Further results on the existence of super-simple pairwise balanced designs with block sizes 3 and 4

In statistical planning of experiments, super-simple designs are the ones providing samples with maximum intersection as small as possible. Super-simple pairwise balanced designs are useful in constructing other types of super-simple designs which can be applied to codes and designs. In this paper, the super-simple pairwise balanced designs with block sizes 3 and 4 are investigated and it is proved that the necessary conditions for the existence of a super-simple \begin{document}$(v, \{3,4\}, λ)$\end{document} -PBD for \begin{document}$λ = 7,9$\end{document} and \begin{document}$λ = 2k$\end{document} , \begin{document}$k≥1$\end{document} , are sufficient with seven possible exceptions. In the end, several optical orthogonal codes and superimposed codes are given.


Introduction
A pairwise balanced design (or PBD) is a pair (X , A) such that X is a set of elements called points, and A is a set of subsets (called blocks) of X , each of cardinality at least two, such that every pair of points occurs in exactly λ blocks of A. If v is a positive integer and K is a set of positive integers, each of which is greater than one, then we say that (X , A) is a (v, K, λ)-PBD if |X | = v, and |A| ∈ K for every A ∈ A. We denote B(K, λ) = {v : there exists a (v, K, λ)-PBD}. A set K is said to be PBD-closed if B(K, λ) = K.
A PBD is resolvable if its blocks can be partitioned into parallel classes; a parallel class is a set of point-disjoint blocks whose union is the set of all points. The notation (v, K, λ)-RPBD is used for a resolvable PBD. When K = {k}, a (v, K, λ)-PBD is a balanced incomplete block design, the notations (v, k, λ)-BIBD and (v, k, λ)-RBIBD are sometimes used in this case.
A design is said to be simple if it contains no repeated blocks. A design is said to be super-simple if the intersection of any two blocks has at most two elements. When k = 3, a super-simple design is just a simple design. When λ = 1, the designs are necessarily super-simple. A super-simple (v, K 1 , λ)-PBD is also a super-simple (v, K 2 , λ)-PBD if K 1 ⊆ K 2 .
In this paper, the existence of a super-simple (v, {3, 4}, λ)-PBD for λ = 7, 9 and λ = 2k, k ≥ 4, is investigated. The necessary conditions for the existence of such a super-simple design are v ≥ λ + 2 and λv(v − 1) ≡ 0 (mod 3). We shall use direct and recursive constructions to show that the necessary conditions are also sufficient with some possible exceptions. Specifically, we shall prove the following theorem.
The paper is organized as follows. Some recursive constructions are provided in Section 2. Some ingredient super-simple designs are given directly by computer search in Section 3. The proof of our main theorem is given in Section 4. Some applications in optical orthogonal codes and superimposed codes are mentioned in Section 5.

Recursive constructions
In this section, the auxiliary design (group divisible design) is introduced and some known results stated for later use, and we also give some standard recursive constructions.
A group divisible design (or GDD) is a triple (X , G, B) which satisfies the following properties: (i) G is a partition of a set X (of points) into subsets called groups.
(ii) B is a set of subsets of X (called blocks) such that a group and a block contain at most one common point. (iii) Every pair of points from distinct groups occurs in exactly λ blocks.
The group type (or type) of GDD is the multiset {|G| : G ∈ G}. We usually use an "exponential" notation to describe types: so type g u1 1 g u2 2 · · · g u k k denotes u i occurrences of g i , 1 ≤ i ≤ k, in the multiset. A GDD with block sizes from a set of positive integers K is called a (K, λ)-GDD. When λ = 1, we simply write K-GDD. When K = {k}, we simply write k for K. Taking the groups of a GDD as blocks yields a PBD, and taking a parallel class of blocks of a PBD as groups also yields a GDD.
A (k, λ)-GDD of group type v k is called a transversal design and denoted by TD λ (k, v) for short. The known result on super-simple TD λ (4, v) is listed in the following which is used in Section 4.
The following results are obvious but very useful. Their proofs are omitted here.

