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: c.suen@csuohio.edu
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 geome-
try . can be partitioned into

1,PG mns
1,PG mns
11
mn n
ss

–1n-flats such that each
–1n-
flat is associated with a point in . An orthogonal array
1, n
PG ms

11
mn n
mn
ss
n
s
Ls

can be constructed by
using

11s
mn n
s points in
1, n
P
Gm s. A set of
11s
t
s
points in
n
1,
P
Gms is called a
–1t-
flat over GF(s) if it is isomorphic to . If there exists a
1,PG ts
–1t-flat over GF(s) in , then we
can replace the corresponding
1, n
m s
PG


11
t
ss
sn-level columns in


11
mn n
mn
ss
n
s
Ls



by


11ss



n
st-level columns and obtain a mixed orthogonal array. Many new mixed orthogonal arrays can be obtained by this pro-
cedure. In this paper, we study methods for finding disjoint
–1t-flats over GF(s) in in order to con-
struct more mixed orthogonal arrays of strength two. In particular, if m and n are relatively prime then we can con-
struct an
1,PG ms
n




11
1is
m
s
1
is
s
1
nm
n
s






1
mn
m
mn
s
s
s
Ls
for any
 
11
11
n
ss
ss
0i
mn
m

. New orthogonal arrays of sizes 256, 512,
and 1024 are obtained by using PG(7, 2), PG(8, 2) and PG(9, 2) respectively.
Keywords: Finite Field; Finite Projective Geometry;
–1t-Flat over GF(s) in
1, n
PG ms; 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 interac-
tions of all factors are negligible.
Let
1
k
Ls s
1, ,i

denote an orthogonal arrays of strength
two with N rows, k columns, and si levels in the ith col-
umn for . In every N × 2 subarray of k
1
k
Ls s, all possible combinations of levels occur
equally often as rows. It is known that
1
i
Ns 
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 construct-
ing mixed orthogonal arrays of strength two have been
developed in [5-9], and many other authors. These meth-
ods use Hadamard matrices, difference schemes, gener-
alized Kronecker sums, finite projective geometries, and
orthogonal projection matrices. We refer to [10] for more
constructions and applications of orthogonal arrays.
1
in an
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 or-
thogonal arrays having two-level and four-level columns.
A systematic method [12] was developed for identifying
1
k
sLs and the orthogonal array is called
tight if the equality holds. Orthogonal array
1
k
Ls s
is called symmetric if 1k
s
s, otherwise it is called
C
opyright © 2012 SciRes. OJS
C.-Y. SUEN 189
disjoint sets of three two-level columns for constructing
LN(2m4n). The method was generalized in [6] for con-
structing , where s is a prime


1
1t
t
k
n
nr
r
m
s
Lss s
power. Mixed orthogonal arrays of strength t were con-
structed 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 con-
structed by this method are called geometric. Geometric
orthogonal arrays L64(8647), L64(83414), L64(8 441025) 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 geome-
tries 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
array


11
mn n
mn
ss
n
s
Ls

, 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 dif-
ferent from the grouping method in [6]. Hence we are
able to obtain some new series of mixed orthogonal ar-
rays.
2. Geometric Orthogonal Arrays
For r 1 and s a prime power, let denote
the -dimensional finite projective geometry over
the Galois field GF(s). A point in is de-
noted by an r-tuple
1,PG rs
1,PG rs
–1r
1,,
r
x
x
PGr
, where xi’s are elements
of GF(s) and at least one xi is not 0. Two r-tuples repre-
sent the same point in if one is a multiple


1, s
of the other. Hence there are 11ss
r points in
1,PG rs. A
–1t-flat in
1,PG rs is a set of


1
t
s1s
points whiche linear combinations of t ar
independent points. A spread of -flats of
is a set of -flats which partition
. It is known [15] that there exists a spread
of -flats of if and only if t divides r.
–1t
1,PG rs
1,PG rs

