The 3-way intersection problem for S(2, 4, v) designs

0.21 MB·English

by  Saeedeh Rashidi

The 3-way intersection problem for S(2,4,v)
designs
Saeedeh Rashidi AND Nasrin Soltankhah
3
1
0 Department of Mathematics
2
Alzahra University

n
Vanak Square 19834 Tehran, I.R. Iran
a
J [email protected]

1 [email protected]
2

]
O
C Abstract
.
h Inthispaperthe3-wayintersectionproblemforS(2,4,v)designs
at is investigated. Let bv = v(v1−21) and I3[v] = {0,1,...,bv}\{bv −
m 7,bv−6,bv−5,bv−4,bv−3,bv−2,bv−1}. Let J3[v]={k| there
existthreeS(2,4,v)designswithk samecommonblocks}. Weshow
[
that J3[v] ⊆ I3[v] for any positive integer v ≡ 1,4 (mod 12) and

1 J3[v] = I3[v], for v ≥ 49 and v = 13. We find J3[16] completely.
v Also we determinesome values of J [v] for v=25,28,37 and 40.
3
4
6 KEYWORDS: 3-way intersection; S(2,4,v) design; GDD; trade
7
4
1 Introduction
.
1
0
3 A Stiener system S(2,4,v) is a pair (V,B) where V is a v-element set
1 and B is a family of 4-element subsets of V called blocks, such that each
: 2-element subsets of V is contained in exactly one block.
v
i TwoSteinersystemsS(2,4,v), (V,B)and (V,B1)aresaidtointersect
X
in s blocks if |B∩B | = s. The intersection problem for S(2,4,v) designs
1
r can be extended in this way: determine the sets J [v](J [v]) of allintegers
a µ µ
s such that there exists a collection of µ (≥ 2) S(2,4,v) designs mutually
intersecting in s blocks (in the same set of s blocks). This generalization
is called µ-way intersection problem. Clearly J [v] = J [v] = J[v] and
2 2
J [v]⊆J [v]⊆J[v].
µ µ
The intersection problem for µ = 2 was considered by Colbourn, Hoff-
man, and Lindner in [8]. They determined the set J[v](J [v]) completely
2
1
for all values v ≡ 1,4 (mod 12), with some possible exceptions for v =
25, 28 and 37. Let [a,b] = {a,a + 1,...,b − 1,b}, b = v(v−1), and
v 12
I[v]=[0,b ]\([b −5,b −1]∪{b −7}). It is shown in [8]; that:
v v v v
(1) J[v]⊆I[v] for all v ≡1,4 (mod 12).
(2) J[v]=I[v] for all admissible v ≥40.
(3) J[13]=I[13] and J[16]=I[16]\{7,9,10,11,14}.
(4)I[25]\{31,33,34,37,39,40,41,42,44}⊆J[25]and{42,44}6⊆J[25].
(5) I[28]\{44,46,49,50,52,53,54,57}⊆J[28].
(6) I[37]\({64,66,76,82,84,85,88}∪[90,94]∪[96,101])⊆J[37].
Also Chang,Feng, and Lo Faro investigateanother type of intersection
which is called triangle intersection (See [3]). Milici and Quattrocchi [15]
determined J [v] for STSs. Other results about the intersection problem
3
can be found in [1, 4, 5, 6, 10, 7, 13]. In this paper we investigate the
threewayintersectionproblemforS(2,4,v)designs. WesetI [v]=[0,b ]\
3 v
[b −7,b −1]. As our main result, we prove the following theorem.
v v
Theorem 1.1 (1) J [v]⊆I [v] for all v ≡1,4 (mod 12).
3 3
(2) J [v]=I [v] for all admissible v ≥49.
3 3
(3) I [40]\{b −15,b −14}⊆J [40].
3 40 40 3
(4) J [13]=I [13] and J [16]=I [16]\{7,9,10,11,12}..
3 3 3 3
(5) [0,11]∪{13,15,17,20,29,50}∪[22,24]⊆J [25] and {42}6∈J [25].
3 3
(6) [1,24]∪{27,28,33,37,39,63}⊆J [28].
3
(7) {18,19,78,79,81,87,102,103,111}∪ [21,32] ∪ [34,36] ∪ [38,43] ∪
[45,48]∪[52,54]∪[58,63]∪[67,71]⊆J [37].
