabled and organized by Projekt DEAL.
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