On a Grouping Method for Constructing Mixed

Open Journal of Statistics, 2012, 2, 188-197 http://dx.doi.org/10.4236/ojs.2012.22022 Published Online April 2012 (http://www.SciRP.org/journal/ojs)

On a Grouping Method for Constructing Mixed Orthogonal Arrays Chung-Yi Suen Department of Mathematics, Cleveland State University, Cleveland, USA Email: [email protected] Received January 20, 2012; revised February 19, 2012; accepted March 8, 2012

ABSTRACT Mixed orthogonal arrays of strength two and size smn are constructed by grouping points in the finite projective geometry PG  mn  1, s  . PG  mn  1, s  can be partitioned into  s mn  1 s n  1   n –1 -flats such that each  n –1  s mn 1  sn 1   flat is associated with a point in PG m  1, s n . An orthogonal array Ls mn  s n  can be constructed by  







 s

using s mn  1

n











 



 1 points in PG m  1, s n . A set of

 s  1  s  1



points in PG m  1, s n

t

flat over GF(s) if it is isomorphic to PG  t  1, s  . If there exists a

 t –1 -flat over GF(s) in



is called a



 t – 1 -



PG m  1, s n , then we

 smn 1  sn 1   n sn-level columns in Ls mn  s n  by  s  1  s  1    st-level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this procedure. In this paper, we study methods for finding disjoint  t –1 -flats over GF(s) in PG m  1, s n in order to con-



  s  1



 

can replace the corresponding  s t  1







struct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can conn i  s m 1   s mn 1 i  s 1 s mn  1  s  1 m s m 1  s 1 n   1 s  struct an Ls mn s for any 0  i  m . New orthogonal arrays of sizes 256, 512, s   s 1 sn 1   and 1024 are obtained by using PG(7, 2), PG(8, 2) and PG(9, 2) respectively.

 

 

 

Keywords: Finite Field; Finite Projective Geometry;

 



 t – 1 -Flat over GF(s) in





PG m  1, s n ; Geometric Orthogonal

Array; Matrix Representation; Minimal Polynomial; Orthogonal Main-Effect Plan; Primitive Element; Tight

1. Introduction Orthogonal arrays of strength two are used as orthogonal main-effect plans in fractional factorial experiments. In an orthogonal main-effect plan, the main effects of each factor can be optimally estimated assuming the interactions of all factors are negligible. Let LN  s1  sk  denote an orthogonal arrays of strength two with N rows, k columns, and si levels in the ith column for i  1, , k . In every N × 2 subarray of LN  s1  sk  , all possible combinations of levels occur equally often as rows. It is known that N  1    si  1 in an LN  s1  sk  and the orthogonal array is called tight if the equality holds. Orthogonal array LN  s1  sk  is called symmetric if s1    sk , otherwise it is called Copyright © 2012 SciRes.

asymmetric or mixed. Symmetric orthogonal arrays have been constructed in [1-3]. Mixed orthogonal arrays were introduced in [4], and they have drawn the attentions of many researchers in recent years. Methods for constructing mixed orthogonal arrays of strength two have been developed in [5-9], and many other authors. These methods use Hadamard matrices, difference schemes, generalized Kronecker sums, finite projective geometries, and orthogonal projection matrices. We refer to [10] for more constructions and applications of orthogonal arrays. The method of grouping was used in [11] to replace three two-level columns in symmetric orthogonal arrays by one four-level column for constructing mixed orthogonal arrays having two-level and four-level columns. A systematic method [12] was developed for identifying OJS

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disjoint sets of three two-level columns for constructing LN(2m4n). The method was generalized in [6] for con-



 

structing Ls k s m s r1

n1

 

 s rt

nt

 , where s is a prime

power. Mixed orthogonal arrays of strength t were constructed by using mixed spreads of strength t in finite geometries in [13]. This method was also independently discovered in [14] for constructing mixed orthogonal arrays of strength three and four. Orthogonal arrays constructed by this method are called geometric. Geometric orthogonal arrays L64(8647), L64(83414), L64(8441025) and L64(8141725) were constructed in [13]. However, the method is restricted to constructing mixed orthogonal arrays with the number of levels in each column a power of 2. In this paper, we shall use finite projective geometries to develop a general procedure for constructing more mixed orthogonal arrays. Moreover, the procedure allows us to construct mixed orthogonal arrays with the number of levels in each column a power of any given prime number. We start with a symmetric orthogonal  s mn 1  sn 1   array Ls mn  s n  , and then construct mixed   orthogonal arrays by replacing a group of columns with another group of columns. Our grouping method uses properties of finite projective geometries, which is different from the grouping method in [6]. Hence we are able to obtain some new series of mixed orthogonal arrays.