3
2 Necessary conditions
InthissectionweestablishnecessaryconditionsforJ [v]. Forthispurpose,
3
we use another concept that is relative to intersection problem: A (v,k,t)
trade of volume s consists of two disjoint collections T and T , each of s
1 2
blocks,suchthatforeveryt-subsetofblocks,thenumberofblockscontain-
ing these elements (t-subset) are the same in both T and T . A (v,k,t)
1 2
trade ofvolume s is Steiner when for everyt-subset ofblocks,the number
ofblockscontainingtheseelementsareatmostone. Aµ-way(v,k,t)trade
T ={T ,T ,...,T }, µ≥2 is a set of pairwise disjoint (v,k,t) trade.
1 2 µ
Ineverycollectiontheunionofblocksmustcoverthesamesetofelements.
This set of elements is called the foundation of the trade. Its notation is
found (T)andr =no. ofblocks ina collectionwhichcontainthe element
x
x.
Bydefinitionofthetrade,ifb −sisinJ [v],thenitisclearthatthereexists
v 3
a3-waySteiner(v,4,2)tradeofvolumes. ConsiderthreeS(2,4,v)designs
(systems)intersectinginb −ssameblocks(ofsizefour). Theremainingset
v
of blocks (of size four) form disjoint partial quadruple systems, containing
2
precisely the same pairs, and each has s blocks. Rashidi and Soltankhah
in [16] established that there do not exist a 3-way Steiner (v,4,2) trade of
volume s, for s∈{1,2,3,4,5,6,7}. So we have the following lemma:
Lemma 2.1 J [v]⊆I [v].
3 3
3 Recursive constructions
In this section we give some recursive constructions for the 3-way inter-
section problem. The concept of GDDs plays an important role in these
constructions. Our aim of common blocks is the same common blocks in
the sequel.
Let K be a set of positive integers . A group divisible design K-GDD
(as GDD for short) is a triple (X,G,A) satisfying the following properties:
(1) G is a partition of a finite set X into subsets (called groups); (2) A is a
set of subsets of X (called blocks), each of cardinality from K, such that a
group and a block contain at most one common element; (3) every pair of
elements from distinct groups occurs in exactly one block.
If G contains u groups of size g , for 1 ≤ i ≤ s, then we denote
i i
by gu1gu2...gus the group type (or type) of the GDD. If K = {k}, we
1 2 s
write {k}-GDD as k-GDD. A K-GDD of type 1v is commonly called a
pairwise balanced design, denoted by (v,K,1)-PBD. When K = {k} a
PBD is just a Steiner system S(2,k,v).
The following construction is a variation of Willson’s Fundamental Con-
struction.
Theorem 3.1 (Weighting construction). Let (X,G,A) be a GDD with
groups G , G ,..., G . Suppose that there exists a function w : X →
1 2 s
Z+∪{0} (a weight function) so that for each block A = {x ,...,x } ∈ A
1 k
there exist three K-GDDs of type [w(x ), ... w(x )] with b common
1 k A
blocks. ThenthereexistthreeK-GDDsoftype[P w(x),...,P w(x)]
x∈G1 x∈Gs
which intersect in P b blocks.
A∈A A
proof. For every x ∈X, let S(x) be a set of w(x) “copies” of x. For any
Y ⊂ X, let S(Y) = S S(y). For every block A ∈ A, there exist three
y∈Y
K-GDDs: (S(A), {S(x) : x ∈ A}, B ), (S(A), {S(x) : x ∈ A}, B˙ ),
A A
(S(A), {S(x) : x ∈ A}, B¨ ), which intersect in b blocks. Then it is
A A
readily checked that there exist three, K-GDDs: (S(X), {S(G) : G ∈
G}, ∪ B ), (S(X), {S(G) : G∈G}, ∪ B˙ ), (S(X), {S(G) : G∈
A∈A A A∈A A
G}, ∪ B¨ ), which intersect in P b blocks.
A∈A A A∈A A
Theorem 3.2 (Filling construction (i)). Suppose that there exist three
4-GDDs of type g g ...g which intersect in b blocks. If there exist three
1 2 s
3
S(2,4,g +1) designs with b common blocks for 1≤i≤s, then there exist
i i
three S(2,4,Ps g +1) designs with b+Ps b common blocks.
i=1 i i=1 i
proof. It is obvious.
