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Smallest defining sets of super-simple 2 - (v, 4,1) directed designs

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Abstract

A $2-(v,k,\lambda)$ directed design (or simply a $2-(v,k,\lambda)DD$) is super-simple if its underlying $2-(v,k,2\lambda)BIBD$ is super-simple, that is, any two blocks of the $BIBD$ intersect in at most two points. A $2-(v,k,\lambda)DD$ is simple if its underlying $2-(v,k,2\lambda)BIBD$ is simple, that is, it has no repeated blocks. A set of blocks which is a subset of a unique $2-(v,k,\lambda)DD$ is said to be a defining set of the directed design. A smallest defining set, is a defining set which has smallest cardinality. In this paper simultaneously we show that the necessary and sufficient condition for the existence of a super-simple $2-(v,4,1)DD$ is $v\equiv1\ ({\rm mod}\ 3)$ and for these values except $v=7$, there exists a super-simple $2-(v,4,1)DD$ whose smallest defining sets have at least a half of the blocks. And also for all $\epsilon > 0$ there exists $v_0(\epsilon)$ such that for all admissible $v>v_0$ there exists a $2-(v,4,1)DD$ whose smallest defining sets have at least $(5/8-\frac{c}{v})\mid \mathcal{B}\mid$ blocks, for suitable positive constant c.
Smallest deﬁning sets of super-simple
2(v, 4,1) directed designs
Department of Mathematics
Alzahra University
Vanak Square 19834 Tehran, I.R. Iran
Abstract
A 2 (v, k, λ) directed design (or simply a 2 (v, k, λ)DD) is
super-simple if its underlying 2 (v, k, 2λ)B IB D is super-simple,
that is, any two blocks of the BIBD intersect in at most two points.
A 2 (v, k , λ)DD is simple if its underlying 2 (v, k, 2λ)B IB D is
simple, that is, it has no repeated blocks.
A set of blocks which is a subset of a unique 2(v, k , λ)DD is said
to be a deﬁning set of the directed design. A smallest deﬁning set, is
a deﬁning set which has smallest cardinality. In this paper simulta-
neously we show that the necessary and suﬃcient condition for the
existence of a super-simple 2 (v, 4,1)DD is v1 (mod 3) and for
these values except v= 7 , there exists a super-simple 2(v, 4,1)DD
whose smallest deﬁning sets have at least a half of the blocks. And
also for all ² > 0 there exists v0(²) such that for all admissible v > v0
there exists a 2 (v, 4,1)D D whose smallest deﬁning sets have at
least ( 5
8c
v)| B | blocks, for suitable positive constant c.
KEYWORDS: super simple directed designs, smallest deﬁning set
1 Introduction
At(v, k, λ) directed design (or simply a t(v , k, λ)DD) is a pair (V, B),
where Vis v-set, and Bis a collection of ordered ktuples of distinct
1
PLEASE CITE AS: F. Amirzade and N. Soltankhah. Smallest defining sets of
super-simple 2- (v,4,1) directed designs. Utilitas Mathematica, 96(2015), 331-344.
elements of V(called blocks), such that each ordered t-tuple of distinct
elements of Vappears in precisely λblocks. We say that a t-tuple appears
in a k-tuple, if its components appear in that k-tuple as a set, and they
appear with the same order. For example the following 4-tuples form a
2(10,4,1)DD, D.
0132 0467 0589 2680 2791 7843 6942 4510
1498 3861 3970 8752 9653 2354 1576.
Here, for example, the 4-tuple 0132 contains the ordered pairs 01, 03, 02,
13, 12, and 32.
A (v, k, t) directed trade of volume sconsists of two disjoint collections
T1and T2, each of sblocks, such that every t-tuple of distinct elements of
Vis covered by precisely the same number of blocks of T1as of T2. Such
a directed trade is usually denoted by T=T1T2. Blocks in T1(T2) are
called the positive (respectively, negative) blocks of T. If D= (V, B) is a
directed design, and if T1⊆ B, we say that Dcontains the directed trade T.