 

2. Geometric Orthogonal Arrays For r ≥ 1 and s a prime power, let PG  r  1, s  denote the  r –1 -dimensional finite projective geometry over the Galois field GF(s). A point in PG  r  1, s  is denoted by an r-tuple  x1 , , xr  , where xi’s are elements of GF(s) and at least one xi is not 0. Two r-tuples represent the same point in PG  r  1, s  if one is a multiple of the other. Hence there are s r  1  s  1 points in





PG  r  1, s  . A  t – 1 -flat in PG  r  1, s  is a set of s t  1  s  1 points which are linear combinations of t





independent points. A spread ℱ of  t –1 -flats of PG  r  1, s  is a set of  t –1 -flats which partition PG  r  1, s  . It is known [15] that there exists a spread ℱ of  t –1 -flats of PG  r  1, s  if and only if t divides r. We call a set of flats    F1 , , Fk  a mixed spread of PG  r  1, s  if it partitions PG  r  1, s  and at least two flats in ℱ have different dimensions. Mixed spreads are useful for constructing mixed orthogonal arrays of strength two. Specifically, we give the following theorem for constructing an orthogonal array from a (mixed) spread. The theorem is the special case of strength two of Theorem 2.1 [14] in finite projective geometry’s language. Copyright © 2012 SciRes.

Theorem 1. Let    F1 , , Fk  be a (mixed) spread of PG  r  1, s  , where Fi is a  ti – 1 -flat for i  1, , k . Then we can construct an orthogonal array Ls r

  s    s   . tk

t1

We now describe the procedure to construct the orthogonal array in Theorem 1. For i  1, , k , let Gi be an r × ti matrix such that the ti columns are any choice of ti independent points of the  ti – 1 -flat Fi. Let G be the r   ti matrix G1 , , Gk  . The Ls r

 s    s   t1

tk

con-

r

sists of s rows which are the elements of the row space of G, where the ti s-level columns of Gi is replaced by an s ti -level column for each i  1, , k . We call orthogonal arrays geometric if they can be obtained by Theorem 1. Geometric orthogonal arrays have been constructed in [1, 9,13,16]. Examples of geometric orthogonal arrays are:   s r 1  s 1  1) Ls r  s ;  

 sr 1  st 1   2) Ls r  s t  if t divides r;   r t 1 s   3) Ls r  s r  t s t  if r ≥ 2t; and  

 

  

4) Ls r

 s  s  , where, t

k

l

   s  1  s  1 , l  s  s  1  s  1 , r  it  j , 0  j  t . k  s j s it  1

t

t

j

j

3. Main Results It is proved in Lemma 12 [13] that if V1, V2, V3 are three disjoint  n – 1 -flats of PG  2n  1, 2  then their union can be regrouped into 2n – 1 disjoint 1-flats. Hence three 2n 1   2n-level columns in an L22 n  2n  can be replaced   n by 2  1 4-level columns. By applying this result to a spread of 2-flats of PG(5, 2), L64(8647) and L64(83414) were constructed. Generalizing the idea, we would like to find a sufficient condition that a set of  s t  1  s  1   n – 1 -flats in PG  mn  1, s  can be regrouped into a set of  s n  1  s  1   t – 1 -flats. Since there exists a spread of  n – 1 -flats of PG  mn  1, s  , we can, by Theorem 1, construct an  smn 1  sn 1   t Ls mn  s n  . If there exist  s  1  s  1   

 

















 

 n – 1 -flats in the spread such that their union can be regrouped into  s n  1  s  1   t – 1 -flats, then we can replace the corresponding  s t  1  s  1  sn-level









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 smn 1  sn 1   columns in the Ls mn  s n  by    s n  1  s  1  st-level columns and obtain an   mn t s  1 s 1 s n 1    Ls mn  s n s n 1 s 1 s t s 1  . By repeating this process,     many orthogonal arrays can be obtained. First we would like to establish a one-to-one correspondence between the s mn  1 s n  1 disjoint  n – 1 -

 





 

 





flats in PG  mn  1, s  and the

s

mn

 s

1

n



1







points in PG mn  1, s n . Let ω be a

primitive element of GF(sn) and let the minimum polynomial of GF s n be  n   n 1 n 1    1   0 , where  0 , ,  n 1 are elements of GF(s). The companion matrix of the minimum polynomial is an n × n matrix

 

 0  0  W     0   0

1

0

0

1



 0

 0

 

1



 2 

  PG  m  1, s  n

 

PG  mn  1, s  which consists of points that are linear combinations of row vectors of the n × mn matrix W  x1  , , W  xm   over GF(s). It can be shown that

 

 

 

the  s mn  1 s n  1   n – 1 -flats corresponding to the  s mn  1 s n  1  points of PG m  1, s n partition PG  mn  1, s  . This establishes a one-to-one correspondence between the s mn  1 s n  1 disjoint

 n – 1 -flats in

s

mn



1









PG  mn  1, s  and the s  1 points in PG m  1, s n . n



Definition 1. A set of









 s  1  s  1 t



points in

PG m  1, s is said to be a  t – 1 -flat over GF(s) if it is possible to find coordinates for this set of s t  1  s 1 points such that it is isomorphic to



n



PG  t  1, s  over GF(s).