–1t
–1t
1,r sPG
We call a set of flats
,,
1k
F
F a mixed spread
of ,s if it partitions and at least
two flats in 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 lan-
guage.
1PGr
Theorem 1. Let
1,,
k
F
F be a (mixed) spread
of
1,PG rs
1, ,ik
, where Fi is a -flat for
–1
i
t
. Then we can construct an orthogonal array
1k
r
t
t
s
Ls s .
We now describe the procedure to construct the or-
thogonal array in Theorem 1. For , let Gi be an
r × ti matrix such that the ti columns are any choice of ti
independent points of the -flat Fi. Let G be the
1, ,i
k
t
–1
i
t
i
r
matrix
1,,
k
GG . The


k
r
t
s
Ls s
1
t con-
sists of sr rows which are the elements of the row space
of G, where the ti s-level columns of Gi is replaced by an
i
t
s
-level column for each . 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:
1, ,ik
1)

11
r
r
ss
s
Ls




;
2)

11
rt
r
ss
t
s
Ls

if t divides r;
3) if
r 2t; and

1rt
r
s
rt t
s
Ls s

4)

r
k
tl
s
Lss
, where,
 
11
jit t j
kssss 1

,
11
tj
lss s
, , . ritj 0 jt
3. Main Results
It is proved in Lemma 12 [13] that if V1, V2, V3 are three
disjoint
–1n-flats of then their union
can be regrouped into 2n – 1 disjoint 1-flats. Hence three
21,2PGn
2n-level columns in an can be replaced

2
21
22
n
n
n
L

by
21
n
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


11
t
ss



–1n-flats in
1,PG mns can be regrouped into a
set of

11
n
ss

–1t-flats.
Since there exists a spread of -flats of
–1n
1,PG mns, we can, by Theorem 1, construct an
 
1,PGr s

11
mn
n
mn
ss
n
s
Ls


. If there exist

11
t
ss



–1n-flats in the spread such that their union can be
regrouped into

11
n
ss

-flats, then we
can replace the corresponding
–1t

11
t
ss



sn-level
Copyright © 2012 SciRes. OJS
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190
columns in the


11
mn n
mn
ss
n
s
Ls



by


11
n
ss

st-level columns and obtain an
 
111
11
1
mn tn
n
mn
ss s
nt
ss
s
s
Ls s




. By repeating this process,
many orthogonal arrays can be obtained.
First we would like to establish a one-to-one corre-
spondence between the

1
mn n
ss1
disjoint
–1n-
flats in and the
1,PG mns

1
mn n
ss1
points in . Let ω be a
1, n
PG mns
primitive element of GF(sn) and let the minimum poly-
nomial of be 0

n
GF s1
11
nn
n

,
where 01
,,
n
are elements of GF(s). The compan-
ion matrix of the minimum polynomial is an n × n matrix
012 1
010 0
001 0
000 1
n
 




 
W

.
If ω is a primitive element of , then
are the sn elements of . The
elements of can be represented by n × n ma-
trices with entries from GF(s). The element ωi is repre-
sented by Wi, and the elements 0 and 1 are represented by
the zero matrix and the identity matrix respectively. De-
note the matrix representation of an element x in

n
GF s
2
0,1, ,,n
s

GF

n
GF s

n
s

n
GF s by W(x). Let each point

1,,
m
x
x

–1n
in
correspond to the -flat in
1,PGm
n
s
1,PG mns which consists of points that are linear
combinations of row vectors of the n × mn matrix


1,, m
Wx Wx
over GF(s). It can be shown that
the
11
mn n
ss


–1n-flats corresponding to
the
11
mn n
ss



points of parti-
1, n
PG ms
tion . This establishes a one-to-one corre-
1,PG mns
spondence between the

1
mn n
ss1
disjoint
–1n

-flats in and the
1,PG mns

11sPG
mn
sn
points in .

1, n
m s
Definition 1. A set of

1
t
ss1
points in
1, n
PG ms is said to be a
–1t-flat over GF(s) if it
is possible to find coordinates for this set of


1
t
ss1
points such that it is isomorphic to
1,PG ts over GF(s).
Note that whether a set of

1
t
ss1 points in
1, n
PG ms is isomorphic to over GF(s)
1,PG ts
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

33
11,, ,,,1S
 
in Example 1
(given after Theorem 2) is an 1-flat over GF(2) in
1, 8PG since it is isomorphic to
1, 2PG over GF(2).
But if we choose different coordinates for

