Construction of 3-Designs Using (1,\sigma)-Resolution

The paper deals with recursive constructions for simple 3-designs based on other 3-designs having $(1, \sigma)$-resolution. The concept of $(1, \sigma)$-resolution may be viewed as a generalization of the parallelism for designs. We show the constructions and their applications to produce many previously unknown infinite families of simple 3-designs. We also include a discussion of $(1,\sigma)$-resolvability of the constructed designs.


Introduction
In our previous papers [16,17] we have presented several recursive constructions for simple 3-designs. In [16], among others, generalizations of the well-known doubling construction of Steiner quadruple systems for 3-designs are introduced. In [17] more general recursive constructions of simple 3-designs are described, whereby ingredient designs may have repeated blocks. The methods in these papers are based on the existence of 3-designs having a parallelism, i.e. the blocks of the design can be partitioned into classes of mutually disjoint blocks such that every point is in exactly one block of each class. Designs with parallelism have shown to be useful for constructing designs in the literature [13], [7], [10], [12], [9], [11], [15], [16,17].
The concept of (1, σ)-resolvability for t − (v, k, λ) designs may be viewed as a generalization of that of parallelism. For the latter means that the design is (1, 1)resolvable. It should be mentioned that if a t − (v, k, λ) design has a parallelism we necessarily have k|v; this condition does no longer hold for (1, σ)-resolvability in general. Thus, the natural question is that whether or not the methods in our previous papers [16,17] can be extended to (1, σ)-resolvable 3-designs. We show that this is in fact the case. Our aim in this paper is to present this generalization. The result provides a general method for constructing simple 3-designs which largely extends the use of complete designs as ingredients for the construction. We show the strength of the method by giving some simple applications to construct a number of families of simple 3-designs, which, to our knowledge, were not previously known to exist. We also include a discussion of (1, σ)-resolvability of the constructed designs.
For notation and general definitions of t-designs we refer to [3,8].

Constructions of 3-Designs using (1, σ)-Resolution
In this section we present recursive constructions of simple 3-designs using (1, σ)resolution of their ingredients.

Preliminaries
We begin with a few definitions and set up necessary conditions for the ingredients used in the constructions.
is said to be (s, σ)-resolvable for a given s ∈ {1, . . . , t}, if its block set B can be partitioned into w classes π 1 , . . . , π w such that (X, π i ) is a s −(v, k, σ) design for all i = 1, . . . , w. Each π i is called a resolution class.
It should be remarked that each t − (v, k, λ) design always has a trivial (s, λ s )resolution consisting of a single class, i.e. w = 1, for all 1 ≤ s ≤ t. Throughout the paper when we speak of (s, σ)-resolution we mean that w ≥ 2. Note that w = Definition 2.2 Let D be a t − (v, k, λ) design admitting a (s, σ)-resolution with π 1 , . . . , π w as resolution classes. Define a distance between any two classes π i and π j by d(π i , π j ) = min{|i − j|, w − |i − j|}.
For the constructions in this paper we employ designs having a (1, σ)-resolution. We now describe the detailed assumption and notation used throughout the paper.
Assume that there exist 3 − (v, k i , λ (i) ) designs D i = (X, B i ) having a (1, σ (i) )resolution such that w i = w n+i for all i = 1, . . . , n, where w j denotes the number of classes in a (1, σ (j) )-resolution of D j , i.e. D i and D n+i have the same number of resolution classes.
It is also assumed that 1. For each pair (D i , D n+i ), 1 ≤ i ≤ n, either D i or D n+i has to be simple.
2. If a D j , j ∈ {i, n + i}, is not simple, then D j is a union of a j copies of a simple 3 − (v, k j , α (j) ) design C j , wherein C j admits a (1, σ (j) )-resolution. Thus, Note that the trivial 2 − (v, 2, 1) design will be considered as a 3 − (v, 2, λ) design with λ = 0.
Further we need to specify the way of setting up (1, σ (j) )-resolution classes for D j , when D j is the union of a j copies C j . Let t j } be a (1, σ (j) )-resolution of the simple design C j . The corresponding (1, σ (j) )-resolution of D j is chosen to be the "concatenation" of a j sets P (j) . This means that the w j = a j t j resolution classes of D j are arranged in the following way π t j . Finally, we also assume that there exists a 3 − (v, k, Λ) design D = (X, B), when it is needed in our construction.
• The distance defined on the resolution classes of D ℓ is then • b (j) = σ (j) v/k denotes the number of blocks in each class of a (1, σ (j) )-resolution of D j .
• u j := σ (j) denotes the number of blocks containing a point in each class of a (1, σ (j) )-resolution of D j .
• λ denotes the number of blocks of D j containing two points.