Copyright © 2012 SciRes.

n



points in

is isomorphic to PG  t  1, s  over GF(s)

depends not only on the choice of the points but also on the choice of the coordinates for these points. For exam-









ple, the set S1  1,   ,  ,  3 ,  3 ,1

in Example 1

(given after Theorem 2) is an 1-flat over GF(2) in PG 1,8  since it is isomorphic to PG 1, 2  over GF(2). But if we choose different coordinates for







S1  1,   , 1,  2 , 1,  4

 , then it is not isomorphic to

PG 1, 2  over GF(2). Hence it is important to specify the correct coordinates when a  t – 1 -flat over GF(s) in PG m  1, s n is mentioned. Also we note that it is pos-









sible to have t > m for a  t – 1 -flat over GF(s) in PG m  1, s n . For example, S1 and S2 in Example 2



can be regrouped into a set of

 n – 1 -flat in

correspond to the



PG m  1, s



If ω is a primitive element of GF s n , then s n 2 are the sn elements of GF s n . The 0,1,  , ,  elements of GF s n can be represented by n × n matrices with entries from GF(s). The element ωi is represented by Wi, and the elements 0 and 1 are represented by the zero matrix and the identity matrix respectively. Denote the matrix representation of an element x in GF s n by W(x). Let each point  x1 , , xm  in

 

t

(given after Theorem 2) are 2-flats over GF(2) in PG 1, 16  . We now give a sufficient condition that a set of s t  1  s  1 disjoint  n – 1 -flats in PG  mn  1, s 

0  0   .  1   n 1 

 

 s  1  s  1

Note that whether a set of

 t – 1 -flats. Theorem 2. A set of

s

n

  s  1

disjoint

disjoint

 n – 1 -

1

 s  1  s  1 t

flats in PG  mn  1, s  can be regrouped into a set of s n  1  s  1 disjoints  t – 1 -flats, if the set of

   s  1  s  1



corresponding points in PG m  1, s n

t

is a  t – 1 -flat over GF(s). Proof. Let the coordinates of the

 s  1  s  1 t



cor-

responding points of the  t – 1 -flat over GF(s) in PG m  1, s n be  x1j , , xmj  for







  s  1 . Also let L be an

j  1, , s  1









t

 s n  1  s  1   n matrix such that the rows are the   points of PG  n  1, s  . Then the  n – 1 -flat in PG  mn  1, s  corresponding to the point  x1j , , xmj  consists of points which are the rows of the  s n  1  s  1   mn matrix  

M j  L W  x1j  , , W  xmj   , where W(x) is the n × n matrix representation of x. We can verify that for each i  1, , s n  1  s  1 , the set of s t  1  s  1 points









which consists of the ith rows of M 1 , , M st 1 s 1 is a     t – 1 -flat in PG(mn-1, s). □ Note that in general there are more ways of regrouping a set of s t  1  s  1 disjoint  n – 1 -flats in





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PG  mn  1, s  into disjoint flats if the





 s  1  s  1

T

t

0 1 0  1 0 0 1 0 1 1    W   0 0 1  ; L  0 1 0 1 1 1 0  . 1 1 0  0 0 1 0 0 1 1 

 t – 1 -flat PG  mn  1, s  with

corresponding points in PG m  1, s n

is a

over GF(s). Let Pij be the point in the ith row of Mj as its coordinates. The  s n  1  s  1    s t  1  s  1  array of points     P   Pij  has the following properties:









1) Each row (column) of P is a  t – 1 -flat (  n – 1 flat). 2) If s u  1  s  1 points in a given row (column)



form a



 u  1 -flat,

then the

s

u

  s 1

1

points at

the same positions in any other row (column) also form a  u  1 -flat. For example, if there exists a 2-flat over GF(2) in PG(1, 16), then each of the 7 points in the 2-flat over GF(2) corresponds to 15 points in PG(7, 2). The 105 points in PG(7, 2) corresponding to the 2-flat over GF(2) in PG(1, 16) can be arranged into a 15 × 7 array such that each row is a 2-flat and each column is a 3-flat. Since a 3-flat can be partitioned into five 1-flats, the 15 × 7 array of points can be partitioned into five 3 × 7 subarrays such that each column is a 1-flat and each row is a 2-flat. Also, consider a 15 × 3 subarray of the 15 × 7 array such that each row is a 1-flat. We can select a 7 × 3 subarray such that each column is a 2-flat. Each of the remaining eight rows is a 1-flat. Hence the 15 × 3 subarray can be partitioned into three 2-flat and eight 1-flats. Therefore, these 105 points can be grouped into: 1)  7  i  3-flats and 5i 1-flats for i  0, , 7 ; 2) 15  3i  2-flats and 7i 1-flats for i  0, ,5 ; or 3) four 3-flats, three 2-flats, and eight 1-flats. Example 1. Let 0,1,  , ,  6 be the 8 elements of GF(8) with  3    1 . Consider PG(1, 8) with nine points (0, 1), (1, 0), (1, 1), (1, ω), (1, ω2), (1, ω3), (1, ω4), (1, ω5) and (1, ω6). Each point of PG(1, 8) corresponds to a 2-flat in PG(5, 2), and the nine 2-flats partition PG(5, 2). We can construct an L64(89) by Theorem 1. Let

      1,   ,  ,   ,  ,1 , and

S1  1,   ,  ,  3 ,  3 ,1 , S2

3

3

S3   0,1 , 1, 0  , 1,1 .