For example the 2 (10,4,1)DD, Dabove contains the following directed
0132 1032
4510 4501
Given a t(v, k, λ) directed design D, a subset of the blocks of Dthat
occurs in no other t(v, k , λ) directed design is called a deﬁning set of
D. A deﬁning set for which no other deﬁning set has a smaller cardinal-
ity, is called a smallest deﬁning set. For example the set of blocks S=
{0132,2354,9653,8752,2791,2680,1498,1576}is a deﬁning set of the 2
(10,4,1)DD above. But the set of blocks R={0132,2354,9653,8752,2791,
2680,1498,0467}can be completed to a 2(10,4,1)DD in two ways: by ad-
joining either {4510,1576,0589,3861,3970,7843,6942}or {4150,5176,0589,
3861,3970,7843,6942}.
Deﬁning sets for directed designs (as suggested by A. P. Street in [12])
are strongly related to trades. This relation is illustrated by the following
result.
PROPOSITION 1.1 [14] Let D= (V, B)be a t(v, k, λ)directed design
and let S⊆ B, then Sis a deﬁning set of Dif only if Scontains a block of
every (v, k, t)directed trade T=T1T2such that Tis contained in D.
Each deﬁning set of a t(v, k, λ)DD , Dcontains at least one block in every
then the smallest deﬁning set of Dmust contain at least mblocks.
2
The concept of directed trades and deﬁning sets for directed designs
were investigated in articles [14,16].
A 2 (v, k, λ) directed design (or simply a 2 (v, k, λ)DD) is super-
simple if its underlying 2 (v , k, 2λ)BIBD is super-simple, that is, any
two blocks of the BIBD intersect in at most two points. A 2 (v, k, λ)DD
is simple if its underlying 2 (v, k , 2λ)BIBD is simple, that is, it has no
repeated blocks.
The concept of super-simple designs was introduced by Gronau and
Mullin [10]. The existence of super-simple designs is an interesting problem
by itself, but there are also some useful applications.
There are known results for the existence of super simple designs, es-
pecially for existence of super simple 2 (v, k, λ)BIBDs. When k= 4 the
necessary conditions for super simple 2 (v, k , λ)BIBDs are known to be
suﬃcient for 2 λ6 with few possible exceptions. These known results
can be found in articles [1,3,4,5,6,9,11].
In [14], Mahmoodian, Soltankhah and Street have proved that if Dbe
a 2 (v, 3,1)DD then a deﬁning set of Dhas at least v
2blocks. In [8]
Grannel, Griggs and Quinn have shown that for each admissible value of v,
there exists a 2 (v, 3,1)DD and a simple 2 (v, 3,1)DD whose smallest
deﬁning sets have at least |B|
2blocks. And they have also proved that, for
all ² > 0 and all suﬃciently large admissible v, there exists a 2(v , 3,1)DD
whose smallest deﬁning sets have at least (2
3²)| B | blocks.
In this paper we show that for all admissible values, there is a super-
simple 2 (v, 4,1)DD (except v= 7) whose smallest deﬁning sets have at
least a half of the blocks.
In other words we are interested in the quantity
f=number of 4tuples in a smallest deﬁning set in D
number of 4tuples in D.
And we show that, there exists a super-simple 2 (v, 4,1)DD,Dwith
f1
2.
The proofs in this paper use various types of combinatorial objects. The
deﬁnitions of these objects are either given in the paper or can be found in
the references.
Several proofs depend on the following result, which involves pairwise
balanced designs (PBDs) and is a special case of a result (the Replacement
Lemma [14]) that is used in several earlier papers on directed designs.
LEMMA 1.1 If there exist a 2(v, K, 1) design and a super-simple (k, 4,1)DD
3
for each kK, then there exists a super-simple 2(v, 4,1)DD.
PROOF. Replacing each block of the 2 (v, K, 1) design with a copy of
a super-simple 2 (k, 4,1)DD with point set the points of that block gives
a super-simple 2 (v, 4,1)DD.
A lower bound for ffor the super-simple 2 (v , 4,1)DD constructed in
Lemma 1.1 can be calculated from lower bounds for ffor the various super-
simple 2 (k, 4,1)DDs. In particular, if there is a constant c such that
each of the super-simple 2 (k, 4,1)DDs has fc, then the resulting
super-simple 2 (v, 4,1)DD also has fc.
2 Main result
A necessary and suﬃcient condition for the existence of a 2 (v, 4,1)DD
is v1 (mod 3) [17].
In this section simultaneously we show that the necessary and suﬃcient
condition for the existence of a super-simple 2(v , 4,1)DD is v1 (mod 3)
and for these values except v= 7, there exists a super-simple 2(v , 4,1)DD
with f1
2.