Construction I
In this section we describe the first construction by using the set-up above for the case k n = k/2.
LetD i = (X,B i ) be a copy of D i defined on the point setX such that X ∩X = ∅. Also letD = (X,B) be a copy of D.
Define blocks on the point set X ∪X as follows: I. blocks of D and blocks ofD; II. blocks of the form A∪B for any A ∈ π III. blocks of the formÃ∪B for anyÃ ∈π Here, and in the sequel, the non-negative integers s h , h = 1, . . . , n, denote the parameters that have to be determined, for which the defined blocks of types I, II and III form a 3-design. Thus, s h , should not be confused with s in (s, σ)-resolution as defined above.
Any 3 points a, b, c ∈ X, resp.ã,b,c ∈X are contained in • Λ blocks of type I, • z h λ (n+h) b (h) blocks of type III for h = 1, . . . , n.
Thus a, b, c appear together in blocks of D n+h ; further, the pointc is in u h (resp. u n+h ) blocks of each resolution class ofD h (resp.D n+h ).
So a, b,c appear in 2 u n+h blocks of type II for h = 1, . . . , n, • z h λ (n+h) 2 u h blocks of type III for h = 1, . . . , n.
Thus a, b,c are contained together in Therefore the blocks defined in I, II and III will form a 3-design if Note that Λ = Θ − ∆ ≥ 0. The case Λ = Θ − ∆ = 0 implies that D andD are not needed in the construction. In both cases either Θ − ∆ > 0 or Θ − ∆ = 0 the constructed blocks form a simple 3 − (2v, k, Θ) design with What remains to be verified is the simplicity of the resulting design when either D h or D n+h is non-simple. Evidently, if both D h and D n+h are simple for all 1 ≤ h ≤ n, then the constructed design is simple.
To start with we observe that two blocks constructed from two pairs (D i , D n+i ) and (D j , D n+j ), i = j, are always distinct. Further any two blocks of different types are also distinct. Thus, we need to consider two blocks of the same type, in particular, of type II or type III constructed from a pair (D j , D n+j ). W.l.o.g. we may assume that D j is a union of a j copies of a simple 3 − (v, k j , α (j) ) design C j and D n+j is simple.
The following argument is the same for blocks of types II and III. So let E = A 1 ∪B 1 and F = A 2 ∪B 2 be two blocks of type II of the resulting design, where In the first case, E and F are the same block. In the second case, E and F are repeated blocks; this can happen only if |i 2 − i 1 | is a multiple of t j , i.e. t j | |i 2 − i 1 |, this is because the resolution classes of D j are chosen to be the concatenation of a j copies of a given set P (j) of resolution classes of C j . Now, as h 1 ) ≤ s j , it follows that z j > t j . Therefore, the second case will not occur if z j ≤ t j .
Hence, if z j ≤ t j for all non-simple D j 's, the resulting design remains simple.
With the notation above, we summarize Construction I in the following theorem.
Theorem 2.1 Let {k 1 , . . . , k n , k n+1 , . . . , k 2n } and k be integers with 2 ≤ k 1 < · · · < k n < k/2 and k i + k n+i = k for i = 1, . . . , n. Assume that there exist where w j is the number of resolution classes of D j . Assume further that at least one design from each pair . Let t j denote the number of resolution classes of C j . Let (ii) Assume that Then there exists a simple 3 − (2v, k, Θ) design D.