Direct constructions
In this section, direct constructions are used and some super-simple (v, {3, 4}, λ)-PBDs for small values of v are obtained, which will be used as master designs or input designs in the recursive constructions. All these designs are obtained by computer.
Usually, it is difficult to find all the blocks of a design directly. So, a technique of "+d (mod v)" is used, which means that we try to find a subset S ⊆ B and an element d ∈ Z v such that {B + kd : B ∈ S, k ∈ Z} = B. The blocks of S are called base blocks. The "+d" is omitted when d = 1 and then the design is cyclic.
Sometimes S is divided into two parts: P and R, and we try to find an element m ∈ Z v and an integer s such that there is a subset P 1 ⊆ P satisfying Here m is a partial multiplier of order s of the design. In this article, m is taken to be some unit of the ring Z v , i.e., m satisfies that gcd(m, v) = 1.
Further, the founded base blocks of S are shuffled when the program takes too much time to find a design. Most of these ideas come from the previous papers such as [16,18,20]. Proof. For v ∈ {22, 34, 46}, we take the point set X = Z v , the base blocks are listed below and all the required blocks can be generated from them by +2 (mod v).  Proof. For v ∈ {37, 49, 73}, we take the point set as X = Z v . With a computer program we found the required base blocks, which are divided into two parts, P and R, where P consists of some base blocks with a partial multiplier m of order s (i.e., each base block of P has to be multiplied by m i for 0 ≤ i ≤ s − 1), and R is the set of the remaining base blocks. We list P, m, s and R below. The desired super-simple design is generated by developing the base blocks (mod v). For v = 40, we take the point set as X = Z 40 . The base blocks are also divided into two parts, P and R, which are listed below and all the required blocks can be generated from them by +2 (mod v). Proof. For each v ∈ M , we take the point set X = Z v . With a computer program we found the required base blocks, which are divided into two parts, P and R, where P consists of some base blocks with a partial multiplier m of order s (i.e., each base block of P has to be multiplied by m i for 0 ≤ i ≤ s−1), and R is the set of the remaining base blocks. We list P, m, s and R below. The desired super-simple design is generated by developing the base blocks +2 (mod v).  To prove Theorem 1.5, we shall divide it into two cases by the remaining value of λ = 7, 9.       When λ = 9, the necessary conditions for a super-simple (v, {3, 4}, 9)-PBD become v ≥ 11. We shall prove that such a necessary condition is also sufficient except possibly for v ∈ {12, 16}.

Concluding remarks
Super-simple cyclic designs with small values are believed to be useful not only in constructing new larger super-simple cyclic designs, but also in constructing optical orthogonal codes with index two and superimposed codes.
As defined in Chung [22]. A (v, k, ρ) optical orthogonal code (OOC), C, is a family of (0, 1) sequences (called codewords) of length v and weight k which satisfy the following two properties (all subscripts are reduced modulo v).
(1) (The Autocorrelation Property) 0≤t≤v x t x t+i ≤ ρ for any x = (x 0 , x 1 , . . . , x v−1 ) ∈ C and any integer i ≡ 0( (mod v)); (2) (The Cross-Correlation Property) 0≤t≤v x t y t+i ≤ ρ for any x = (x 0 , x 1 , . . . , x v−1 ) ∈ C and y = (y 0 , y 1 , . . . , y v−1 ) ∈ C with x = y, and any integer i. The parameter ρ is the index of the OOC. It is well known that the number of codewords of a (v, k, ρ)-OOC can not exceed 1 [30]). The OOC is said to be optimal when its size reaches this bound.
Suppose that there exists a super-simple (v, k, λ)-CBIBD. We construct a (0, 1)sequence of length v from each of the base blocks of the super-simple CBIBD such that the i-th position is 1 if and only if i is an element of the base block. According to the definitions of a super-simple CBIBD and an OOC, it is easy to see that the derived (0, 1)-sequences constitute a (v, 4, 2)-OOC with λ(v−1) k(k−1) codewords. So we have the following by Lemma 3.2 and Lemma 4.3. The main problem in the study of superimposed codes is to find the minimal length N (T ; w, r) of a (w, r) superimposed code for a given cardinality T . The following result can be found in [32].  [37] Y. Zhang, K. Chen