24
11,, 1,, 1,S
 
, then it is not isomorphic to
1, 2PG over GF(2). Hence it is important to specify
the correct coordinates when a
–1t-flat over GF(s) in
1, n
PG ms is mentioned. Also we note that it is pos-
sible to have t > m for a
–1t-flat over GF(s) in
1, n
PG ms. For example, S1 and S2 in Example 2
(given after Theorem 2) are 2-flats over GF(2) in
1, 16PG.
We now give a sufficient condition that a set of
1
t
ss1
disjoint
–1n-flats in
1,PG mns

can be regrouped into a set of

11s
n
s disjoint
–1t-flats.
Theorem 2. A set of

1
t
ss1
disjoint
–1n-
flats in
1,PG mns can be regrouped into a set of
11s
disjoints
–1t-flats, if the set of
n
s
1
t
ss1
corresponding points in
1, n
PG ms
is a
–1t-flat over GF(s).
Proof. Let the coordinates of the


1
t
ss1 cor-
responding points of the
–1t-flat over GF(s) in
1, n
PG ms be
,,
1
j
mj
x
x for
1, ,1
t
js 1s
. Also let L be an

11
n
s
sn

matrix such that the rows are the
points of
1,PG ns. Then the
–1n-flat in
1, sPG mn corresponding to the point

1,,
j
mj
x
x
consists of points which are the rows of the

11
n
s
sm

n
matrix
1,,
jj
M LWxWx
mj
, where W(x) is the n × n
matrix representation of x. We can verify that for each
1, ,11
n
iss
, the set of


1
t
ss1
points
which consists of the ith rows of


111
,, t
ss
MM
is a
–1t-flat in PG(mn-1, s).
Note that in general there are more ways of regrouping
a set of
1
t
ss1
disjoint -flats in
–1n
Copyright © 2012 SciRes. OJS
C.-Y. SUEN
Copyright © 2012 SciRes. OJS
191
1,PG mns into disjoint flats if the

11
t
ss
corresponding points in
1, n
PG ms is a
-flat
010
001
110
W;
T
1001011
0101110
0010011
L.
–1t
over GF(s). Let Pij be the point in with
the ith row of Mj as its coordinates. The

1,PG mns
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.


11 11
nt
ss ss



array of points
has the following properties:
ij
P


P

100010
010001
001110
1, 110011
011111
111101
101100
WW













1
ML


3
010110
001011
110111
,011101
111100
101010
100001
WW















2
ML


3
110100
011010
111001
,1 101110
100011
010111
001101
WW














3
ML
,
,
.
1) Each row (column) of P is a -flat (
–1t
–1n-
flat).
2) If


1
u
ss1
points in a given row (column)
form a -flat, then the
1u

1
u
ss1
points at
the same positions in any other row (column) also form a
-flat.
1u
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 parti-
tioned into three 2-flat and eight 1-flats. Therefore, these
105 points can be grouped into: 1) 3-flats and 5i
1-flats for ; 2) 2-flats and 7i 1-flats
for ; or 3) four 3-flats, three 2-flats, and eight
1-flats.
7i
0,, 7i
,5
15 3i
0,i
We observe that for each , 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 corre-
sponding 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).
1,, 7i
Example 1. Let 6
0,1, ,,
1
be the 8 elements of
GF(8) with . 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
3




33
11,,,,, 1S
 
 


33
21,,,,,1S


,
, and

30,1 , 1,0,1,1S.
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 14
0,1, ,,
1

be the 16 elements of
GF(16) with 4
. 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 con-
struct an L256(1617) by Theorem 1. Let
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
 

792 1245 88210 11
11,,,,,,,1,,,,, ,S

,
 

123 2410 59813108
21, ,,,,,, ,,,,,,S

,
C.-Y. SUEN
192



 
 


 

44 2288
12 3
44 8822
45
0, 1,1,0,1, 1,1,,,,,1,1,,,,,1,
1,, ,,,1,and1,,,, ,1.
TT T
TT
  
 
 

We can verify that S1 and S2 are disjoint 2-flats and
are disjoint 1-flats over GF(2) in PG(1, 16).
6
l arrays. Let
let the m ×
15
Moreover, S1, S2, and T1 partition PG(1, 16). By the dis-
following Theorem 2, we can replace the subar-
ray 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 15
,,TT
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(16118 416), , can be obtained by this procedur
4. Construction of More Orthogonal Arrays
,,TT
cussion
e.
In this section, methods for finding disjoint flats over
GF(s) in

1, n
PG ms are developed to construct
more orthogona a primitive element of α be
GF(sm), andm matrix representation of α in
GF(s) be W. Since