In [13], Soltankhah and Mahmoodian have shown that, up to isomor-
phism, there exist two 2(7,4,1)DDs that they are super-simple and each
of them has smallest deﬁning set of cardinality 2. Consequently, if v= 7
then f=2
7.
In proofs we handle with a special type of directed trade, named a
Let T=T1T2be a (v, 4,2) directed trade of volume s, where T1con-
tains blocks b1, . . . , bssuch that each pair of consecutive 4-tuples (blocks)
of T1,bibi+1 i= 1, . . . , s (mod s) is a trade of volume 2. Therefore if a
directed design Dcontains T1, then any deﬁning set for Dmust contain at
least [s+1
2] blocks of T1.
The following 4-tuples form a super-simple 2 (10,4,1)DD.
0467 0589 2680 2791 7843 6942 0132
1576 1498 3861 3970 8752 9653 4510
2354
Each of the ﬁrst six columns above contains a trade, hence any deﬁning set
for this directed design must contain at least one 4-tuple of each column.
4
The last column above is a cyclical trade of volume 3. Hence any deﬁning
set for this directed design must contain at least two 4-tuples of the last col-
umn. We can use integer programming problem [2] to ﬁnd a smallest deﬁn-
ing set for this directed design and S={0132,2354,9653,8752,2791,2680,
1498,1576}is a deﬁning set of size 8. Therefore this super-simple 2
(10,4,1)DD has f=8
15 >1
2.
The following base blocks by (+1 mod 13) form a super-simple 2
(13,4,1)DD with f1
2.
0139,1 0 11 5
The following 4-tuples form a super-simple 2 (16,4,1)DD.
1 3 2 4 1 5 10 14 1 6 12 13 3 8 10 15 1 8 11 16
16 14 3 1 13 9 5 1 15 10 6 1 14 10 8 9 16 11 10 2
5 6 7 8 2 6 11 15 2 5 9 16 4 7 11 14 2 7 10 13
16 13 7 6 14 11 6 5 15 5 2 3 10 7 4 5 8 7 2 1
9 10 12 11 3 7 12 16 2 8 12 14 1 7 9 15 3 6 9 14
13 12 10 3 16 12 8 5 15 14 12 7 11 9 7 3 12 9 6 2
13 14 15 16 4 9 8 13 3 5 11 13 4 6 10 16 5 4 12 15
14 13 4 2 16 15 9 4 15 13 11 8 8 6 4 3 11 12 4 1
This super-simple 2 (16,4,1)DD contains 20 disjoint directed trades of
volume 2. Hence any deﬁning set of this super-simple directed design must
contain at least one 4-tuple of each trade. So for this super-simple 2
(16,4,1)DD we have f20
40 =1
2.
In our future results we need some super-simple directed group divisible
designs with block size four (super-simple 4 DGDDs). We may use a
super-simple 4 DGDD of type (34), a super-simple 4 DGDD of type
(35), a super-simple 4 DGDD of type (36) and a super-simple 4DGDD
of type (24). Now we can construct a super-simple 4 DGDD of type (34)
with groups {0,4,8},{1,5,9},{2,6,10},{3,7,11}, and with the following
blocks.
4 3 10 9 9 2 3 8 2 1 4 11 11 10 5 4 7 6 0 1
91007 11029 3416 4572 6794
10103 3205
8 1 2 7 5 0 6 11 7 8 5 10
10 11 1 8 8 9 11 6 6 5 8 3
5
Each of the ﬁrst two columns above is a cyclical trade of volume 3, hence
any deﬁning set of this super-simple directed group divisible design must
contain at least two 4-tuples from each of the ﬁrst two columns and 4-tuples
in the last three columns above form six disjoint directed trades of volume
2 and one 4-tuple from each of them must be in any deﬁning set. Therefore
any deﬁning set of this super-simple 4 DGDD must contain at least 10
blocks, so it has f10
18 >1
2.
A super-simple 4 DGDD of type (35) can be constructed with groups
{0,5,10},{1,6,11},{2,7,12},{3,8,13},{4,9,14}
and with the following base blocks by (+1 mod 15).
1037,0 1 13 9
This super-simple directed group divisible design has 15 disjoint directed
trades of volume 2 and one 4-tuple from each of them must be in any
deﬁning set. Therefore any deﬁning set of this super-simple 4 DGDD
must contain at least 15 blocks, so it has f15
30 =1
2.