We can verify that S1, S2, and S3 are disjoint 1-flats over GF(2) in PG(1, 8). The 3 × 3 matrix representation W of ω and the 7 × 3 matrix L given in the proof of Theorem 2 are

The three points of S1 correspond to the three 2-flats in PG(5, 2) which are rows of the following three matrices M1, M2, and M3 respectively. 1 0 0 0 1 0  0 1 0 0 0 1    0 0 1 1 1 0   M1  L W 1 , W     1 1 0 0 1 1  , 0 1 1 1 1 1    1 1 1 1 0 1  1 0 1 1 0 0    0 0  1  3 M 2  L W   , W     0   1  1 1 

1 0 1 1 1 0 0

0 1 0 1 1 1 0

1 0 1 1 1 0 0

1 1 1 0 0 1 0

0 1  1  1 , 0  0 1 

1 0  1  3 M 3  L W  , W 1   1   1  0 0 

1 1 1 0 0 1 0

0 1 1 1 0 0 1

1 0 0 1 0 1 1

0 1 0 1 1 1 0

0 0  1  0 . 1  1 1 

 

 

We observe that for each i  1, , 7 , the ith rows of M1, M2, and M3 are three points of a 1-flat in PG(5, 2). Hence we can replace the three 8-level columns corresponding to S1 in L64(89) by seven 4-level columns to obtain an L(8647). Continuing this procedure, we can replace the three 8-level columns corresponding to S2 in L64(8647) by seven 4-level columns to obtain an L64(83414). □ Note that L64(8647) and L64(83414) were also construct in [13] using a different method. However, Theorem 2 is more versatile as shown in following example. Example 2. Let 0,1,  , , 14 be the 16 elements of GF(16) with  4    1 . Consider PG(1, 16) with 17 points (0, 1), (1, 0), (1, 1), (1, ω),  , (1, ω14). Each point of PG(1, 16) corresponds to a 3-flat in PG(7, 2), and the seventeen 3-flats partition PG(7, 2). We can construct an L256(1617) by Theorem 1. Let

           1,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,   ,

S1  1,  7 ,  ,  9 ,  2 , 12 ,  4 ,1 ,  5 ,  8 ,  8 ,  2 , 10 , 11 , S2 Copyright © 2012 SciRes.

12

3

2

4

10

5

9

8

13

10

8

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

        1,   ,  ,   ,  ,1 .  xW   xW  x   W   W   xW





T1   0,1 , 1, 0  , 1,1 , T2  1,   ,  ,  4 ,  4 ,1 , T3  1,  2 ,  2 ,  8 ,  8 ,1 ,









T4  1,  4 ,  4 ,  ,  ,1 , and T5

We can verify that S1 and S2 are disjoint 2-flats and T1 , , T5 are disjoint 1-flats over GF(2) in PG(1, 16). Moreover, S1, S2, and T1 partition PG(1, 16). By the discussion following Theorem 2, we can replace the subarray L256(167) corresponding to S1 or S2 in L256(1617) by L256(1648348) or L256(815-3i47i), 0 ≤ i ≤ 5. Similarly, we can replace the subarray L256(163) corresponding to T1 , , T5 in L256(1617) by L256(8348). Many mixed orthogonal arrays such as L256(1610815), L256(163830), L256(161081247), L256(161089414), L256(16148348), L256(16781848), L256(161186416),  , can be obtained by this procedure. □

8

in



Sx is



 sm 1  s 1

xW

t

  s  1

1



s

t

 t –1 -flat over GF(s)   if and only if the number of points in

  s  1

1

for some integer t. Moreover, for







either Sx = Sy can be parti-

0 1 W  . 1 1 Then

2

3

      1,   ,  ,   ,  ,1 ,  

S1,   1,   ,  ,  ,  ,1 , 4

Copyright © 2012 SciRes.



2

S 0,1   0,1 , 1,1 , 1, 0  ,

2

and

 0,1 ,  0,1W ,  0,1W    0,1 , 1,1 , 1, 0 ,   1,   , 1,   W , 1,   W   1,   ,  ,   ,  ,1 , and  1,   , 1,   W , 1,   W   1,   ,  ,   ,  ,1   3

3

are three disjoint 1-flats over GF(2) in PG(1, 8). □ Example 4. Let ω be a primitive element of GF(16) with ω4 = ω + 1, and let α be a primitive element of GF(4) with α2 = α + 1 and matrix representation W given in Example 3. The 17 points of PG(1, 16) can be partitioned into the following flats over GF(2):

2



tioned into disjoint sets of Sx’s. Example 3. We illustrate how we obtain the three disjoint 1-flats over GF(2) in PG(1, 8) in Example 1. Let ω be a primitive element of GF(8) with ω3 = ω + 1, and let α be a primitive element of GF(4) with α2 = α + 1 and matrix representation

 sm 1  s 1

S 1, 3







i

 sm 1  s 1

S1,

for some integer t. Now if



  s  1 points

S 0,1 





or S x  S y   . Hence PG m  1, s n

(  x Im   x ) and x represent the same point. Moreover, if β and γ are any elements of GF(s) and xWi and xWj are elements of Sx, then

S 1, 2

since linear combinations of points in

any two points x and y in PG m  1, s n

S x  xW : i  0 contains at most s m  1 in PG m  1, s n

 is  s

in PG m  1, s n

Im is the m × m identity matrix. Then for any fixed point x   x1 , , xm  in PG m  1, s n , the set



n

S x  xW : i  0 . Then Sx is a

 s 1  s 1 is an element of GF(s)

  since

for some l, since

S x  S y   , then there exist i and j such that xWi = yWj. We have y  xW i  j  S x , hence Sx = Sy. Theorem 3. Let x be a point in PG m  1, s n , and let

 s 1  s 1 is the matrix representation of  sm 1  s 1 , we have W  s m 1  s 1    sm 1  s 1 I , where  m