11
m
ss

is an element of GF(s)

and

11
m
ss
W is tepresentation of


he matrix r
1
m
s
have
1s
, we



1
m
ss
m is the m × m identity matrix. Then for any fixed point
1 11
m
ss
m
WI
 
, where
I

1,,
m
x
x
x
SxWi
x in PG

1, n
m s
ins at mos
, the set

:0
inta

cot
1s
since
1s points
m
in
1, n
PG ms

11
xW 
(
m
ss
ij ijl
x
WxWxWWxW
 
 
i j
for some l, since
βW + γW is the matrix representation of ti
+ γαj of
he element βα
m
GF s. S has the structure o
xf a flat over GF(s)
in
1, n
P
Gm since linear combinations of points in
Sx are also poiSx. In fact, Sx is a -flat over
GF
s

–1tnts in
(s) in
1, n
s if and only if the number of PG m
points in Sx is
11s
t
s
for some integer t. Now if
x and y are two

1,
points in n
P
Gm s and
xy
SS
, then there ethat xWi j
We have ij
xist i and j such = yW .
x
y
xW S
y. , hence Sx = S
Theorem 3. Let x be a point in

1, n
PG ms, and let
:0xWi
i
x
S
. Then Sx is a GF(s)

–1t-flat over
in
1, n
P
Gm s
if and only if the points in
Sx is
number of
11
t
ss
for some integer t. Moreover, for
any two n

1, n
PG ms either Sx = Sy
or xy
SS
points x and y i
. Hence

1, n
P
Gm s can be parti-
Ee illustrate how we obtain the three dis-
tioned into disjoint sets of Sxs.
xample 3. W
, 8) in Example 1. Let ω
be
joint 1-flats over GF(2) in PG (1
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
01
W


11ss
m
11ss
mm
x
Ix

 
 rep
xW then
) and xresent the same
point. Moreover, if β and γ are any elements of GF(s) and
i and xWj are elements of Sx,
11
.
Then

2
0,1 0,1 ,0,1,0,1SWW, 0,1, 0


, 1,1 , 1

23
1, 1,, 1,, 1,1,,,,SWW
 


3
,1
, and

3
33 323
1, 1, ,1,,1,1, ,,SWW
 

3
, ,
1
are three disjoint 1-flats over GF(2) in PG(1, 8).
Example 4. Let ω be a primitive element of GF(16)
GF(4) with ω4 = ω + 1, and let α be a primitive element of
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):

0,10,1,1,1,1,0S,


44
1, 1,, ,,S


,
1,
2
228
1, 1,,,,S

8
, ,1


4
44
1, 1,,,,,1S

,

8
8822
1, 1,, ,,,1S


,
5
5
1, 1,S
, and

10
1,
S
ve disjoint 1-flats over GF(2
ple 2.
Theorem 4. If s is a prime power a
relatively prime, then we can construct
10
e
nd m an
d orth
1, .
The fi in
Exam
d n are
ogonal
arrays
) ar
mixe
15
,,TT
Copyright © 2012 SciRes. OJS
C.-Y. SUEN 193






1 1
mn nn
mn
sss iss
nm
s
Ls s
 


1 111
m
is

for

0,,i

1 111
mnm n
ssss
 
.

 
Proof. We can construct an


11
mn n
mn
ss
n
Ls


s
from . From the proof of Theorem.6
[15], if m and n are relatively prime then S is an
-flat over GF(s) in

1, n
PG ms 4.3
x

1, n
m1
P
Gm s
1,PG ms
foevery
. Hence can be
r
point x in
pa

1, n
PG ms

n
rtitioned into

11
mn
ss


1mF(s). Each Sx represents
1
m n
ss
1

-flats over G

11
m
ss



sn-level columns in


11
mn n
mn
ss
n
s


placed by
Ls

, and by Theorem 2 it can be re-

11
n
ss sm-level columns.
The eorem 4 is
a general
Corollary 1. If d is the greatest
common divisor of integers m and n, then we can con-
struct mixed orthogon
following result which follows from Th
ization of Theorem 4.
s is a prime power and
al arrays




