Construction of Optimal Supersaturated Designs by The

Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143

Construction of optimal supersaturated designs by

the packing method

¾ ¿ FANG Kaitai½ , GE Gennian & LIU Minqian

1. Department of Mathematics, Hong Kong Baptist University, Hong Kong, China (email: [email protected]); 2. Department of Mathematics, Zhejiang University, Hangzhou 310027, China (email: [email protected]); 3. Department of Statistics, Nankai University, Tianjin 300071, China (email: [email protected]) Received July 31, 2002

Abstract A supersaturated design is essentially a factorial design with the equal oc- currence of levels property and no fully aliased factors in which the number of main effects is greater than the number of runs. It has received much recent interest because of its potential in factor screening experiments. A packing design is an important object in combinatorial design theory. In this paper, a strong link between the two apparently un- related kinds of designs is shown. Several criteria for comparing supersaturated designs are proposed, their properties and connections with other existing criteria are discussed. A combinatorial approach, called the packing method, for constructing optimal supersaturated designs is presented, and properties of the resulting designs are also investigated. Comparisons between the new designs and other existing designs are given, which show that our construction method and the newly constructed designs have good properties. Keywords: Kirkman triple systems, orthogonality, packing design, resolvability, supersaturated design. DOI: 10.1360/02ys0271

Many preliminary studies in industrial and scientific experiments contain a large number of potentially relevant factors, but often only a few are believed to have significant effects. When experiments are expensive, the problem is how to identify these few significant effects with a relatively small number of runs. One approach is to use a so-called supersaturated design, namely, a factorial design in which the number of main effects is greater than the number of runs. Developing such screening designs has received a great deal of attention: Satterthwaite [½] introduced the supersaturated design as a random bal- ance design. Booth and Cox [¾] first examined this problem systematically. Since then, most studies, such as refs. [3—14], have focused on two-level supersaturated designs. These two-level designs can be used for screening the factors in simple linear models. When the relationship between a set of factors and a response is nonlinear, or approxi- mated by a polynomial response surface model, designs with multi-level and mixed-level

are often required, e.g., to explore nonlinear effects of the factors. Recently, Yamada and [½6] Lin[½5] and Yamada et al. presented two construction methods for three-level supersaturated designs; Fang, Lin and Ma [½7] obtained a new class of multi-level supersaturated

Copyright by Science in China Press 2004 Optimal supersaturated designs 129 designs through a global optimization algorithm, while Fang, Lin and Liu [½8] provided a

class of universally optimal mixed-level supersaturated designs.

´Ò; Õ µ Ò Ñ Õ

Let Í denote a design with runs and -level factors, which corresponds

= ´ ; ¡¡¡; µ Ò ¢ Ñ X Ü Ü ½

to an matrix Ñ and satisﬁes the following conditions:

; ¾; ¡¡¡;Õ

1. all the levels, denoted by ½ , of each factor appear equally often; X 2. no two columns of X are fully aliased, i.e., no column of can be obtained from

another column of X by permuting levels.

´Ò; Õ µ Ò

The set of all such designs is denoted by Í . Obviously, must be a multiple

Ô = Ò=Õ

of Õ . Throughout the paper let . Two columns are called orthogonal if all of

´Ò; Õ µ

their level-combinations appear equally often. And a design in Í is called an

Ä ´Õ µ

orthogonal array of strength 2, denoted by Ò , if any two columns are orthogonal.

´Õ ½µ 6 Ò ½

In this case, Ñ . A comprehensive study of orthogonal arrays can be

´Õ ½µ = Ò ½ Ñ´Õ ½µ >

found in ref. [19]. When Ñ , the design is saturated. When

Ò , the design does not have enough degrees of freedom to estimate all the factorial

´Ò; Õ µ

main effects simultaneously, it is called a supersaturated design, denoted by Ë .

Ò; Õ ; Ñµ Ë ´Ò; Õ µ

For given ´ , different ’s may have different performances in statistical

´f µ

inference. In this paper we employ the E criterion proposed by Fang, Lin and

ÆÇD

Ñ [½8]

´Ò; Õ µ E ´f µ

Liu for evaluating such designs. An Ë is called -optimal, denoted by

ÆÇD

Ñ Ñ

Ë ´Õ µ E ´f µ Í ´Ò; Õ µ

Ò , if it minimizes over all possible designs in . And we will

ÆÇD

ÑaÜ f Æ

propose two additional criteria, i.e. and ÑaÜ , for making further comparisons

ÆÇD

´f µ

among E -optimal designs.