  PG  m  1, s 

x and y are two points in PG m  1, s n

m



l

j

points in Sx





j

Sx are also points in Sx. In fact, Sx is a  t –1 -flat over GF(s) in PG m  1, s n if and only if the number of

m

i

i

βW + γW is the matrix representation of the element βαi + γαj of GF s m . Sx has the structure of a flat over GF(s)

more orthogonal arrays. Let α be a primitive element of GF(sm), and let the m × m matrix representation of α in

and W

2

j

i

In this section, methods for finding disjoint flats over GF(s) in PG m  1, s n are developed to construct

GF(s) be W. Since 

2

i

4. Construction of More Orthogonal Arrays



8

4

8

8

3

2

3

3

3

      1,   ,  ,   ,  ,1 ,    1,   , and S  1,   .     

S 1, 4  1,  4 ,  4 ,  ,  ,1 ,



S 1,8



S 1,5





8

8

2

2

5

10

1,10

The five disjoint 1-flats over GF(2) are T1 , , T5 in Example 2. □ Theorem 4. If s is a prime power and m and n are relatively prime, then we can construct mixed orthogonal arrays OJS

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 

 Ls mn  s n 

  sn 1 i  sm 1  s 1

s  

s mn 1







 

 





for i  0, ,  s mn  1  s  1   s m  1 s n  1  .  s mn 1  Proof. We can construct an Ls mn  s n 

 sn 1 

 



4  s12 1 i  s 1 6 s 6 1  s 2 1  4) Ls12 s s4   0 ≤ i ≤ s4 – s2 + 1; 2  s14 1 i  s 1  7 5) Ls14  s s7 1 s 1 s 2  

  s 1 

i s n 1

m

 

 



from PG m  1, s n . From the proof of Theorem 4.3.6









point x in PG m  1, s n . Hence PG m  1, s n













m



 

s

n

  s  1

1

 

 Ls mn d  s n 

s

  s 1 i  s 1  s 1 n

1



m

d



s   m

n

  s 1 

i s 1

 



d

 



for i  0, ,  s mn d  1 s d  1   s m  1 s n  1  . Proof. If d is the greatest common divisor of m and n, then m d and n d are relatively prime. By substituting m, n, and s with m d , n d , and sd respectively in Theorem 4, we obtain the mixed orthogonal arrays. □ By using Theorem 4 and Corollary 1, we obtain the following new series of tight orthogonal arrays for any prime power s. 2 i  s 3 1   s 6 1 i  s 1 3 s3 1  s 1 2  1) Ls6 s s s 1  ,     0 ≤ i ≤ s2 – s + 1; 2 i  s 5 1   s10 1 i  s 1 5 s 5 1  s 1 2  2) Ls10 s s s 1  ,    

 

 

 

 



  s  1 ;

0  i  s5  1

 3) Ls12  s 4  

 





3 s12 1 i s 1  4 s 1 s 1



s  3



s 1



,  



0  i  s4  s2  1 s2  s  1 ;

Copyright © 2012 SciRes.

,  





i s 7 1

,  

s 1

  s  1 ; and





3 s15 1 i s 1  5 s 1 s 1

s 







i s5 1

3

,  

s 1

 s

2



 s 1 .

The following theorem gives a set of s – 1 disjoint n  – 1 -flats over GF(s) in PG(1, sn). Theorem 5. For i  0, , s  2 , let





   GF  s  . Then T , , T

Ti   ,  i  s :   GF s n \ 0 , where ω is a primin

s 2

0

, are s – 1 dis-

joint  n – 1 -flats over GF(s) in PG(1, s ). Proof. Ti is a set of s n  1  s  1 points in PG(1, n



n

s ), since

  ,     (    ,    ) represents the s

i

i

s

same point for each nonzero element α of GF(s). To show that Ti is an  n – 1 -flat over GF(s), we prove that any linear combination of elements in Ti is again in Ti. If  1 ,  i  1s ,  2 ,  i  2 s  Ti and 1 ,  2  GF s n , then







 

1   1 ,  i  1s    2   2 ,  i  2 s 





 1 1   2 2 ,  i 1 1   2 2   Ti . s

For 0 ≤ i < j ≤ s – 2, if



2



 ,     T i

1

s 1

i

and

,  j  2 s  T j represent the same points in PG(1, sn),

then   i

 1  2 

s –1 1 s 1

  j  2s 1 . Hence  j i    1  2 



k  s 1



s 1

. But

  s  1  1 ,

for some 0  k  s  1 n

which contradicts 0 ≤ i < j ≤ s – 2. Hence Ti and Tj are disjoint for all 0 ≤ i < j ≤ s – 2. □ s n 1 i  s n 1 1  s 1 i  s n 1  s 1   Corollary 2. Ls 2 n  s n s n 1    can be constructed for any integer n, prime power s, and i  1, , s  1 . s n 1   Proof. We can construct an Ls 2 n  s n  from PG(1,   n  s ). For each i  0, , s  2 , let Ti  Ti be an  n  2  flat over GF(s) in PG(1, sn). Ti : i  0, , s  2 is a set of s – 1 disjoint (n-2)-flats over GF(s) in PG(1, sn). Then for each Ti  we replace the corresponding