A super-simple 4 DGDD of type (36) can be constructed with groups
{0,6,12},{1,7,13},{2,8,14},{3,9,15},{4,10,16},{5,11,17}
and with the following base blocks by (+2 mod 18).
51020 ,14 1 10 5
15 0 4 11 ,11 4 3 1
0132
This super-simple directed group divisible design has 18 disjoint di-
rected trades of volume 2 in the ﬁrst two rows above and the last row
is a cyclical trade of volume 9. So any deﬁning set for this super-simple
4DGDD must contain at least ﬁve 4-tuples of the last row. Hence any
deﬁning set must contain at least 18+5 = 23 blocks, so for this super-simple
4DGDD, we have f23
45 >1
2.
A super-simple 4 DGDD of type (24) with f1
2can be constructed
with groups {1,2},{3,4},{5,6},{7,8}and with the following blocks
4 1 6 7 ,8613
6 8 2 4 ,7542
5 7 3 1 ,2376
1 4 5 8 ,3285.
6
Our principal tool is to apply Wilson’s Fundamental construction(weighting),
that is described in the following lemma.
LEMMA 2.2 If there is a {K} − GDD of type g1u1g2u2. . . gN
uN, there
are super-simple 2(αgi+1,4,1)DDs for each i, i = 1,2, . . . , N and there
are super-simple 4DGDDs of type αkfor each kKthen, there exists
a super-simple 2(αPN
i=1 giui+ 1,4,1)DD.
PROOF. Let (G, B) be a group divisible design with element set U,
blocks of size kKand groups of size g1, g2, . . . , and gN. We form a
super-simple 2(αPN
i=1 giui+1,4,1)DD on the element set U×ZαS{∞}.
Give weight αto all of points. That is replace each point xUwith
αnew points {x1, x2, . . . , xα}. Now replace each block b∈ B of size
kKwith a super-simple 4 DGDD of type αksuch that its groups
are {x1, x2, . . . , xα:xU}to get a super-simple 4 DGDD of type
αg1u1αg2u2. . . αgN
uN. Finally ﬁll in the holes with a new point , using
super-simple 2 (αgi+ 1,4,1)DDs for i= 1,2, . . . , N .
In particular, if there is a constant c such that each of the super-simple
4DGDD of type αkand the super-simple 2 (αgi+ 1,4,1)DDs has
fc, then the resulting super-simple 2 (v , 4,1)DD in Lemma 2.2 also
has fc.
THEOREM 2.1 For all v1 (mod 3) except v= 7 there exists a super-
simple 2(v, 4,1)DD with f1
2.
PROOF. For all nexcept nA={7,8,9,10,11,12,14,15,18,19,23}
there exists a P BD (n, {4,5,6}) [15]. We can delete one point from this
P BD to form a {4,5,6} − GDD of order n1 of type 3a4b5c, where a,
b, c are non-negative integers. We then apply lemma 2.2 to this GDD
using a weight of 3 to get a super-simple 4 DGDD of type 9a12b15cwith
f1
2. Finally ﬁll in the holes with a new point , using super-simple
2(m, 4,1)DDs for m= 10,13,16 with f1
2.
Now for the remaining values v∈ {19,22,25,28,31,34,40,43,52,55,67}
we construct a super-simple 2 (v, 4,1)DD with f1
2as follows.
v= 19: the following 4-tuples form a super-simple 2 (19,4,1)DD.