ÆÇD

Ë ´Õ µ

Then, how to generate Ò ’s? In fact, one can employ some powerful optimization

Ñ Ñ [½¼]

Ë ´Õ µ Í ´Ò; Õ µ

methods to search for Ò ’s over , like Yamada et al. and Fang, Lin and

[½7] Ñ Ma . However, when Ò; Õ and are moderate, it is difﬁcult to search a complete list

of designs. The main purpose of the paper is to propose a new way for constructing

Ñ Ñ

Ë ´Õ µ Í ´Ò; Õ µ

Ò ’s. We will establish a connection between designs in a subset of and resolvable packing designs which are well-studied in combinatorial design theory. Then

the construction approach just follows from this connection.

< Õ

When Ò , the orthogonality between any two columns could not be satisﬁed.

; Ü Ü i

Naturally, we hope that the factorial level-combinations in any two columns j of

´Ò; Õ µ

an Ë will be distributed as evenly as possible. Therefore, we deﬁne a subset of

Ñ Ñ Ñ ¾

´Ò; Õ µ Í ´Ò; Õ ; Ö µ Í ´Ò; Õ ; Ö µ Õ

Í , denoted by , such that for each design in , any of the Ö

level-combinations in any two columns appears at most Ö times. The number is called

> ½ ´Ò; Õ ; Ñµ

the largest frequency of the design. Obviously Ö . For given , we expect to

Ñ ¾

´Ò; Õ ; Ö µ Ö Ò = Õ

choose a design in Í such that is as small as possible. When , for given

´Ò; Õ ; Ñµ Ä ´Õ µ

, we will choose an orthogonal array Ò if it exists as its largest frequency

½ Ö = ½ Ò 6 Õ Ö takes the minimum value , i.e. . When , the following natural questions

arise:

Ò; Õ ; Ö µ Í ´Ò; Õ ; Ö µ Question 1. For given ´ , we can ﬁnd many designs belonging to www.scichina.com

130 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143 Ñ

with different number of columns, Ñ. What is the largest value of ? Or equivalently,

Í ´Ò; Õ ; Ö µ ´Ò; Õ ; Ñµ

what is the smallest value of Ö in for given ?

´f µ

Question 2. How do we construct E -optimal supersaturated designs for given

ÆÇD

Ò; Õ ; Ñ; Ö µ Ö = ½; ¾ Ñ

´ , especially for and achieving its largest value in Questions 1?

´Ò; Õ ; Ö µ With the connection between designs in Í and resolvable packing designs,

the exact answer for Question 1 is obtained (see Theorem 3 in Section 2.1), and a class

Ë ´Õ µ

of Ò ’s that satisfy the requirement in Question 2 is constructed without any com-

4×

Ë ´¿ µ × = ½; ¡¡¡; 7

puter search. For example, a class of 9 for are obtained by the new

¾8

Ë ´¿ µ

approach, which are listed in table 1. There are 7 parts each having 4 columns in 9 .

Ä ´¿ µ Ö = ½

This design has the following properties: (a) each part is an 9 , i.e., ; (b) for

4×

× = ¾; ¡¡¡; 7; 4× Ë ´¿ µ

any the design formed by the ﬁrst columns is an 9 with the largest

= ¾ ´Ò; Õ ; Ö µ = ´9; ¿; ¾µ Ñ

frequency Ö ; (c) for given , the largest value of is 28, which is

¾8

Ë ´¿ µ ´Ò; Õ ; Ñµ = ´9; ¿; 4×µ;× = ¾; ¡¡¡; 7

attained by this 9 ; and (d) for given , the small-

4×

Ö Ë ´¿ µ E ´f µ ÑaÜ f

est value of is 2 which is attained by the 9 . Under criteria ,

ÆÇD ÆÇD Æ

and ÑaÜ , these designs have a slightly better performance than that obtained by Yamada [½7] et al. [½6] and Fang, Lin and Ma through computer searches. More detailed comparisons

will be given in Section 3. In fact, many new designs are obtained by our new approach.