 





 



i s 4 1



s 2 1

0  i  s10  s 5  1

tive element of

sm-level columns. □

The following result which follows from Theorem 4 is a generalization of Theorem 4. Corollary 1. If s is a prime power and d is the greatest common divisor of integers m and n, then we can construct mixed orthogonal arrays mn

 



n



placed by

 6) Ls15  s 5  

can be

partitioned into  s  1  s  1   s  1 s  1   m  1 -flats over GF(s). Each Sx represents  s m  1  s  1  sn-level columns in    n  s mn 1  s n 1  Ls mn  s  , and by Theorem 2 it can be re  mn

 



0  i  s7  1

[15], if m and n are relatively prime then Sx is an  m 1 -flat over GF(s) in PG m  1, s n for every



 

i s 6 1

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C.-Y. SUEN

  s  1

 s n 1  1 

 sn-level columns in Ls 2 n  s n 

 

  

s n 1





by  s n  1  s  1  sn-1-level columns to obtain the orthogonal array. □

Example 5. Let ω be the primitive element of GF(16) with ω4 = ω + 1.



 



















T0   ,  2 :   GF 16  \ 0  1,1 ,  ,  2 ,  2 ,  4 ,  3 ,  6 ,  4 ,  8 ,  5 , 10 ,  6 , 12 ,  7 , 14 ,

 ,   ,  ,   ,  8

9

3

10







, ,  , ,  , ,  , 5

11

7

12

9

13

11

 , 

14

,

13



is a 3-flat over GF(2) in PG(1, 16) and















T0  1,1 ,  ,  2 ,  2 ,  4 ,  4 ,  8 ,  5 , 10 ,  8 ,  , 10 ,  5

is a 2-flat over GF(2) in PG(1, 16). □ Note that we are able to find two disjoint 2-flats over GF(2) in PG(1, 16) in Example 2 by trial and error. However, we do not have a method to find more than s-1 disjoint (n-2)-flats over GF(s) in PG(1, sn). With n = 4, 5, 6 and 7 in Corollary 2, we obtain the following new series of tight orthogonal arrays for any prime power s and i  1, , s  1 . s 4 1 i  s3 1  s 1 i  s 4 1   s3 1) Ls8  s 4 ;  

 

 

 

  T

0



  s 1

s  





  s 1

s  



 2) Ls10  s 5 

s 5 1 i s 4 1

 3) Ls12  s 6 

s 6 1 i s 5 1

   



6 5 i s 1

; 

 ; and 

s 7 1 i  s 6 1  s 1 i  s 7 1   4) Ls14  s 7 s6 .   The following theorem gives an n-flat over GF(s) in PG(2, sn). The proof is omitted since it is similar to that of Theorem 5.

 

Theorem 6. For any integer n ≥ 2 and   GF s n \ 0 ,



5 4 i s 1

 



 

T   ,  s ,  :   GF s n ,   GF  s  ,  ,     0, 0 

is an n-flat over GF(s) in PG(2, sn). However, for β1 ≠ β2 the n-flats over GF(s) T1 and T 2 are not disjoint in PG(2, sn). But if s = 2, we can

find more disjoint n-flats over GF(2) in PG(2, 2n). Theorem 7. Let ω be a primitive element of GF(2n), and let

     T    ,  ,   :   GF  2  ,   GF  2  ,  ,     0, 0  , U    ,  ,   :   GF  2  ,   GF  2  ,  ,     0, 0  , and V   ,  ,   :   GF  2  ,   GF  2  ,  ,     0, 0  . S   ,  ,  2 :   GF 2n ,   GF  2  ,  ,     0, 0  , 2

n

2

n

2

2

n

Then we have 1) S and T are disjoint n-flats over GF(2) in PG(2, 2n) for n ≥ 2. 2) T, U and V are three disjoint n-flats over GF(2) in PG(2, 2n) if n is even. Proof. By Theorem 6, S, T, U and V are n-flats over GF(2) in PG(2, 2n). We now prove that S and T are disjoint. Assume that  1 , 1 ,  12  S and



2







,  2 2 ,  2  T represent the same point in PG(2, 2n),

   ,  ,  

where 1 ,  2  GF  2  and  1 ,  2  GF 2n . Clearly,

α1, α2, γ1, γ2 ≠ 0, hence α1 = α2 = 1 and

1

2 1

and

 2 ,  22 ,1 represent the same point. We have

  1   2 and  1  1  2 , which imply ω = 1, a contradiction. Hence S and T are disjoint. Now we show that T and U are disjoint if n is even. If n is even then 3 divides Copyright © 2012 SciRes.