11 1
mn nd
ss s
n
s
 
11 1
m dn
mn d
issis
m
s
L s
 

for
0,,11
mn dd
iss
1 1ss

.
Proof. If d is the greatest common divisor of m and n,
then
m n

md and nd are relativ
ing m, n, and s with
ely prime. By substitut-
md, nd,
xed ort
and sd respectively in
Theorem 4, we obtain the mihogonal arrays.
By using Theorem 4 and Corollary 1, we obtain the
following new series of tight orthogonal arrays for any
e power s.
prim
1)



23
6
3
11
1
32
11
is is
s
s
s
ss s







,
0 i s2s + 1;
61s
L
2)



25
10
5
10
11
1
52
11
1
isis
s
ss
s
s
Ls s





,

5
01is s ; 1
3)



34
12
12
11
1
1
is is
s
s
s
L






,
4
43
1
1s
s
s s

422
01iss ss ;
4)



46
12
62 2
12
11
1
64
11 1
is is
s
ss s
s
Lss

 





,
0 i s4s2 + 1;
5)



27
14
7
14
11
1
72
11
1
isis
s
ss
s
s
Ls s







,
7
01iss1
 ; and
6)



35
15
5
15
11
1
53
11
1
is is
s
ss
s
s
Ls s







,
 
10 52
01is s1ss
.
The following theorem gives a set of s – 1 disjoint
–1
n-flats over GF(s) in PG(1, sn).
Theorem 5. For 0, ,2
is
, let

,: \0
is n
i
TGFs
 

tive element of
, where ω is a primi-
n
GFs. Then 02
,,
s
TT
, are s – 1 dis-
joint
–1n-
Proof. T
flats over GF(s) in PG ).
i is a set of
(1, sn

11
n
ss points in PG(1,
sn), since

,
s
i
 
(

,is

) represents the
sa
sh
me point for each nonzero element α of GF(s). To
ow that Ti is an
–1n-flat over GF(
binati ts in T
s), we prove that
any linear comelemen is again in T. If
on of i i
112 2
,,,
isis
i
T
 
and

12
,n
GF s

, then
 
112 2
,,
isi s
s
i
 

11 22
,.
i
T
 
1112 22


For 0 i <
j s – 2, if
1
is
i
T

1
,d an
22
,js
j
T

represent the same points in PG(1, sn),
then –1 1
12
is js

. Hence

1
12
s
ji

. But


1s1
sk
 
1
12
for so
me
0 11 , 1
n
ks s
which contradicts 0 i < j s – 2. Hence Ti and Tj are
disjoint for all 0 i < j s – 2.
Corollary 2.





1
2
111 1
n
s1
1
nn
n
sis sis
nn
s
Ls s
 



any integer n, wer s, and can be constructed for prime po
1, ,1is
.
Pro truct an om PG(1,of. We can cons
fr
sn). For each

1
n
s
n

2n
s
Ls
0,is, 2
, let be an
ii
TT
2n
-
flat over GF(s) in PG(1, sn).

:0,,2
i
Ti s
 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
Copyright © 2012 SciRes. OJS
C.-Y. SUEN
Copyright © 2012 SciRes. OJS
194


111
n
ss



sn-level columns in
by

11
n
ss

sn-1-level columns to obtain the

2
1
n
n
s
n
s
Ls
orthogonal array.
ith ω4 = ω + 1.
Example 5. Let ω be the primitive element of GF(16) w





  
 
22 24 36 48510 612 714
810511 712913 1114 13
,:16\01,1,,,,,,,,,,,,,,,
,,,,,,,,,,,,,
TGF
 
   
 

0
9 3
 
is a 3-flat over GF(2) in PG(1

, 16) and
224485108
0 1,1,,,,,,,,,,T
  
105
0
,,

T
is a 2-flat over GF(2)
Note that we arnt 2-flats ove
FPG 1, 16n Example 2 by trial andror.
H
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 se-
ries of tight orthogonal arrays for any prime power s and
1)
in PG(1, 16).
e able to find two disjoir




54 5
10
111 1
54
sis sis
s
Ls s
 



;
G(2) in () i er
owever, we do not have a method to find more than s-1
1, , 1
is.