4×

Ë ´¿ µ´½ 6 × 6 7µ

Table 1 9 Row1234567 1 1321 3111 2231 2123 1213 2312 1132 2 1132 1321 3111 2231 2123 1213 2312 3 2312 1132 1321 3111 2231 2123 1213 4 1213 2312 1132 1321 3111 2231 2123 5 2123 1213 2312 1132 1321 3111 2231 6 2231 2123 1213 2312 1132 1321 3111 7 3111 2231 2123 1213 2312 1132 1321 8 3222 3222 3222 3222 3222 3222 3222

9 3333 3333 3333 3333 3333 3333 3333

´f µ ÑaÜ f

The paper is organized as follows. In Section 1, the deﬁnitions of E ,

ÆÇD ÆÇD Æ and ÑaÜ are introduced, some justiﬁcation for using them as design criteria is provided,

and some properties of these three criteria are obtained. In Section 2, the connection

´Ò; Õ ; Ö µ

between designs in Í and resolvable packing designs is established, and the

´f µ

combinatorial approach, i.e. the packing method, for constructing E -optimal su-

ÆÇD

´Ò; Õ ; Ö µ

persaturated designs in Í is proposed. In Section 2, some inﬁnite classes for

Ñ Ñ

Ë ´Õ µ Í ´Ò; Õ ; Ö µ the existence of Ò ’s in are also obtained, along with the discussion of properties of the resulting designs, and several such designs are tabulated. Comparisons between the new designs and other existing designs are made in the last section. Note that for the brevity of the main presentation, all proofs in this paper are given in the Appendix.

1 Design criteria

¾ Í ´Ò; Õ µ E ´f µ

For a design X , the criterion is deﬁned as minimizing

ÆÇD

E ´f µ = f

; where

ÆÇD

½ i

ij ´ij µ

; f = Ò

ÙÚ

ÆÇD

Ù;Ú =½

¾ ´ij µ

´Ù; Ú µ ´Ü ; Ü µ Ò=Õ Ò j

is the number of -pairs in columns i , and stands for the average

ÙÚ

´Ü ; Ü µ f j

frequency of level-combinations in each pair of i . The value gives a non-

ÆÇD

´Ü ; Ü µ Ü Ü

j i j

orthogonality measure for i . It is obvious that columns and are orthogonal

f E ´f µ ¼

if and only if = . A strict lower bound of is obtained by Fang, Lin and

ÆÇD ÆÇD

Liu[½8] as follows.

¾ Í ´Ò; Õ µ

Lemma 1. For any design X ,

k;Ð=½;k 6=Ð kÐ

· ´Ñ Ôµ Ô ´f µ =

E (1)

ÆÇD

Ñ´Ñ ½µ Ñ ½

Ò ÑÒ

¾ ¾

´Ô ½µ · ´Ñ Ôµ Ô ;

> (2)

´Ñ ½µ´Ò ½µ Ñ ½

Ô = Ò=Õ k Ð

where and kÐ is the number of coincidences between the -th and -th rows.

´f µ

The lower bound of E on the right-hand side of (2) can be achieved if and only if

ÆÇD

= Ñ´Ô ½µ=´Ò ½µ k 6= Ð

is a positive integer and all the kÐ ’s for are equal to .

´Ò; Õ µ E ´f µ

Obviously, a design in Í with its -value achieving the lower bound

ÆÇD

Ë ´Õ µ E ´f µ

in (2) is an Ò . For making further comparisons among -optimal designs,

ÆÇD

ÑaÜ f Æ

the two additional criteria, i.e. and ÑaÜ , are deﬁned as minimizing

ÆÇD

f = ÑaÜff j½ 6 i < j 6 Ñg;

ÑaÜ and

ÆÇD

Æ = ff = ÑaÜ f j½ 6 i < j 6 Ñg

ÑaÜ the frequency of

ÆÇD ÆÇD

respectively.

E ´f µ ÑaÜ f Æ

In order to show some justiﬁcation for using , and ÑaÜ as de-

ÆÇD ÆÇD sign criteria for supersaturated designs, let’s recall some existing criteria for comparing

supersaturated designs. For a two-level design X , the two levels are commonly denoted

½ ½ × ´i; j µ X E ´× µ X by and . Let ij be the -element of . The popular criterion, pro-

posed by Booth and Cox [¾] , is to minimize

¾ ¾

: × E ´× µ =

½ i

ÑaÜ j×j = ÑaÜfj× jj½ 6 i < j 6 Ñg f× = ij

The criteria ij and the frequency of

[¿;4] [5] [½4]

ÑaÜ j×jg ¦ are also widely used (e.g., Lin ;Wu ; Liu and Zhang ).