 

2n – 1. For any   GF 2n \ 0 , γ3 = ω3k for some









0  k  2  1 3  1 . Assume that  2 ,  2 2 ,  2  T and



2 3

n



,  3 ,  3  U represent the same point in PG(2, 2n),

   ,  ,1

where  2 ,  3  GF  2  and  2 ,  3  GF 2n . Clearly, α2,

α3, γ2, γ3 ≠ 0, hence α2 = α3 = 1 and



2 3

,,  3



2

2 2

and

represent the same point. We have γ2 = γ3

and     3 , which imply    23   3k for some 0  k  2n  1 3  1 , a contradiction. Hence T and U are 2 2





disjoint. We can similarly prove that T and V are disjoint and that U and V are disjoint if n is even. □ s 2 n  s n 1   An Ls3 n  s n  can be constructed from PG(2,   sn). By applying Theorems 2, 6, and 7, we obtain the fol-

 

OJS

195

C.-Y. SUEN

lowing orthogonal arrays. Corollary 3. For any prime power s, we can construct  sn 1  s 1 n s2 n  sn  s   s 1   1) Ls3 n  s n 1 s , n  2;  





 2) L23 n  2n 1 



 



2n1  2

3) L26 n   22 n 1 

2 

322 n  3



n

22 n  3 2n  3

2 

  , n  2 ; and 

4n n 2 n 2  5 2  4

, n 1  

Example 6. Let ω be the primitive element of GF(8) with ω3 = ω + 1. Let



       and 1,1,1 ,  ,  ,1 ,  ,  ,1 ,  ,  ,1 ,  ,  ,1 ,  ,  ,1 ,  ,  ,1 T  1, 0,1 ,  , 0,   ,  , 0,   ,  , 0,   ,  , 0,   ,  , 0,   ,  , 0,   ,  0,  , 0  , 1,  ,1 ,  ,  ,   ,  ,  ,   ,  ,  ,   ,  ,  ,   ,  ,  ,   ,  ,  ,   S  1,1, 0  ,  ,  2 , 0 ,  2 ,  4 , 0 ,  3 ,  6 , 0 ,  4 ,  , 0 ,  5 ,  3 , 0 ,  6 ,  5 , 0 ,  0, 0,1 , 2

2

2

4

2

2

3

4

6

3

2

4

4

6

4

3

6

 

3

5

3

6

5

5

3

6

5

    B  1,  ,   ,  0,1,   , 1,  , 0  , C  1,  ,   ,  0,1,   , 1,  , 0  , A1  1,  ,  2 , 0,1,  2 , 1,  3 , 0 , 2

4

4

4

5

1

and W be the 3 × 3 matrix defined in Example 1. For i  2, , 7 , let Ai (or Bi, Ci) be the set obtained by multiplying each element in Ai (or Bi, Ci) by W. For Example,





A2  1,  ,  2 W , 0,1,  2 W , 1,  3 , 0 W   2 ,  6 ,  ,  2 ,  2 ,1 , 0,1,  3

A5  1,  5 ,  5 ,  2 ,  2 ,1 ,

GF(2) in PG(2, 8). An L512(873) can be constructed from PG(2, 8). We can replace the subarray L512(83) corresponding to each 1-flat over GF(2) in PG(2, 8) by an L512(47) to obtain L512(873-3i47i) for i  1, , 22 . □ Example 8. Let ω be the primitive element of GF(32) with ω5 = ω2 + 1. An L1024(3233) can be constructed from PG(1, 32). 1)

4

10

20

20

9

9

16

16

10

10

4

14

14

13

13

4

8

9

16

16

9

9

19

19

11

11

10

11

     A  1,   ,  ,   ,  ,1 , A  1,   ,  ,   ,  ,1 , 4

8

7

A2  1,   ,  , 18 , 18 ,1 , 5

8

6

A1  1, 0  ,  0,1 , 1,1 ,

2

 .

      A  1,   ,  ,   ,  ,1 , A  1,   ,  ,   ,  ,1 , A  1,   ,  ,   ,  ,1 , A  1,   ,  ,   ,  ,1 , A  1,   ,  ,   ,  ,1 , and A  1,   ,  ,   ,  ,1

It can be verified that A1 , , A7 , B1 , , B7 , C1 , , C7 , and 1, 0, 0  ,  0,1, 0  , 1,1, 0  are 22 disjoint 1-flats over

2

6

1

  

 

6

5

4

be two disjoint 3-flats over GF(2) in PG(2,8). An L512(873) can be constructed from PG(2,8). We can replace the subarray L512(815) corresponding to S or T by an L512(167) to obtain L512(167858) and L512(1614843). □ The following two examples are obtained by applying Theorems 3 and 5 and by trial and error. Example 7. Let ω be the primitive element of GF(8) with ω3 = ω + 1. Let



5

are eleven disjoint 1-flats over GF(2) in PG(1, 32). We can replace the subarray L1024(323) corresponding to each 1-flat over GF(2) in the L1024(3233) by an L1024(163416) to obtain L1024(3233-3i163i416i) for i  1, ,11 .

5

3

10

4

2)

     ,       1,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,   ,  1,   ,  ,   ,  ,1 ,  ,   ,  ,   ,  ,   ,  ,   , and  1,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,   ,  ,  

B1  1,  21 ,  ,  23 , 18 ,  26 ,  2 ,  25 ,  5 ,1 , 19 ,  28 , 11 , 12 B2

B3 B4

25

8

2

5

6

25

18

7

18

18

2

2

17

16

2

are four disjoint 2-flats over GF(2) in PG(1,32). We can replace the subarray L1024(327) corresponding to each 2Copyright © 2012 SciRes.