Theorem 3.3 (Filling construction (ii)). Suppose that there exist three
4-GDDs of type g g ...g which intersect in b blocks. If there exist three
1 2 s
S(2,4,g +4) designs containing b common blocks for 1 ≤ i ≤ s. Also
i i
s
all designs containing a block y. Then there exist three S(2,4,P g +4)
i=1 i
designs with b+Ps b −(s−1) common blocks.
i=1 i
proof. Let(X,G,A ),(X,G,A )and(X,G,A )bethree4-GDDsoftype
1 2 3
g g ...g which intersect in b blocks. Let Y = {y ,y ,y ,y } be a set of
1 2 s 1 2 3 4
cardinality 4 such that X ∩Y =φ.
For1≤i≤s,thereexistthreeS(2,4,g +4)designs(g ∪Y,ε ),(g ∪Y,ε )
i i 1i i 2i
and(g ∪Y,ε )containingthesameblocky =y ,y ,y ,y withb common
i 3i 1 2 3 4 i
blocks. It is easyto see that(X ∪Y,A ∪(S (ε −y))∪ε ),(X ∪
1 1≤i≤s−1 1i 1s
Y,A ∪(S (ε −y))∪ε )and(X∪Y,A ∪(S (ε −y))∪
2 1≤i≤s−1 2i 2s 3 1≤i≤s−1 3i
s s
ε )arethreeS(2,4,P g +4)designswithb+P b −(s−1)common
3s i=1 i i=1 i
blocks.
We apply another type of recursive constructions that explained in the
following.
Let there be three S(2,4,v) designs with a common parallel class, then
J [v] for v ≡ 4 (mod 12) denotes the number of blocks shared by these
p3
S(2,4,v) designs , in addition to those shared in the parallel class.
Lemma 3.4 Let G be a GDD on v = 3s+6t elements with b blocks of
size 4 and group type 3s6t, s ≥ 1. For 1 ≤ i ≤ b, let a ∈ J [16]. For
i p3
1 ≤ i ≤ s−1, let c +1 ∈ J [16] and let c ∈ J [16]. For 1 ≤ i ≤ t, let
i 3 s 3
d +1∈J [28]. Then there exist three S(2,4,4v+4) designs with precisely
i 3
Pb a +Ps c +Pt d blocks in common.
i=1 i i=1 i i=1 i
proof. The proof is similar to Lemma 3.3 in [8].
The flower ofan elementis the set ofblocks containing thatelement. Let
J [v] denote the number of blocks shared by three S(2,4,v) designs, in
f3
addition to those in a required common flower.
Lemma 3.5 Let G, B be a GDD of order v with b blocks of size 4, b
4 5
blocks of size 5 and group type 4s5t. For 1 ≤ i ≤ b , let a ∈ J [13]. For
4 i f3
1≤i≤b , let c ∈J [16]. For 1≤i≤s, let d ∈J [13]and for 1≤i≤t,
5 i f3 i 3
let e ∈J [16]. Then there exist three S(2,4,3v+1) designs intersecting in
i 3
precisely Pb4 a +Pb5 c +Ps d +Pt e blocks.
i=1 i i=1 i i=1 i i=1 i
4
proof. The proof is similar to Lemma 3.5 in [8].
Lemma 3.6 [9]. Thenecessaryandsufficientconditionsfortheexistence
of a 4-GDD of type gn are: (1) n ≥ 4, (2) (n−1)g ≡ 0 (mod 3), (3)
n(n−1)g2 ≡ 0 (mod 12), with the exception of (g,n)∈{(2,4),(6,4)}, in
which case no such GDD exists.
Lemma 3.7 [2]. Thereexistsa(v,{4,7∗},1)-PBDwithexactlyoneblock
of size 7 for any positive integer v ≡7,10 (mod 12) and v 6=10,19.
Lemma 3.8 [9]. A 4-GDD of type 12um1 exists if and only if either
u=3 and m=12, or u≥4 and m≡0 (mod 3) with 0≤m≤6(u−1).
4 Ingredients
Inthissectionwediscusssomesmallcasesneededforgeneralconstructions.
Lemma 4.1 J [13]=I [13].
3 3
proof. ConstructanS(2,4,13)design,(V,B)withV =Z ∪{a,b,c}. All
10
blocksofB arelistedinthe following,whichcanbe foundinExample1.26
in [9].
0 0 0 0 1 1 1 2 2 3 3 4 5
1 2 4 6 2 5 7 3 6 4 7 8 9
3 8 5 a 4 6 b 5 7 6 8 9 a
9 c 7 b a 8 c b 9 c a b c
Consider the following permutations on V.