For three-level supersaturated designs, Yamada and Lin [½5] deﬁned a measure for de- www.scichina.com

132 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143

Ü Ü j

pendency between two factors i and by

´ij µ ¾

µ = Ò Ò=9 ´Ò=9µ: ; ´ Ü Ü

j i

ÙÚ

Ù;Ú =½

Then they deﬁned the following two criteria to evaluate the whole design X :

¾ ¾

; aÚe = ´Ü ; Ü µ

i j

½ i

¾ ¾

ÑaÜ = ÑaÜf ´Ü ; Ü µj½ 6 i < j 6 Ñg:

i j

Ñ ¾ [½7]

´Ò; Õ µ aÚe´f µ

For Ë ’s, Fang, Lin and Ma recommended the use of criterion

´f µ

which is similar to E , it is deﬁned by

ÆÇD

¾ ¾

; aÚe´f µ = f

½ i

where

¬ ¬

´ij µ

¬ ¬

Ò f =

ÙÚ

¬ ¬

Ù;Ú =½

ÑaÜ f = ÑaÜff j½ 6 i < j 6 Ñg

They also used ij as an additional criterion for their

algorithm.

¾ ¾ [½8]

´× µ

Fang, Lin and Liu showed that the E and ave criteria are in fact spe-

[½7]

´f µ

cial cases of E criterion, and Theorem 2.1 of Fang, Lin and Ma proved that

ÆÇD

¾ ¾ ¾

Úe´f µ = E ´× µ E ´f µ E ´× µ

a for a two level design. So is an extension of and

ÆÇD

¾ ¾

Úe´f µ aÚe ÑaÜ f

a for two-level designs, and for three-level designs. Similarly,

ÆÇD

j×j ÑaÜ f ÑaÜ

is an extension of ÑaÜ , and . These provide a strong justiﬁcation for

E ´f µ ÑaÜ f Æ

using , and also ÑaÜ as design criteria for supersaturated designs.

ÆÇD ÆÇD

These three criteria also have the following properties.

¾ Í ´Ò; Õ µ Ô = Ò=Õ Theorem 1. For any design X , let . Then the following

results are true.

E ´f µ ÑaÜ f Æ

(i) All three criteria , and ÑaÜ are invariant under exchanging

ÆÇD ÆÇD X

rows and columns of X and permuting levels of each column of .

ij ¾

< Ô ´Õ ½µ ½ 6 i 6= j 6 Ñ

(ii) f , for , and the upper bound is achieved when ÆÇD

the two columns are fully aliased.

¾ ij ¾

> Ò Ô ½ 6 i 6= j 6 Ñ Ò 6 Õ

(iii) f , for and , and the equality holds if

ÆÇD

Õ Ü Ü j

and only if each of the level-combinations appears at most once in columns i and .

¾ ij

Ò Õ Ü Ü f > 4 j

Furthermore, if is a multiple of , and i , are not orthogonal, then . ÆÇD

2 Design construction

´× µ

Many E -optimal designs have been constructed by a number of authors, for ex- [½6]

ample, refs. [3, 6, 9, 14]. Yamada and Lin [½5] and Yamada et al. presented two construc-

¾ ¾

Úe ÑaÜ tion methods for three-level supersaturated designs under a and criteria. It

can be veriﬁed that only the two designs with 9 runs of Yamada et al. [½6] achieve the lower

¾ [½7]

Úe E ´f µ

bound of a (also ). Fang, Lin and Ma obtained a new class of multi-

ÆÇD

Úe´f µ ÑaÜ f

level supersaturated designs under a and criteria. Many of their designs

´f µ

are E -optimal also. In this section, we will present a combinatorial approach for

ÆÇD

´f µ Ñ

constructing E -optimal supersaturated designs with the number of columns, , ÆÇD

achieving its upper bound shown in Theorem 3 and study the properties of the new de-

Í ´Ò; Õ ; Ö µ signs. Note that the upper bound of Ñ is obtained by relating a design in to

an important combinatorial design, namely a resolvable packing design.