7
3 6 4 15 2 7 4 17 10 3 11 13 11 16 5 12 14 12 17 8
6 3 16 1 5 4 7 16 9 3 10 14 13 15 12 5 6 17 12 13
5 8 17 6
6 11 0 7 17 9 4 5 3 9 12 7 10 12 15 6
11 6 8 2 16 9 17 0 1 8 7 12 12 10 9 2 2 1 14 6
13 4 1 2
4 0 18 12 13 16 7 11 8 1 9 11 2 9 13 8 18 14 1 4
18 0 2 11 16 13 6 14 9 1 15 16 15 13 9 18
0 5 1 13
18 7 9 6 7 14 18 13 8 18 16 15 8 4 13 10 2 5 0 15
0 4 6 9 17 14 7 15 2 12 16 18 16 4 8 3 15 0 8 14
7 10 8 0
12 1 3 0 5 14 11 9 15 11 17 1 14 0 10 16 10 7 1 5
13 0 3 17 12 4 11 14 11 15 10 4
1 18 10 17 14 5 3 2 17 11 3 18 17 16 2 10
6 5 10 18 15 7 2 3 18 3 8 5
This super-simple 2(19,4,1)DD has 22 disjoint directed trades of volume
2 in the ﬁrst four columns above and the last column has three cyclical
trades of volume 3, 3 and 5, respectively. So any deﬁning set for this DD
must contain at least two 4-tuples from each of the cyclical trades of volume
3 and three 4-tuples of the cyclical trade of volume 5. Hence any deﬁning
set must contain at least 22+ 2×2 + 3 = 29 blocks, so for this super-simple
2(19,4,1)DD, we havef29
57 >1
2.
v= 22: the following base blocks by (,+1 mod 11) form a super-simple
2(22,4,1)DD.
(0,0)(0,3)(0,9)(0,10) ,(1,0)(1,4)(0,9)(0,3)
(0,0)(1,0)(1,7)(1,2) ,(1,0)(0,0)(1,9)(1,10)
(0,0)(0,2)(1,5)(1,8) ,(1,0)(1,5)(0,2)(0,6)
(1,7)(0,3)(0,0)(1,4)
This super-simple 2(22,4,1)DD has 33 disjoint directed trades of volume
2 in the ﬁrst three rows above and the last row is a cyclical trade of volume
11. So any deﬁning set for this DD must contain at least six 4-tuples of the
last row. Hence any deﬁning set must contain at least 33 + 6 = 39 blocks,
so for this super-simple 2 (22,4,1)DD, we have f39
77 >1
2.
v= 25: given base blocks in [10] form a super-simple 2 (25,4,1)DD
with f1
2.
v= 28: the following 4-tuples form a super-simple 2 (28,4,1)DD.
8
1 2 14 16 1 8 22 23 2 3 19 22 6 8 9 14 4 7 14 15
21 9 2 1 25 4 8 1 25 14 3 2 27 8 6 3 23 21 15 14
12 14 20 24 23 22 9 3 13 14 19 26 3 6 13 17 2 6 21 23
16 14 12 10 10 9 22 24 11 14 13 9 27 25 17 13 11 10 6 2
6 7 10 12 8 4 21 24 7 13 8 11 7 16 25 27 10 11 23 25
24 15 10 7 25 24 21 5 22 19 11 8 17 16 7 4 23 11 7 1
2 8 10 15 1 3 24 25 2 7 18 24 16 17 20 22 1 7 9 17
17 9 10 8 18 10 3 1 12 8 7 2 20 17 14 1 26 9 7 5
6 15 24 27 0 3 10 14 9 11 21 26 14 17 21 25 5 7 19 20
27 24 16 9 24 11 3 0 27 21 11 4 26 21 17 6 20 19 9 4
4 3 9 16 1 6 20 26 5 6 22 25 3 12 21 27 13 16 23 24
26 15 3 4 22 16 6 1 19 14 6 5 19 17 12 3 24 23 17 2
0 2 12 17 1 4 11 12 2 4 25 26 11 17 19 24 3 5 11 15
20 16 2 0 26 20 12 11 26 25 22 10 17 11 18 5 16 13 5 3
4 6 18 19 0 5 24 26 3 8 18 20 18 15 17 23 0 4 20 23
24 12 6 4 26 24 14 8 24 22 20 18 22 17 15 0 25 23 20 6
0 8 19 25 5 8 12 16 2 11 20 27 4 13 22 27 18 14 4 0
25 19 18 7 23 8 5 0 27 20 10 5 27 22 14 7 14 18 11 22
8 17 26 27 3 7 23 26 0 1 13 15 10 13 20 21 9 15 20 25
27 26 1 0 26 23 19 16 24 19 13 1 21 20 7 3 20 15 8 13
15 18 9 6 2 5 9 13 6 0 11 16 12 15 22 26 13 7 0 6
0 9 18 27 22 4 5 2 25 15 16 11 15 12 5 1 0 7 21 22
16 15 19 21 12 13 18 25 1 10 19 27 5 14 23 27 5 4 10 17
21 19 10 0 22 21 13 12 27 19 15 2 27 23 18 12 23 13 10 4
1 5 18 21 10 16 18 26 25 12 9 0
21 18 16 8 26 18 13 2 9 12 19 23
This super-simple 2 (28,4,1)DD contains 63 disjoint directed trades.
