The Construction of Pairwise Additive Minimal BIB Designs with Asymptotic Results

An asymptotic existence of balanced incomplete block (BIB) designs and pairwise balanced designs (PBD) has been discussed in [1]-[3]. On the other hand, the existence of additive BIB designs and pairwise additive BIB designs with 2 k = and 1 λ = has been discussed with direct and recursive constructions in [4]-[8]. In this paper, an asymptotic existence of pairwise additive BIB designs is proved by use of Wilson’s theorem on PBD, and also for some  and k the exact existence of  pairwise additive BIB designs with block size k and 1 λ = is discussed.


Introduction
A pairwise balanced design (PBD) of order v with block sizes in a set K is a system ( ) , V  , where V is a finite set (the point set) of cardinality v and  is a family of subsets (blocks) of V such that 1) if B ∈  , then B K ∈ and 2) every pair of distinct elements of V occurs in λ blocks of  [9].This is denoted by , this is especially called additive BIB designs [6] [7].
It is clear by the definition that the existence of  pairwise additive ( ) Hence, for given parameters , , v k λ , the larger  is, the more difficult a construction problem of  pairwise additive BIB designs is.
In pairwise additive ( ) v k λ , since a sum of any two incidence matrices yields a BIB design, it is seen [7]  that 2) It follows from (1.2) that the existence of  pairwise additive ( ) ) It is clear by the definition that v k ≥  .The purpose of this paper is to show that, for a given odd prime power k and a given positive integer , the necessary conditions (1.1) for the existence of  pairwise additive minimal In particular, for the existence of  pairwise additive minimal ( )

PBD v, K, λ
The existence of ( ) PBD , , v K λ is reviewed along with necessary and asymptotically sufficient conditions.
Let K be a set of positive integers and Necessary conditions for the existence of a ( ) PBD , , v K λ are known as follows.Lemma 2.1 [2] Necessary conditions for the existence of a ( ) Wilson [3] proved the asymptotic existence as Theorem 2.

PBD v, K, λ
In this section, a method of constructing pairwise additive BIB designs through The following simple method is useful to construct pairwise additive BIB designs.Lemma 3. 1 The existence of a ( ) all block size k′ be formed by the  pairwise additive ( ) for each i .Then it follows that the ( ) As the next case of block sizes, 3 ,3 m + has been discussed as a compatibly nested minimal partition in [12], which shows the existence of pairwise additive ( )

Some Class of Pairwise Additive
( ) In this section, a necessary condition for the existence of pairwise additive being minimal is provided and then some classes of ( ) are constructed.Now (1.1) implies that necessary conditions for the existence of pairwise additive Furthermore, the following is given.Theorem 4.1 When k is an odd prime power, necessary conditions for the existence of pairwise additive Proof.Since ( ) ( ) When k is an odd prime power, a class of pairwise additive is obtained as follows.This observation shows a generalization of Lemma 3.  ( ) It can be shown that a development of the following initial blocks on ( ) of the required BIB design: where α is a primitive element of ( ) Furthermore, the following is known to be provided by recursive constructions with affine resolvable BIB designs.This result will be used in the next section.
Theorem 4.3 [7] Let k be an odd prime power.Then there are additive  Lemma 5.2 [13] For any positive even integer m , there are primes p and q for which ( ) In the proof of Lemma 5.2 (i.e., Lemma 3.4 in [13]), primes p and q are obtained by using Theorem 5.1.Thus Lemma 5.2 implies the existence of sufficiently large primes p and q as follows.
Lemma 5. 3 For a given odd prime power k , there are primes p and q such that (a) Proof.Let k be an odd prime power.Then, for an even integer 2k , Lemma 5.2 provides primes p and q such that (a) Hence it is seen that ( ) which imply (c) and (d).
, since an additive Next, for a given odd prime power k and a given positive integer  , even if is discussed for sufficiently large ( ) 5 For a given odd prime power k and a given positive integer  , there are primes p and q such that (a) p q k > >  , (b) ( ) Proof.Let k be an odd prime power and  be a positive integer.Then, for a positive integer 2k , Lemma 5.2 provides primes p and q such that (a) p > q > k , (b) Hence it is seen that ( ) gcd 1, 1 2 p q k − − = and (c) holds.


Thus the following result is obtained through conditions (a), (b) and (c) as in Lemma 5.5.