´Ò; Õ ; Ö µ

2.1 Connection between designs in Í and resolvable packing designs

> Ô > Ø Ø ´Ò; Ô; ½µ

Let’s introduce some properties of packing designs. Let Ò .A -

Î ; B µ Î Ò B

packing design is a pair ´ , where is a set of elements and is a collection

Î Ø Î

of Ô-element subsets of , called blocks, such that every -element subset of occurs

Ø ´Ò; Ô; ½µ ´Î ; B µ B in at most one block of B .A- packing design is resolvable if can

be partitioned into parallel classes, each of which consists of Ò=Ô disjoint blocks. The

´Ò; Ô; Øµ Ø ´Ò; Ô; ½µ

packing number D is the maximum number of blocks in any - packing

´Ò; Ô; ½µ ´Î ; B µ jB j = D ´Ò; Ô; Øµ jB j design. And a Ø- packing design is optimal if , where

denotes the cardinality of the set B . For more discussions about packing designs, please

[¾¼]

Î ; B µ

refer to Stinson . Table 2 shows us a resolvable 2-(6,2,1) packing design ´ , where

= f½; ¾; ¿; 4; 5; 6g B Î , is partitioned into 5 parallel classes, each of which consists of 3

disjoint blocks of size 2, and every unordered pair of elements occurs in exactly one block B

of B . Note that this design is an optimal 2-(6,2,1) packing design, because contains all

k; Ðg B

the different blocks of size 2 and adding one more block, e.g. f ,to will cause pair

k; Ðg B

f appearing in two blocks of .

; ¾; ½µ

Table 2 A resolvable 2-´6 packing design

f g

Î 1, 2, 3, 4, 5, 6

B È È È È È

½ ¾ ¿ 4 5

f g f g f g f g f g

b 1,2 1,3 1,4 1,5 1,6

g f g f g f g f g f

b 3,4 2,5 2,6 2,4 2,3

g f g f g f g f g f

b 5,6 4,6 3,5 3,6 4,5

i j

b : -th block in the -th parallel class.

¾ Í ´Ò; Õ ; Ö µ Ü X Î =

Let X . For any column of , there exists a partition of

Õ Ô = Ò=Õ k Ü ½; ¾; ¡¡¡;Òg

f , blocks of size , such that if the -th element of takes level

k Ù Õ

Ù, then is contained in the -th block. Obviously, these blocks form a parallel class.

= ´½ ¾ ¿ 4 ½ ¾ ¿ 4 ½ ¾ ¿ 4µ = ½¾ Õ = 4 Ü

Take Ò , , and a column as an example,

½; 5; 9g f¾; 6; ½¼g f¿; 7; ½½g

this column corresponds to four blocks of size 3, f , , , and

4; 8; ½¾g Ñ X

f , which form a parallel class. Corresponding to the columns of , there are

Ô Ö ¾ 6 Ô 6 Õ

altogether ÕÑ blocks of size . The largest frequency being , where and

6 Ö < Ô ´Ö · ½µ Î

½ , means that any -element subset of appears in at most one of the

Ô ´Ö · ½µ ÕÑ blocks of size . Otherwise, if an -element subset appears in two blocks, then

www.scichina.com

134 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143

Ö · ½µ

there exists a level-combination that appears ´ times, which is a contradiction. Let

ÕÑ Ô

B be a collection of the blocks of size , then from the deﬁnition of resolvable packing

Î ; B µ ´Ö · ½µ ´Ò; Ô; ½µ

design, it is easy to see that ´ is a resolvable - packing design.

Ö · ½µ ´Ò; Ô; ½µ ´Î ; B µ

On the contrary, given a resolvable ´ - packing design , where

Ë Ë

¡¡¡ È Î = f½; ¾; ¡¡¡;Òg B = È È ´j = ½; ¡¡¡;Ñµ Ñ

Ñ j

, ½ , and are the parallel

Í ´Ò; Õ µ

classes of B ,a can be constructed as follows:

½; ¡¡¡;Õ Õ È j =

Step 1. Give a natural order to the blocks in each parallel class j ,

; ¡¡¡;Ñ

½ .

È Õ Ü = ´Ü µ Ü =

j kj kj

Step 2. For each j , construct a -level column as follows: set

k Ù È Ù = ½; ¾; ¡¡¡;Õ Ñ Õ Ù

,if is contained in the -th block in j , . The -level columns

È ´j = ½; ¡¡¡;Ñµ Í ´Ò; Õ µ

constructed from j form a .