14

5

5

19

2

15 5

19

19

20

4

11

24

19

11

15

29

9

11

22

flat over GF(2) in the L1024(3233) by an L1024(167816) or an L1024(831) to obtain L1024(3233-7i-7j167i816i+31j) for 1 ≤ i + j ≤ 4. OJS

196

C.-Y. SUEN

3)

   B   ,   , 

  , 

  , 

  , 

   ,  ,   , 

 ,   , 

 and ,  

C1  B1   3 ,  27 ,  29 , 17 ,  6 ,  2 ,  27 , 13 ,  20 ,  30 ,  8 ,  6 , 12 , 14 ,  23 ,  5 C2

3

18

29

2

,9

6

, 12

27

,  26

are two disjoint 3-flats over GF(2) in PG(1,32), where B1 and B2 are 2-flats over GF(2) in 2). Moreover, C1, C2, and A1 in 1) partition PG(1, 32). We can replace the subarray L1024(3215) corresponding to C1 or C2 in the L1024(3233) by an L1024(1624815) or an L1024(1631) to obtain L1024(32181624815), L1024(3231648830), L1024(32181631), L1024(3231662), and L1024(3231655815). □

5. Discussion



We use t-flats over GF(s) in PG m  1, s n



20

,  24

8

11

12

17

23

6

proach of this paper is the construction of orthogonal arrays having sn rows, where n is a prime number. For example, it is known that L128(161816) can be constructed by a mixed spread of PG(6, 2), which consists of a 3-flat and 16 2-flats . But it is not known that if it is possible to find, among those 16 2-flats, disjoint sets of three 2-flats such that each set can be regrouped into seven 1-flats. We could construct an L128(161816-3i47i) if there exist i such disjoint sets of three 2-flats.

to find dif-

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ferent ways to regroup a set of  n – 1 -flats in PG m  1, s n into disjoint flats. However, many prob-

[1]

lems remain unsolved. For example, we do not know how many disjoint  n – 2  -flats over GF(s) exist in PG(1, sn). Since there are sn + 1 (  s 2  s s n 1  1  s  1  s  1 ) points in PG(1, sn),

R. C. Bose and K. A. Bush, “Orthogonal Arrays of Strength Two and Three,” The Annals of Mathematical Statistics, Vol. 23, No. 4, 1952, pp. 508-524. doi:10.1214/aoms/1177729331

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Y. S. Zhang, Y. Q. Lu and S. Q. Pang, “Orthogonal Arrays Obtained by Orthogonal Decomposition of Projection Matrices,” Statistica Sinica, Vol. 9, No. 2, 1999, pp. 595-604.

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C. Suen and W. F. Kuhfeld, “On the Construction of Mixed Orthogonal Arrays of Strength Two,” Journal of Statistical Planning and Inference, Vol. 133, No. 2, 2005, pp. 555-560. doi:10.1016/j.jspi.2004.03.018











the upper bound for the number of disjoint  n – 2  -flats over GF(s) equals s2-s if n ≥ 4 and equals s2 – s + 1 if n = 3. An obvious conjecture is that PG(1, sn) can be partitioned into s 2  s  n – 2  -flats and one 1-flat over GF(s). This conjecture is true for n = 3, since PG(1, s3) can be partitioned into s 2  s  1 1-flats over GF(s) by Theorem 4. It is also true for s = 2 and n = 4, 5, which are shown in Example 2 for n = 4 and shown in Example 8(3) for n = 5. If the conjecture is true, we can construct an s n 1 i  s n 1 1  s 1 i  s n 1  s 1   Ls 2 n  s n s n 1  for n ≥ 3 and   i  1, , s 2  s , which would be a significant improvement of Corollary 2. Also we do not know how many disjoint n-flats over GF(s) exist in PG(2, sn). The number of points in PG(2, sn) is







 











  s  1  s

s 2n  s n  1  s n  s n 1 s n 1  1

n

 s n 1  1 .

Hence an upper bound for the number of disjoint n-flats over GF(s) in PG(2, sn) is sn – sn–1 if n ≥ 3 and is s2 – s + 1 if n = 2. The upper bound is attained for n = 2, since PG(2, sn) can be partitioned into s 2  s  1 2-flats over GF(s) by Theorem 4. In general, the difference between the upper bound and what can be obtained in Theorems 6 and 7 is considerably large for n ≥ 3. There may be better ways to find disjoint n-flats over GF(s) in PG(2, sn) than the approach used in Theorems 6 and 7. So far, we do not know any example having sn – sn–1 disjoint n-flats over GF(s) in PG(2, sn) for n ≥ 3. Another problem which cannot be solved by the ap-



Copyright © 2012 SciRes.



[10] A. S. Hedayat, N. J. A. Sloane and J. Stufken, “Orthogonal Arrays,” Springer, New York, 1999. doi:10.1007/978-1-4612-1478-6 [11] S. Addelman, “Orthogonal Main Effect Plans for AsymOJS

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[15] J. W. P. Hirschefld, “Projective Geometries over Finite Fields,” Oxford University Press, Oxford, 1979.

[13] E. M. Rains, N. J. A. Sloane and J. Stufken, “The Lattice of N-Run Orthogonal Arrays,” Journal of Statistical Planning and Inference, Vol. 102, No. 2, 2002, pp. 477-500. doi:10.1016/S0378-3758(01)00119-7

[16] C. Suen and A. Dey, “Construction of Asymmetric Orthogonal Arrays through Finite Geometries,” Journal of Statistical Planning and Inference, Vol. 115, No. 2, 2003, pp. 623-635. doi:10.1016/S0378-3758(02)00165-9

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