3-Group Divisible Designs with 3 Groups and Block Size 5

A 3-GDD ( n , 2, k , λ 1 , λ 2 ) was defned by combining the defnitions of a group divisible design and a t-design. In this paper, we extend the defnitions to 3 groups and block size 5, and we denote such GDD by 3-GDD ( n , 3, 5, μ 1 , μ 2 ). Some necessary conditions for the existence of these GDDs are developed, and several new constructions and specifc instances of nonexistence are given.


Introduction
A group divisible design, GDD (n, m, k, λ 1 , λ 2 ), is an ordered triple (V, G, B), where V is a mn-set of symbols, G is a partition of V into m sets called groups of size n each, and B is a collection of k-subsets called blocks of V, such that each pair of symbols from the same group occurs in exactly λ 1 blocks and each pair of symbols from diferent groups occurs in exactly λ 2 blocks [1][2][3][4][5].Any two symbols occurring together in the same group are called frst associates, and pairs of symbols occurring in diferent groups are called second associates.
Group divisible designs (GDDs) have been studied for their usefulness in statistics, coding and for their universal application to constructions of new designs (for instance balanced incomplete design, orthogonal arrays, and transversal designs etc.) [6][7][8].Te existence of such GDDs has been of interest over the years, going back to at least the work of Bose and Shimamoto in 1952 that began classifying such designs [9].GDDs and t-designs have been studied by many authors [10][11][12][13] and the references therein.Recently, a 3-GDD (n, 2, k, λ 1 , λ 2 ) was defned by extending the defnitions of a group divisible designs and a t-design, and some necessary conditions for its existence were given [14,15].
In this paper, these recent works are extended to include more than two groups.We mainly continue to focus on the defnition of 3-GDDs, and we explicitly consider the case when the required designs have three groups of size n each and block size 5. Troughout this paper, such GDD is denoted by 3-GDD (n, 3, 5, μ 1 , μ 2 ).In this work, some necessary conditions for the existence of such GDDs are determined, the existence of some GDDs will be proved and their constructions are produced.Furthermore, several specifc instances of nonexistence are proved.
Tis work is organized as follows.In Section 2, we present some well-known defnitions and examples that will be used to proof the main results.In Section 3, some necessary conditions for the existence of such designs together with their proofs are given.In Section 4, the proofs of some constructions especially when μ 2 � 0 are presented.Finally, in Section 5, an infnite families of existences for the 3-GDD when n � 3 are given.

Preliminaries
In this section, we present some well-known defnitions and concepts which will be used in the subsequent sections.
Defnition 1 (see [14]).A t-(v, k, λ) design, or a t-design is a pair (X, B), where X is a v-set of points and B is a collection of k-subsets (blocks) of X with the property that every t-subset of X is contained in exactly λ blocks.Te parameter λ is called the index of the design.
A necessary condition for the existence of a t-(v, k, λ) design is known [13] to be the following equation: Te problem of determining the values of k, t, and X for which this condition is also sufcient is not yet solved completely, and less is known about confgurations with t � 3. Te concepts of a GDD and a 3-design can be merged to defne a 3-GDD as follows.
is a set X of mn symbols partitioned into m parts of size n called groups together with a collection of k-subsets of X called blocks, such that (i) Every 3-subset of each group occurs in Λ 1 blocks (ii) Every mixed 3-subset, meaning either two symbols are from one group and one symbol from the other group or all three symbols are from diferent groups, occurs in Λ 2 blocks Te above mentioned defnition was given in [14,15], in which case a 3-subset of X has only two choices, all symbols from one of the two groups or one symbol from a group and two symbols from the other group.Such GDD was denoted by a 3-GDD (n, 2, 4, λ 1 , λ 2 ).
One may relax condition (ii) in Defnition 2 to defne a 3-PBIBD as follows.
Defnition 4. A 3-PBIBD (partially balanced incomplete block design), 3-PBIBD (n, m, k, θ 1 , θ 2 , θ 3 ) is a collection of k-subsets of an mn set X, where X is partitioned in to m groups of order n such that (i) Every triple formed from symbols of only a single group occurs in θ 1 blocks (ii) Every triple formed from symbols of only two groups occurs in θ 2 blocks (iii) Every triple formed from symbols of all three groups occurs in θ 3 blocks where X is a set of mn elements partitioned into mn-subsets (groups) and B is a collection of k-subsets (blocks) of X such that (i) Every triple occurs in exactly μ 1 blocks if it contains elements from at most 2 groups (ii) It occurs in exactly μ 2 blocks if it has all three elements from diferent groups Defnition 5 is the subject matter of this paper and we explicitly consider the case in which m � 3 and k � 5.Such GDD is denoted by 3-GDD (n, 3, 5, μ 1 , μ 2 ).
Te next three sections of this paper discuss our research fndings.
Let λ 1 (frst associate pair) denote the number of blocks containing x 1 , x 2  , where x 1 and x 2 are from the same group, λ 2 (second associate pair) denote the number of blocks containing x, y   where x and y are from diferent groups, r and b, respectively, denote the replication number and the number of blocks in a 3-GDD. 2

Journal of Mathematics
Proof.Te abovementioned results can be proved as follows: (  Each of the λ 2 blocks containing the pair (x, y) has three triples containing (x, y).

□
As a consequence of Teorem 7, we have the following corollaries.

Some Constructions When μ 2 = 0
In this section, the proofs of some constructions when μ 2 � 0 are given.
1)We count the number of triples containing a fxed element x in the design in two ways.
the form (3, 0) or (2, 1) in the design.Similarly, there are n 2 triples of the type (1, 1, 1) containing x, and are repeated μ 2 times.n 2 μ 2 triples containing x. Second, in every block containing x, there are 6 triples containing x and x occurs in r blocks, which means there are 6r triples containing x in the design.