´Ò; Õ µ Ö

It is easy to see the largest frequency of this Í is , i.e., this design is in

´Ò; Õ ; Ö µ Í . We call the above construction method the packing method. Based on these

discussions, we conclude that

´Ò; Õ ; Ö µ ¾ 6 Ô = Ò=Õ 6 Õ

Theorem 2. The existence of a design in Í , where and

6 Ö < Ô ´Ö · ½µ ´Ò; Ô; ½µ ½ is equivalent to the existence of a resolvable - packing design

with Ñ parallel classes.

[¾¼]

´Ò; Ô; Øµ Theorem 33.5 of Stinson gives us the upper bound of the packing number D .

That is

j kk k j j

Ò ½ Ò Ø·½ Ò

¡¡¡ ´Ò; Ô; Øµ 6

Lemma 2. D .

Ô Ô ½ Ô Ø·½

´Ò; Õ µ Õ Ô Note that each column of a Í corresponds to blocks of size . Then from this lemma and Theorem 2, we have the following theorem which gives the exact answer

for Question 1.

Ò; Õ ; Ö µ ¾ 6 Ô = Ò=Õ 6 Õ ½ 6 Ö < Ô Theorem 3. For given ´ satisfying and . The

upper bound of Ñ is given by

Ò Ö Ò ¾ Ò ½

¡¡¡ ; Ñ 6

Ô ½ Ô ¾ Ô Ö

Üc Ü

where b denotes the integer part of .

´Ò; Õ ; Ö µ

From Theorem 2, we can construct designs in Í from existing resolvable

Ò; Õ ; Ö µ Ñ packing designs. Especially, for given ´ we can construct designs with achiev-

ing its upper bound in Theorem 3 from resolvable optimal packing designs. Note that

´Ò; Õ ; Ö µ most of these designs in Í are supersaturated. Some existence results for re-

solvable optimal packing designs were given by Rees and Stinson [¾½] , Table 33.22 of [¾¾]

Stinson[¾¼] and Ge . From Theorem 2, the existence results for the corresponding de-

´Ò; Õ ; Ö µ signs in Í can be directly obtained, which are shown in table 3.

In the following subsections, we will mainly concentrate on the construction of super-

= ¿ Ö = ½; ¾ saturated designs for Ô and .

Optimal supersaturated designs 135

´Ò; Õ ; Ö µ Ö = ½ Ô = Ò=Õ = ¾; ¿; 4 Ñ = b´Ò ½µ=´Ô ½µc Table 3 Existence of designs in Í for , and

Ô Existence results

2 Ò 0 (mod 2)

Ò 6= 6; ½¾

3 Ò 0 (mod 3),

Ò > ¾4;Ò 6¾ f¾64; ¿7¾g

4 Ò 0 (mod 12),

4 Ò 4 (mod 12)

´Ò; Õ ;½µ

2.2 Designs in Í

´f µ

Here we consider E -optimal designs separately according to different cases

ÆÇD

Ô; Õ µ Ô = Õ Ö = ½

of the ´ values. Note that when and , the corresponding design is an

Ä Ñ 6 Õ · ½ ´Õ µ

orthogonal array Õ . From Theorem 3, , i.e., the maximum number of

Õ · ½ Õ

orthogonal factors that can be examined in Õ runs is less than or equal to .If is a

¾ ´Õ ·½µ ´Õ ·½µ

Í ´Õ ; Õ ;½µ Ä ´Õ µ prime power, then the designs in are in fact saturated Õ ’s. As for the existence of such orthogonal arrays, please see Theorem 3.20 of Hedayat, Sloane

and Stufken [½9] .

6 Ô 6 Õ Í ´Ò; Õ ;½µ Ñ 6 b ´Ò ½µ=´Ô ½µc

When ¾ , for designs in with , from

´f µ

Theorem 1 (iii) and the deﬁnition of E , we can easily obtain the following result.

ÆÇD

¾ Í ´Ò; Õ ;½µ ¾ 6 Ô = Ò=Õ 6 Õ Ñ 6 b ´Ò ½µ=´Ô ½µc

Theorem 4. Let X , and ,

ij Ñ ¾

E ´f µ= f ½ 6 i 6= j 6 Ñ X Ë ´Õ µ = Ò Ô

then for , and is an Ò .

ÆÇD

´Ò; Õ ;½µ E ´f µ

This theorem tells us that any design in Í is -optimal regardless ÆÇD

of whether Ñ achieves the upper bound or not.