π π π int. no.
1 2 3
id (0,1,2,3,4,5) (5,4,3,2,1,0) 0
id (8,5)(a,b)(3,7)(1,6) (8,6)(1,5)(a,3,b,7) 1
id (7,b,c,6) (8,5,6,7) 2
id (4,7)(9,2)(1,8) (3,9,c)(1,8) 3
id (3,7)(c,0,2)(1,6)(9,b) (9,4)(3,7)(0,2)(1,6) 4
id (a,b)(4,5) (a,b)(c,8) 5
id id id 13
Lemma 4.2 J [16]=I [16]\{7,9,10,11,12}.
3 3
5
proof. The proof has three steps:
(1) J [16]⊆J[16]={0,1,2,3,4,5,6,8,12,20}.
3
(2)ConstructanS(2,4,16)design,(V,B)withV =Z ∪{a,b,c,d,e,f}.
10
All 20 blocksof B arelisted in the following,whichcan be foundin Exam-
ple 1.31 in [9].
{0,1,2,3}, {0,4,5,6}, {0,7,8,9}, {0,a,b,c}, {0,d,e,f}, {1,4,7,a},
{1,5,b,d}, {1,6,8,e}, {1,9,c,f}, {2,4,c,e}, {2,5,7,f}, {2,6,9,b},
{2,8,a,d}, {3,4,9,d}, {3,5,8,c}, {3,6,a,f}, {3,7,b,e}, {4,8,b,f},
{5,9,a,e}, {6,7,c,d}.
Consider the following permutations on V.
π π π int. no.
1 2 3
id (0,1,4,9,d)(6,5)(b,f) (2,3,7,6,c)(5,8)(a,b)(f,4) 0
id (0,1,2,3)(8,9,5,b,c) (d,f,e,a)(4,6,7)(c,9) 1
id (1,2)(a,b)(7,f)(c,e)(6,8) (a,b,7,f,c,e,6,8)(4,d) 2
id (c,e)(5,6)(b,f,7,d,9) (8,b,7,a,d,9,f)(1,3) 3
id (0,1,2,3) (4,8,b,f) 4
id (4,f)(d,7)(a,9,5) (7,c,d)(f,b,4) 5
id (0,1,2)(a,e) (f,b)(1,0,2) 6
id (6,7,c) (6,c,7) 8
id id id 20
Hence we have {0,1,2,3,4,5,6,8,20}⊂J [16].
3
(3) b −86∈J [16]:
16 3
If b −8 = 20−8 = 12 ∈ J [16], then a 3-way (v,4,2) trade of volume 8
16 3
is contained in the S(2,4,16) design. Let T be this trade.
If all elements in found (T) appear 3 times in T , then for one block as
i
a a a a , there exist 8 more blocks, so |T | ≥ 9. Hence there exists x ∈
1 2 3 4 i
found(T),withr =2. Withoutlossofgenerality,letxa a a andxb b b
x 1 2 3 1 2 3
be in T . But T is Steiner trade so there exist (for example): xb a a and
1 1 2 3
xa b b in T and there exist xa b a and xb b a in T . Now T must
1 2 3 2 1 3 3 1 2 2 3 1
contains at least 6 pairs: a b , a b , a b , a b , a b , a b which those
1 2 1 3 2 1 3 1 3 3 2 2
come in disjoint blocks, since T is Steiner. So we have:
6
T T T
1 2 3
xa a a xb a a xa b a
1 2 3 1 2 3 1 3 3
xb b b xa b b xb b a
1 2 3 1 2 3 1 2 2
a b −− −−
1 2
a b −− −−
1 3
a b −− −−
2 1
a b −− −−
2 2
a b −− −−
3 1
a b −− −−
3 3
We know that the S(2,4,16) design is unique (See [9]). Without loss of
generality,we can assume x,a ,a ,a =0,1,2,3and x,b ,b ,b =0,4,5,6
1 2 3 1 2 3
(two blocks of the S(2,4,16) design). Hence T has the following form:
T T T
1 2 3
0123 0423 0163
0456 0156 0452
15bd −− −−
168e −− −−
24ce −− −−
257f −− −−
349d −− −−
36af −− −−
Therefore r =1, and by Lemma 3 in [11] this is impossible .
7
Lemma 4.3 {0,23,29,50}⊆J [25].