Theorem 5.6
For a given odd prime power k and a given positive integer  , there are  pairwise addi- tive for sufficiently large ( ) Let p and q be primes as in Lemma 5. 5. Then it follows from (c) that there are { } ( ) , BD , 1 P , v p q for sufficiently large ( ) , on account of Theorem 2.2.Also Theorem 4.2 along with conditions (a) and (b) shows that there are ( ) ( ) p k k − and ( ) ( ) q k k − .Thus the required designs are obtained on account of Lemma 3.1.

Pairwise Additive ( ) B 3,1 v,
In this section, the existence of pairwise additive ( ) At first it is shown that there are  pairwise additive ( ) . Furthermore, the exact existence of 2 pairwise ( ) is discussed by providing direct and recursive constructions of pairwise additive ( ) . Finally, it is shown that there are 2 pairwise additive ( ) Z with the index being fixed yields incidence matrices 1 2 , N N of the required BIB design: , ( ) ( ) and additive ( ) Next, some recursive constructions of pairwise additive ( )  ( ) , , V   such that 1) V is a set of kn elements, 2)  is a partition of V into k classes (groups), each of size n , 3)  is a family of k-subsets (blocks) of V , 4) every unordered pair of elements from the same group is not contained in any block, and 5) every unordered pair of elements from other groups is contained in exactly λ blocks. .A method of construction is presented, similarly to a recursive construction given in [4], by use of ( ) TD 6, n .Theorem 6. 4 The existence of  pairwise additive ( ) ,t ∞ = ∞ : v k λ is especially called a balanced incomplete block (BIB) de- sign, where b =  , each block contains k different points and each point appears in r different blocks [10].This is denoted by ( ) , IBD , , B , v b r k λ or ( ) B , , v k λ .It is well known that necessary conditions for the exis- tence of a ( ) B , , v k λ are the i-th point occurs in the j-th block or otherwise.Hence the incidence matrix N satisfies the conditions: 1) s need not be an integer unlike other parameters.Further let 2 k λ is called  pairwise additive BIB designs if  corresponding incidence matrices 1

3 . 4 . 2
Theorem Let both 2 1 km + and k be odd prime powers for a positive integer m .Then there are( )

Theorem 4 . 4 [ 8 ]
B k k k − .Especially, when 3 k = , the further result is known.There are additive B ( ) In this section, when k is an odd prime power, an asymptotic existence of pairwise additive it is shown that the necessary conditions (4.1) for the existence of  pairwise additive theorem on primes is useful for the discussion.Theorem 5.1 (Dirichlet) If ( ) gcd , 1 a b = , then a set of integers of the following form Now Theorem 5.1 yields the following.

 3 . 5 . 4
Thus one of the main results of this paper is now obtained through conditions (a), (b), (c) and (d) given in Lemma 5.Theorem For a given odd prime power k , (4.1) is a necessary and asymptotically sufficient condition for the existence of k pairwise additive B − .Proof (sufficiency).Let p and q be primes as in Lemma 5.3 with v satisfying (4.1), on account of Theorem 2.2.Conditions (a) and (b) show that there are ( ) ( )

2 B
k k k − , on account of Theorems 4.2 and 4.3.Hence the required designs can be obtained on account of Lemma 3.1.Unfortunately, by use of Theorem 5.4 we cannot show the existence of  pairwise additive

Theorem 6 . 2 B 3 , 3 , 1 n
For a given positive integer  , even if3 > , there are  pairwise additive ( ) Let n be a positive integer satisfying 3 3 n ≥  .Then ( 1 3 n− pairwise) additive ( ) are constructed by Theorem 4.4, and there are primes p and q such that 3 account of Lemma 5.2.Furthermore, since ( ) , denote an element which occurs in the m-th block of a ( )TD 3 ,v and the n-th group.Then it can be shown that the following  incidence matrices yield the re- quired  pairwise additive BIB designs with vv′ elements denoted by ( ) , s t for 1 s v ≤ ≤ and 1 t v′ ≤ ≤ : , denote an element which occurs in both the m-th block of a ( ) TD 3 ,v  and the n-th group.Then the following  incidence ma- trices can yield the required  pairwise additive BIB designs with 1 vv′ + elements denoted by ( ) , s t for 1 s v ≤ ≤ and 1 t v′ ≤ ≤ , and , 15 is the smallest value of v for which the existence of 2 pairwise additive ( ) Hence at first this case is individually considered here.It can be shown that a development of the following initial blocks on 7 The existence of  pairwise additive ( )Let h ′  , 1 h≤ ≤  , be a block set similarly to the proof of Theorem 6.4 and let h  , 1 h ≤ ≤  , be a block set of  pairwise additive ( ) Another recursive method is presented.Theorem 6.5 ∞ .Also let ( ), d m n , ,