= Ô < Õ Ö = ½ Ñ 6 ¾Õ ½

For the case of ¾ and ,wehave . As an example, Table

Ë ´¿ µ ´6; ¾; ½µ

4 gives us an 6 which is constructed from the resolvable optimal 2- packing

design shown in table 2.

Ë ´¿ µ

Table 4 6 derived from table 2 Row12345 1 11111 2 12222 3 21332 4 23123 5 32313

6 33231

= Ô 6 Õ Ö = ½ Ò For the case of ¿ and , from table 3, the three smallest are 9, 15 and 18. The designs with 9 and 15 runs will be given in the next subsection as they are

constructed from another kind of combinatorial design, which can be regarded as a special

6 8

kind of resolvable packing design. For the design with 18 runs, we have Ñ , the

E ´f µ Ë ´6 µ ´½8; ¿; ½µ

-optimal ½8 and the corresponding resolvable optimal 2- packing ÆÇD design (cf. Kotzig and Rosa [¾¿] ) are shown in tables 5 and 6 respectively. Combining Theorem 4 and table 3, we obtain the following result. www.scichina.com

136 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143

Ë ´6 µ

Table 5 ½8 derived from table 6 Row12345678 1 11111144 2 22162521 3 33614223 4 13226351 5 21243235 6 32421632 7 12333463 8 23355112 9 31532316 1044444411 1155435254 1266341556 1346553624 1454516562 1565154365 1645666136 1756622445

1864265643

; ¿; ½µ

Table 6 A resolvable optimal 2-´½8 packing design

f g

Î 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

B È È È È È È È È

½ ¾ ¿ 4 5 6 7 8

f g f g f g f g f g f g f g f g

b 1,4,7 1,5,9 1,2,15 1,3,14 1,6,12 1,8,16 8,9,10 2,4,10

g f g f g f g f g f g f g f g f

b 2,5,8 2,6,7 4,5,18 4,6,17 2,9,17 3,5,11 2,3,13 6,8,14

g f g f g f g f g f g f g f g b

f 3,6,9 3,4,8 7,8,12 7,9,11 5,7,13 4,9,15 5,6,16 3,7,18

g f g f g f g f g f g f g f g b

f 10,13,16 10,14,18 10,11,6 10,12,5 10,15,3 10,17,7 17,18,1 11,13,1

g f g f g f g f g f g f g f g f

b 11,14,17 11,15,16 13,14,9 13,15,8 11,18,8 12,14,2 11,12,4 15,17,5

g f g f g f g f g f g f g f g f

b 12,15,18 12,13,17 16,17,3 16,18,2 14,16,4 13,18,6 14,15,7 12,16,9

i j

b : -th block in the -th parallel class.

Ë ´Õ µ Ñ 6 b ´Ò ½µ=´Ò=Õ ½µc

Corollary 1. There exists an Ò with for all the

Õ; Òµ

pairs ´ satisfying:

= Ò=¾ Ò

(i) Õ , 0 (mod 2) or;

= Ò=¿ Ò Ò 6= 6; ½¾

(ii) Õ , 0 (mod 3), or;

= Ò=4 Ò Ò > ¾4;Ò 6¾ f¾64; ¿7¾g

(iii) Õ , 0 (mod 12), or;

= Ò=4 Ò

(iv) Õ , 4 (mod 12).

´Ò; Õ ;¾µ

2.3 Designs in Í

´f µ Í ´Ò; Õ ;¾µ

In this subsection we will construct E -optimal designs in via ÆÇD large sets of Kirkman triple systems. A large set of Kirkman triple systems is an important object in combinatorial design theory (see Stinson [¾4] ) which can be regarded as a

Copyright by Science in China Press 2004 Optimal supersaturated designs 137 kind of resolvable optimal packing design. It has played a crucial role in the construction

of secret sharing schemes in Cryptography (see Schellenberg and Stinson [¾5] ).

Ò ´Î ; B µ Î

A Steiner triple system of order Ò, denoted by STS( ), is a pair , where is

B Î

a set containing Ò elements and is a collection of 3-element subsets of , called triples B or blocks, such that every unordered pair of Î appears in exactly one triple. If can

be partitioned into disjoint parallel classes, we call the STS(Ò) a Kirkman triple system,

Ò Ò

which is denoted by KTS(Ò). A large set of KTS( ), denoted by LKTS( ), is a collection

Ò ¾µ Ò Î

of ´ pairwise disjoint KTS( )’s on the same set .