3
proof. Construct an S(2,4,25) design, (V,B) with
V =Z ∪{a,b,c,d,e,f,g,h,i,j,k,l,m,n,o}. All 50 blocks of B are listed
10
in the following, which can be found in Example 1.34 in [9].
{0,1,2,i}, {0,l,3,6}, {0,4,8,o}, {0,a,5,9}, {0,7,g,h}, {0,b,d,n},
{0,c,f,g}, {0,k,m,e}, {1,3,a,b}, {1,4,7,m}, {1,5,6,o}, {1,8,f,h},
{1,9,e,l}, {1,c,k,n},{1,d,g,j}, {2,3,7,o}, {2,4,b,9}, {2,8,5,n},
{2,6,f,g}, {2,a,c,m}, {2,d,k,l}, {2,e,h,j}, {3,4,5,j}, {3,8,c,d},
{3,9,k,f}, {3,e,g,n}, {3,h,i,m}, {4,6,d,e}, {4,a,g,k}, {h,4,c,l},
{4,i,n,f}, {5,7,c,e}, {5,b,h,k}, {5,d,f,m}, {5,g,i,l}, {6,7,8,k},
{6,9,c,i}, {6,a,h,n}, {6,b,j,m}, {9,7,j,n}, {7,a,d,i}, {7,b,f,l},
{m,g,8,9}, {8,a,j,l}, {8,b,e,i}, {9,d,h,o}, {a,e,f,o}, {b,c,g,o},
{i,j,k,o}, {l,m,n,o}.
Consider the following permutations on V.
7
π π π int. no.
1 2 3
id (0,1,2,3) (2,1,0,3) 23
id (0,1,2) (2,1,0) 29
id id id 50
Hence we have {23,29,50}⊂J [25].
3
By taking the 5th, 6th and 8th designs, of Table 1.34 in [9], we have 0 ∈
J [25].
3
Lemma 4.4 {1,63}⊆J [28].
3
proof. 63 ∈ J [28] by taking an S(2,4,28) design thrice. We obtain 1 ∈
3
J [28],byapplyingthe followingpermutationsonthe designofLemma6.3
3
(Step 1) in the last section.
π is identity, π = (0,1,3,5,6,7,12,17,15,18,19,20,11)(25,26,27), and
1 2
π =π−1.
3 2
Lemma 4.5 There exist three 4-GDDs of type 44 with i common blocks,
i∈{0,1,2,4,16}.
proof. TaketheS(2,4,16)design,(V,B)constructedinLemma4.2. Con-
sider the parallel class P = {{0,1,2,3},{4,8,b,f},{5,9,a,e},{6,7,c,d}}
as the groups of GDD to obtain a 4-GDD of type 44 (X,G,B′), where
X =V, G =P and B′ =B\P.
Consider the following permutations on X, which keep G invariant.
π π π int. no.
1 2 3
id id id 16
id (6,7,c) (6,c,7) 4
id (0,1,2)(a,e) (f,b)(1,0,2) 2
id (4,f)(d,7)(a,9,5) (7,c,d)(f,b,4) 1
id (0,1,2,3) (4,8,b,f) 0
In fact J [16] is precisely the intersection sizes of three 4-GDDs of group
p3
type 44 having all groups in common.
Corollary 4.6 {0,1,2,4,16}⊆J [16].
p3
Lemma 4.7 There exist three 4-GDDs of type 35 with i common blocks,
i∈{0,1,3,15}.
8
proof. Take the S(2,4,16) design, (V,B) constructed in Lemma 4.2.
Deletetheelement0fromthisdesigntoobtaina4-GDDoftype35(X,G,B′),
where X =V \{0},
G = {{1,2,3},{4,5,6},{7,8,9},{a,b,c},{d,e,f}} and B′ = B\{B ∈ B :
0∈B}. ConsiderthefollowingpermutationsonX,whichkeepG invariant.
π π π int. no.
1 2 3
id id id 15
id (a,c)(1,3)(6,5) (7,9)(d,e)(2,3)(a,c) 1
id (2,3)(5,6)(7,8)(a,c)(d,f) (1,2)(4,6)(8,9)(a,c)(d,e) 0
If we delete d then we have a 4-GDD of type 35 (X,G,B′), where X =
V \{d},
G = {{6,7,c},{2,8,a},{1,5,b},{3,4,9},{0,e,f}} and B′ = B\{B ∈ B :
d ∈ B}. When the following permutations act on X then we obtain 3 as
intersection number.