Hence, any deﬁning set of this super simple directed design must contain
at least one 4-tuple of each trade. So for this super-simple 2 (28,4,1)DD
we have f63
126 =1
2.
9
v= 31: the following base blocks by (+1 mod 31) form a super-simple
2(31,4,1)DD with f1
2.
2 0 4 1 ,1 4 19 9
4 0 20 13 ,13 20 8 30 ,14 28 8 20
This super-simple 2(31,4,1)DD has 31 disjoint directed trades of volume
2 in the ﬁrst row above and the second row is a cyclical trade of volume 93.
So any deﬁning set for this DD must contain at least 47, 4-tuples of the
cyclical trade. Hence any deﬁning set must contain at least 31 + 47 = 78
blocks, so for this super-simple 2 (31,4,1)DD we have f78
155 >1
2.
v= 34: the following base blocks by (+1 mod 34) form a super-simple
2(34,4,1)DD with f1
2.
0 22 1 24 ,31 29 24 1
25 1 19 0 ,19 1 15 22 ,23 15 1 20
0 17 8 25
This super-simple 2 (34,4,1)DD has 34 disjoint directed trades of vol-
ume 2 in the ﬁrst row above and the second row is a cyclical trade of
volume 102. So any deﬁning set for this DD must contain at least 51,
4-tuples of the second row. The last row form a cyclical trade of volume
34. Hence any deﬁning set must contain at least 34 + 51 + 17 = 102
blocks, so for this super-simple 2 (34,4,1)DD we have f102
204 1
2.
v= 40: begin with a 4 GDD(54) [15]. We apply lemma 2.2 using
a weight 2, using a super-simple 4 DGDD of type (24) with f1
2,
to get a super-simple 4 DGDD of type 104with f1
2. Now ﬁll in
the holes of size 10 with a super-simple 2 (10,4,1)DD with f1
2.
v= 43: begin with a 4 GDD(27) [15]. We apply lemma 2.2 using a
weight 3, using a super-simple 4DGDD of type (34) with f5
9, to get a
super-simple 4DGDD of type 67with f1
2. Finally ﬁll in the hole with a
new point , using a super-simple 2(7,4,1)DD. This yield a super-simple
2(43,4,1)DD with f154
301 >1
2.
v= 52: the following base blocks form a super-simple (52,4,1)DD with
f1
2.
0 5 1 3 ,48 5 0 45 (+1 mod 52)
30 6 0 13 ,13 0 23 42 (+1 mod 52)
0 6 21 37 ,4 37 21 45 (+1 mod 52)
8 38 20 0 ,34 9 0 20 (+1 mod 52)
27 1 0 26 ,26 0 25 51 (+2 mod 52)
10
v= 55: a RBIBD(16,4,1) has 5 parallel classes. For two parallel
classes add a new point to each of the blocks in that parallel class. This
yields a {4,5} − GDD of type 4421. We apply lemma 2.2 using a weight 3,
using a super-simple 4DGDD of type (34) with f5
9and a super-simple
4DGDD of type (35) with f1
2, to get a super-simple 4 DGDD of
type 12461with f1
2. Finally ﬁll in the holes with a new point , using
a super-simple (13,4,1)DD with f1
2and a super-simple 2 (7,4,1)DD.
v= 67: begin with a 4 GDD(6491) [15]. Applying lemma 2.2 using a
weight 2, we obtain a super-simple 2 (67,4,1)DD with f1
2.
3 Asymptotic results
In previous section, we showed that for all admissible values of v, except
v= 7, there exists a super-simple 2(v, 4,1)DD with f1
2. In this section
we show that these results can be improved for 2 (v, 4,1)DDs. And we
prove that for all suﬃciently large admissible v, there exists a 2(v, 4,1)DD
with f5
8c
vfor suitable positive constant c. For this result we need a
directed group divisible design with block size four (4 DGDD) of type
(24) with f5
8.