Òµ

Note that from the deﬁnition of Kirkman triple system, a KTS´ is in fact a resolv-

Ò; ¿; ½µ Ò´Ò ½µ=6

able optimal 2-´ packing design. In such a design, there are exactly

Ò ½µ

triples which contain all the unordered pairs, all the triples are partitioned into ´

Ò = ´Ò ¾µ[Ò´Ò ½µ=6] ¾

= parallel classes. As an LKTS( ) contains all the different

´Ò; ¿; ½µ

triples of Î , it is a resolvable optimal 3- packing design. The necessary condition

Ò ¿ for the existence of an LKTS(Ò)is (mod 6), but knowledge on the existence of

LKTS(Ò) is very limited. The known results can be summarized as follows (see Chang [¾7]

and Ge [¾6] ; Denniston and the references therein).

Ñµ k Ñ ¾f Lemma 3. There exists an LKTS´¿ for any positive integer and 1, 5,

11, 17, 25, 35, 43, 67g.

Let’s see an example of LKTS(9) (cf. Teirlinck [¾8] ).

= 9 Î = f½; ¾; ¿; 4; 5; 6; 7; 8; 9g 4

Example 1. For Ò , suppose , and parallel

½ ½ ½ ½

È È È

classes, denoted by È , , and , are

½ ¾ ¿ 4

½ ½

= ff¾; 5; 7g; f4; 6; 8g; f½; ¿; 9gg; = ff½; ¾; 4g; f¿; 5; 6g; f7; 8; 9gg; È È

¾ ½

½ ½

= ff½; 6; 7g; f¾; ¿; 8g; f4; 5; 9gg: = ff¿; 4; 7g; f½; 5; 8g; f¾; 6; 9gg; È È

4 ¿

Ë Ë Ë

½ ½ ½ ½

´Î ; B µ B = È È È È ½

Then ½ , where , is a KTS(9) which serves as an ini-

½ ¾ ¿ 4

tial KTS. Following the common method in the combinatorial design theory, we ap-

; ¾; ¿; 4; 5; 6; 7µ´8µ´9µ

ply the permutation group generated by ´½ to this KTS and obtain

¾ = 7

Ò pairwise disjoint KTS(9)’s, which form an LKTS(9). For example, based

; ¾; ¿; 4; 5; 6; 7µ´8µ´9µ

on the permutation group generated by ´½ , adding one to each ele-

½ ¾ ½

È = ff¾; ¿; 5g; f4; 6; 7g; f½; 8; 9gg È

ment of È gives a parallel class . And from ,

½ ½ ½

¿ 7

= ff¿; 4; 6g; f5; 7; ½g; f¾; 8; 9gg ¡¡¡ È = ff7; ½; ¿g; f¾; 4; 5g; f6; 8; 9gg

È , , can be

½ ½

¾ 7

È ¡¡¡; È

obtained in a similar way. These six parallel classes, ; , are the ﬁrst parallel

½ ½

´Î ; B µ; ¡¡¡; ´Î ; B µ 7

classes of other six KTS(9)’s, ¾ , respectively. Similarly we can get all

½ ½ ½

; È È

the other parallel classes of those six KTS(9)’s from È and . All the 7 KTS(9)’s

¿ ¾ 4

are shown in table 7. In each of the 7 KTS(9)’s, there are 4 parallel classes each composed

Î ¿6

of 3 triples, and each of the = unordered pairs of appears exactly once. And this

Î 84

LKTS(9) contains all the = different triples of .

Î = f½; ¾; ¡¡¡;Òg Ò Î ´Î ; B µ i = ½; ¡¡¡;Ò ¾

Suppose , for an LKTS( )on , let i , ,

Ë Ë

i i

¡¡¡ È ´Ò ¾µ Ò B = È

Ò ½

be the pairwise disjoint KTS( )’s, where i . Following the

½ ¾ www.scichina.com 138 Ë cieÒce iÒ C hiÒa Ser. A Mathematics 2004 Vol.47 No.1 128—143

Table 7 An LKTS(9)

Î f½; ¾; ¿; 4; 5; 6; 7; 8; 9g

i i i i

È È È È

½ ¾ ¿ 4

B ff gf gf gg ff gf gf gg ff gf gf gg ff gf gf gg

½ 1,2,4 3,5,6 7,8,9 2,5,7 4,6,8 1,3,9 3,4,7 1,5,8 2,6,9 1,6,7 2,3,8 4,5,9