8
43
s
Ls s


;
2)
3)




65 6
12
1111
65
sis sis
s
L

s s


; and
4)




76 7
14
111 1
76
sis sis
s
Ls s
 



.
43 4
1111sis sis 

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.
nd
\0
n
GF s
Theorem 6. For any integer n 2 a,
 
,,:
s
TGFs
 

is an n-flat over GF(s) in PG(2, sn).
However, for β1 β2 the n-flats over GF(s) 1
T
,,, 0,0
n
GFs
 

fin ore disjoint n-flats over GF(2) in 2n).
Theorem 7. Let ω be a primitive element of GF(2n),
ant
d mPG(2,
and
2
T
are not disjoint in PG(2, sn). But if s = 2, we can
d le
 
2
,, :2,2,,0,0
n
SGFGF
 

,

2
,, :2,
n
TGFG
 
2,, 0,0F

,
 
2:2,2,, 0,0
n
GFF
 

,,UG , and
 
22
,,:2,2,, 0,0
n
VGFGF
 
.
G(n
fo
2
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 dis-
joint. Assume that and
represent the same point in PG(2, 2n),
and . Clearly,
2n – 1. For any

2\0
n
GF
, γ3 = ω3k for some
Then we have
1) S and T are disjoint n-flats over GF(2) in P2, 2)
r n 2.
) T, U and V are three disjoint n-flats over GF(2) in
02131
n
k
 . Assume that

2
22 2
,, T
 
and
2
33 3
,, U

represent the sam 2n),
where
e point in PG(2,
23
,2GF

γ3 0, hence α2

23
,2
n
GF

1 and

2
22
,,1

and
=
. Clearly, α2,
α3, γ2, α3 = and

2
111
,, S

2
33
,,
representme point. We have γ2 = γ3
and
the sa

2
22 2
,, T
 
where 12

1

,2GF

12
,2
n
GF

α, α2, γ1, γ2 0, hence α1 = α2 = 1 and

2
11
,,
and

2
,,1

22 represent the same point. We have
12
 
and 1
12

, which imply ω = 1, a contra-
diction. Hence S and T are disjoint. Now we show that T
U are disjoint if n is even. If n is even then 3 divides and
2
23

, whichply 33
2
k

 im for some
02131
n
k
 , a contradiction. Hence T and U are
disjoint. We can sim T and V
ilarly prove thatare disjoint
and that U and V are disjoint if n is even.
onstructed fr
sn). By applying Theorems 2, 6, and 7, we obtain the fol-
An
21
nn
ss
can be com PG(2,

3n
n
s
Ls


C.-Y. SUEN 195
lowing orthogonal arrays.

Corollary 3. For any prime power s, we can construct
1)





2
11 1
nnn
ss ssss 

3n
s
Ls
2)
ω3 = ω + 1. Let
1
nn
s


, 2n;
Example 6. Let ω be the primitive element of GF(8) with

1
2
1
2
n
n
L



2
2 2323
2
nn
n

, 2n; and
3
2n
3)
 
24
6
323252 4
21 2
222
nnn
n
nn
L 


, 1n




4 5365
, ,0,,,0,,,
 


2
243 4
1,1,0, ,,0,0,0,0,0,1,
1,1,1,,,1,,,1,,,1,,
 
 

 
 
26
 
5365
,1,,,1,, ,1
 
and
2436
,, ,0,,S
 

   
224364 5365
22 43 65 36 5
1,0,1,,0,, ,0,,,0,, ,0,,,0,, ,0,,0,,0,
,,,,,,,,,,,,
T
 
    
be two disjo
e constru ). We can
rray L(8 o S or T by an
to
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
4
,
1,,1,,,,,
int 3-flats over GF(2) in PG(2,8). An L512(873)
can bcted from PG(2,8replace the
ba 15) corresponding t L(167) su 512 512
obtain L512(167858) and L512(1614843).
223
11,,,0,1,,1,, 0A
 
,
2446
11,,,0,1,,1,,0B
 
,

45
11,,,0,1, ,1,,0C
 
,
and W be the 3 × 3 matrix defined in E 1. For xample
2,, 7i
plying each
, let Ai (or Bi, Ci) be the set obtained by multi-
element in Ai (or Bi, Ci) by W. For Example,
2232622
21,,,0,1,,1,, 0,,,,,1,0,1,AWWW
   

3
.
can be verified thatIt 17
,,
A
A, ,
G(2, 8) by an
L
ment of GF(32)
w
1)
,
1717
and