Construction II
In this section we consider the case k n = k/2. We observe that the resulting designs in Construction I would have repeated blocks if k n = k/2 and the block sets of D n and D 2n are not disjoint. To deal with the case k n = k/2 the blocks constructed from the pair (D n , D 2n ) need to be modified.
Suppose now 2 ≤ k 1 < · · · < k n = k/2. Take D n = D 2n and assume that D n is simple. Now define the blocks on the point set X ∪X as follows: I. blocks of D and blocks ofD; III. blocks of the formÃ∪B for anyÃ ∈π Construction II differs from Construction I only in blocks of type IV. Observe that any three points a, b, c ∈ X (resp.ã,b,c ∈X) are contained in z n λ (n) b (n) blocks of type IV; any three points a, b,c with a, b ∈ X andc ∈X (resp.ã,b, c) are contained in z n λ (n) 2 u n blocks of type IV. All other countings as well as the proof of simplicity of the resulting design remain unchanged as shown in Construction I.
We obtain the following theorem for the case k n = k/2. Theorem 2.2 Let {k 1 , . . . , k n , k n+1 , . . . , k 2n } and k be integers with 2 ≤ k 1 < . . . < k n = k/2 and k i + k n+i = k for i = 1, . . . , n. Assume that there exist 3 − (v, k i , λ (i) ) designs D i = (X, B i ) admitting a (1, σ (i) )-resolution such that w i = w n+i , where w j is the number of resolution classes of D j . Assume further that at least one design from each pair (D i , D n+i ), 1 ≤ i ≤ n, is simple and if a D j , j ∈ {i, n + i}, is not simple, then D j is a union of a j copies of a simple 3 − (v, k j , α (j) ) design C j admitting a (1, σ (j) )-resolution, i.e. λ (j) = a j α (j) . Let t j denote the number of resolution classes of C j . Let with 1 ≤ z h ≤ w h if both D h and D n+h are simple and 1 ≤ z h ≤ t j if D j is non-simple, j ∈ {h, n + h}. Then there exists a simple 3 − (2v, k, Θ * ) design D.

Applications
In this section we show applications of Constructions I and II for some small values of n. It turns out that we can construct many new infinite families of simple 3-designs by merely using complete designs as ingredients. For these applications we implicitly use the following result and observation.
The resolution classes are the block orbits of a fixed point free automorphism of order v.

Applications of Construction I
3.1.1 n = 1 We consider the most simple case of Construction I, namely the case with n = 1, k 1 = 2 and k 2 = 3.

Theorem 3.1 There is a simple
design for any integer v ≡ 2 mod 6.
We can construct another family of 3-designs with moderate value for Θ. Let v = 2 f + 1 with odd f .

n = 2
We construct a family of simple 3-designs with k = 7 by using Construction I with n = 2.

Then we have
It follows that Θ − ∆ = 0 if we have Az 1 = Bz 2 , which reduces to the equation

n = 1
Here is the first example.

Now from Theorem 2.2 we have Θ
We have the following. We consider another example of general form. Let v, k be integers with v > k ≥ 3 and gcd(v, k) = 1.
We have Θ * = λ Construction II yields a simple 3−(2v, 2k, Θ * ) design, when it holds In this case we have We record the result obtained above.
is an integer. This condition is equivalent to v ≡ 1, 2 mod 3 and v ≡ 0, 1 mod 4.
Here we record this result.
We illustrate two special cases with k = 3 and k = 4 of Theorem 3.7.

(1, σ)-resolvability of the constructed designs
In this section, we discuss the question of (1, σ)-resolvability of the designs obtained by Constructions I and II. In particular, we will consider the cases Θ − ∆ = 0 and Θ * −∆ * = 0, i.e. the cases where a 3−(v, k, Λ) design D is not used in the construction.
We make use of the following observation.
• Let (D h , D n+h ) be a pair of designs in Constructions I or II such that k h = k n+h .
For given (i, j) the blocks constructed from the resolution classes (π • For the blocks of type IV in Construction II we have D n = D 2n i.e. k n = k 2n . Let B (i,j) n,n denote the set of blocks constructed from resolution classes of D n and D n corresponding to the pair (i, j). Then we have blocks of B (i,j) n,n .
Observe that the blocks constructed by each pair (D h , D n+h ) is a union of z h w h subsets B h,n+h of equal size. Now assume that m h |z h w h for all h = 1, . . . , n. This is equivalent to say that the blocks constructed by the pair (D h , D n+h ) can be partitioned into z h w h /m h disjoint 1 − (2v, k h + k n+h , σ) = 1 − (2v, k, σ) designs. It is then clear that the constructed design is (1, σ)-resolvable.
We have proven the following.