A 4DGDD of type (24) can be constructed with groups {1,2},{3,4},
{5,6},{7,0}, and with the following blocks.
blocks : 5 1 0 3 3 7 1 5 4 0 2 5
6 0 1 4 4 1 7 6 5 2 7 4
2 3 0 6 6 7 3 2
Each of ﬁrst two columns above is a cyclical trade of volume 3, hence any
deﬁning set of this directed group divisible design must contain at least two
4-tuples from each of the ﬁrst two columns. The 4-tuples in the last column
above form a trade and one of them must be in any deﬁning set. Therefore
any deﬁning set of this 4DGDD must contain at least 5 blocks, so it has
f5
8.
LEMMA 3.3 For all v1 (mod 12),v61, there exists a 2(v, 4,1)DD
with f5
813
8v.
PROOF. For all k > 4 there exists a 4 GDD(6k) [15]. Let (G, B) be
a group divisible design with element set U, blocks of size 4 and groups of
11
size 6. We replace each group gGwith a 2 (13,4,1)DD with f1
2
on g×Z2S{∞} and each block b∈ B with 4 DGDD of type (24) with
f5
8on b×Z2, such that its groups are {x} × Z2,{y} × Z2,{z} × Z2and
{w} × Z2. Since a 4 GDD(6k) has 3k(k1) blocks, so a 2 (v , 4,1)DD
has
f(5
8.24k(k1) + 1
2.26k)/((12k+1)(12k)
6).
Simplifying gives f5
813
8v.
LEMMA 3.4 For all v4 (mod 12),v64, there exists a 2(v, 4,1)DD
with f5
81
2v.
PROOF. For all k > 4 there exists a 4 GDD(23k+1) [15]. Let (G, B)
be a group divisible design with element set U, blocks of size 4 and groups
of size 2. We replace each group gGwith a 2 (4,4,1)DD on g×Z2
and each block b∈ B with 4 DGDD of type (24) with f5
8on b×Z2,
such that its groups are {x} × Z2,{y} × Z2,{z} × Z2and {w} × Z2. Since
a 4 GDD(23k+1 ) has k(3k+ 1) blocks, so a 2 (v, 4,1)DD has
f(5
8.8k(3k+ 1) + 1
2.4(3k+ 1))/((12k+4)(12k+3)
6).
Simplifying gives f5
81
8(4k+1) >5
81
2v.
LEMMA 3.5 For all v7 (mod 12),v67, there exists a 2(v, 4,1)DD
with f5
813
8v21
2v(v+11) .
PROOF. For all k > 4 there exists a 4 GDD(6k91) [7]. Let (G, B) be
a group divisible design with element set U, blocks of size 4 and groups of
size 6 and a single group of size 9. We replace each group gGof size
6 and 9 with a 2 (13,4,1)DD and a 2 (19,4,1)DD with f1
2on
g×Z2S{∞}, respectively, and each block b∈ B with 4 DGDD of type
(24) with f5
8on b×Z2, such that its groups are {x} × Z2,{y} × Z2,
{z} × Z2and {w} × Z2. Since a 4 GDD(6k91) has 3k(k+ 2) blocks, so
a 2 (v, 4,1)DD has
f(5
8.24k(k+ 2) + 1
2.26k+ 29)/((12k+19)(12k+18)
6).
Simplifying gives f5
813
8v21
2v(v1) .
12
LEMMA 3.6 For all v10 (mod 12),v70 there exists a 2(v, 4,1)DD
with f5
81
4v1
2v.
PROOF. For all k > 4 there exists a 4 GDD(23k51) [7]. Let (G, B) be
a group divisible design with element set U, blocks of size 4 and groups of
size 2 and a single group of size 5. We replace each group gGof size 2
and 5 with a 2 (4,4,1)DD and a 2 (10,4,1)DD with f1
2on g×Z2,
respectively, and each block b∈ B with 4 DGDD of type (24) with f5
8
on b×Z2, such that its groups are {xZ2,{yZ2,{zZ2and {wZ2.
Since a 4 GDD(23k51) has k(3k+ 4) blocks, so a 2 (v , 4,1)DD has
f(5
8.8k(3k+ 4) + 1
2.4(3k) + 8)/((12k+10)(12k+9)
6).
Simplifying gives f5
81
4vk+4
2v(4k+3) or f > 5
81
2v.
THEOREM 3.2 For all ² > 0there exists v0(²)such that for all admis-
sible v > v0there exists a 2(v, 4,1)DD with f5
8c
vwhere c=13
8+².
References
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14
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