1, 0, 0,0,1, 0,1,1, 0 are 22 disjoint 1-flats over
GF(2) in PG(2, 8). An L512(873) can be constructed from
PG(2, 8). We can replace the subarray L512(83) corre-
sponding to each 1-flat over GF(2) in P
,,BB,,,CC
512(47) to obtain L512(873-3i47i) for 1,,22i.
Example 8. Let ω be the primitive ele
ith ω5 = ω2 + 1. An L1024(3233) can be constructed from
G(1, 32). P


11, 0,0,1,1,1A

18
, ,

18
21,,A
 
,1 ,
2255
, ,1

,
31,, ,A

44 1010
41,, ,,,1A
 
,
5522
51,, ,,,1A

,
88 2020
61,,,,,1A
 
,
991616
71,, ,,,1A
 
,
1010 44
1,,,, ,1A

8,
1414 1313
91,,,, ,1A

,
1616 99, and
10 1,,,,,1A

1919 11
11 1,,,, ,1A

n disjoint 1-flats over GF(2) in
place the subarray L1024(323) corres
GF(2) in the L1024(3233) by
L1024(3233-3i163i416i) for 1,,i
11
are eleve(1, 32). We
can reding to each
1-flat over L1024(163416)
to obtain
PG
po
a
1
n
n
1.
2)
212318 262
11,, ,,,,,B

255192811 12
,,1 ,,,,
 
,
2581872 145219411 29
21,,,,, ,,,,,, ,,B
 
,
25 18216515 1924 119
31,,,,1 ,,,,,,,,B
 
, and
,
621817 219 5201915112
41,, ,,,,,,,,,,,B
   
5
G(1,32)
2
are four disjoint 2-flats over GF(2) in P. We can
replace the subarray L1024(327) corresponding to each 2-
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.
Copyright © 2012 SciRes. OJS
C.-Y. SUEN
196
3)
32729176227132030861214235
11 ,,,,,,,,,,,,,,,CB
  

and
 
 
 
3182996 12272620248 1112 17236
22 ,,,,,,,,,,,,,,CB
  
,
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 su-
barray L1024(3215) corresponding to C1 or C2 in the
1024(3233) by an L1024(1624815) or an L1024(1631) to obtain
5. Discussion
We use t-flats over GF(s) in
L
L1024(32181624815), L1024(3231648830), L1024(32181631),
L1024(3231662), and L1024(3231655815).

1, n
P
Gm s to find dif-
erent ways to regroup a set of -flats in f

–1n

1, n
P
Gm s into disjoint flats. However, many prob-
lems remain unsolved. For example, we do not know
how many disjoint

–2n-flats over GF(s) exist in
PG(1, sn). Since there are sn + 1
(
 

21
111
n
s
sss s
 ) points in PG(1, sn),
the upper bound for the number of disjoint –2n-fl
over GF(s) equals s2-s if n 4 nd equa
ats
als s2f n =
njecture is that PG(1, sni-

s + 1 i
) can be part3. An obvious co
tioned into
2
s
s

–2n
ss
-flats and over
r n = 3, si
1-
nd h are
shown in Exam
one 1-flat
n
hic
GF(s). This conjecture is true foce PG(1, s3)

2rGF(s) by can be partitioned into flats ove
Theorem 4. It is also true for s = n = 4, 5, w
1
2 a
ple 2 for n = 4 and shown in Example 8(3)
for n = 5. If the conjecture is true, we can construct an





1
2
11 11
nnn
n
sissis s
nn
s
Ls s
 


for n 3 and
1
1
s, which would be
2
1,, si a significant improve-
ment 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


211 1
111
nnnn nnn
ss ssss
 
.
Hence an upper bound for the number of disjoint n-flats
over GF(s) in PG(2, sn) is snsn–1 if n 3 and is s2s +
1 if n = 2. The upper bound is attained for n = 2, since
n
1ss 
PG(2, s) can be partitioned into 2-flats
over GF(s) by Theorem 4. In general
tween the upp
ver GF(s) in PG(2, sn) for n 3.
Another problem which cannot be solved by the ap-
proach of this paper is the cuction of orthogonal
arrays having sn rows, wher a prime number. For
example, it is known that L128
16) can be constructed
by a mixed spread of PG(6, 2), wh consists of a 3-flat
and 16 2-flats . But it is not known that if it is possible to
could constructL128(16 83i47i) if there exist i
such disjoint sets of three 2-flats.
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
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(1618
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Copyright © 2012 SciRes. OJS