π = identity, π =(6,7,c), π =(6,c,7).
1 2 3
We have {0,1,3,15}⊆J [16]because J [16]is precisely the intersec-
f3 f3
tion sizes of three 4-GDDs of group type 35 having all groups in common.
Lemma 4.8 There exist three 4-GDDs of type 34 with i common blocks,
i∈{0,1,9}.
proof. Take the S(2,4,13) design, (V,B) constructed in Lemma 4.1.
Deletetheelement0fromthedesigntoobtaina4-GDDoftype34(X,G,B′),
where X =V \{0},
G = {{1,3,9},{2,8,c},{4,5,7},{6,a,b}} and B′ = B\{B ∈ B : 0 ∈ B}.
Consider the following permutations on X, which keep G invariant.
π π π int. no.
1 2 3
id id id 9
id (a,b)(4,5) (a,b)(8,c) 1
If we delete 8 then we have a 4-GDD of type 34 (X,G,B′), where X =
V \{8},
G = {{0,2,c},{1,5,6},{3,7,a},{b,4,9}} and B′ = B\{B ∈ B : 8 ∈ B}.
WhenthefollowingpermutationsactonX thenweobtain0asintersection
number.
π = identity, π =(3,7)(c,0,2)(1,6)(9,b), π =(9,4)(3,7)(0,2)(1,6).
1 2 3
We obtain {0,1,9} ⊂ J [13] because J [13] is precisely the intersection
f3 f3
9
sizes of three 4-GDDs of group type 34 having all groups in common.
Corollary 4.9 {0,1,9}⊆J [13].
f3
5 Applying the recursions
In this section, we prove the main theorem for all v ≥ 40. First we treat
the (easier) case v ≡1 (mod 12).
Theorem 5.1 For any positive integer v =12u+1, u≡0,1 (mod 4)and
u≥4, J [v]=I [v].
3 3
proof. Startfroma4-GDDoftype3ufromLemma3.6. Giveeachelement
of the GDD weight 4. By Lemma 4.5 there exist three 4-GDDs of type 44
withαcommonblocks,α∈J [16]. Thenapplyconstruction3.1toobtain
p3
three4-GDDsoftype12u withPb α commonblocks,whereb= 3u(u−1)
i=1 i 4
andα ∈J [16],for1≤i≤b. Byconstruction3.2,fillinginthegroupsby
i p3
three S(2,4,13)designs with β (1≤j ≤u) common blocks,we have three
j
b u
S(2,4,12u+ 1) designs with P α + P β common blocks, where
i=1 i j=1 j
β ∈J [13]for1≤j ≤u. Itischeckedthatforanyintegern∈I [v],ncan
j 3 3
b u
bewrittenastheformofP α +P β ,whereα ∈J [16](1≤i≤b)
i=1 i j=1 j i p3
and β ∈J [13](1≤j ≤u).
j 3
Theorem 5.2 For any positive integer v =12u+1, u≡2,3 (mod 4)and
u≥7, J [v]=I [v].
3 3
proof. There exists a (3u +1,{4,7∗},1)-PBD from Lemma 3.7, which
contains exactly one block of size 7. Take an element from the block of
size 7. Delete this element to obtain a 4-GDD of type 3u−261. Give each
elementof the GDD weight4. By Lemma 4.5, there existthree 4-GDDs of
type44withαcommonblocks,α∈J [16]. Thenapplyconstruction3.1to
p3
obtainthree4-GDDsoftype 12u−224withPb α commonblocks,where
i=1 i
b = 3(u2−u−2) and α ∈ J [16] for 1 ≤ i ≤ b. By construction 3.2, filling
4 i p3
in the groups by three S(2,4,13) designs with β (1 ≤ j ≤ u−2) common
j
blocks, and three S(2,4,25)designs with β common blocks, we have three
S(2,4,12u+1)designswithPb α +Pu−2β +β commonblocks,where
i=1 i j=1 j
β ∈ J [13] for 1 ≤ j ≤ u−2 and β ∈ J [25]. It is checked that for any
j 3 3
integer n∈I [v], n can be written as the form of Pb α +Pu−2β +β,
3 i=1 i j=1 j
where α ∈ J [16](1 ≤ i ≤ b), β ∈ J [13](1 ≤ j ≤ u−2) and β ∈ J [25].
i p3 j 3 3
10