75

CHAPTER-IV

CONSTRUCTION OF NEIGHBOUR DESIGNS FOR OS2 SERIES USING

PROJECTIVE GEOMETRY

4.1 Introduction

A finite geometry (Euclidean Geometry and Finite Projective Geometry) is any

geometric system that has only a finite number of points and finite number of dimensions.

Projective geometry was developed by Desargues. In the early 19th

century the work of

Poncelet, Von Staudt and others established projective geometry as an independent field of

Mathematics. Sir Ronald A. Fisher was one of the key people in establishing Statistics as a

Mathematics discipline. He gave many results depended on a statistical sample of s points

as a vector in s- dimensional Euclidean space and found that for every s equal to the power

of a prime there is a finite affine plane of order s and a finite projective plane of order s. He

has quoted that “Design theory encompasses many notations besides affine and projective

geometries. Design theory developed from mathematician’s natural inclination to

generalize and explore and to meet the needs of applications, especially designs for

statistical experiments.”

Designs over finite fields were introduced by Thomas in 1986. He constructed the

first nontrivial 2-designswhich are also called BIBD, over a finite field which is a design

with parameters 2-(v,3,7;2) for v ≥7 &v ≡ ±1 mod 6 and has used a geometric construction

in a projective plane. In 1989, Suzuki extended Thomas family of 2-designs to a family of

designs with parameters 2 -(v,3, s2+s+1; s) admitting a Singer cycle. In 1995 Miyakawa,

Munemasa and Yoshiara gave a classification of 2-(7,3, λ, s) designs for s= 2,3 with small

λ. In 2005 Kerber, Laue and Braun published the first 3-design over a finite field, a 3-(8, 4,

11; 2) design admitting the normalize of a Singer cycle, as well as the smallest 2-design

known as design with parameters 2-(6, 3, 3; 2).

76

A projective plane geometry is a nonempty set (whose elements are called "points"),

along with a nonempty collection of subsets of set (whose elements are called "lines"), such

that:

i) Given any two distinct points, there is exactly one line that contains both points.

ii) The intersection of any two distinct lines contains exactly one point.

iii) There exists a set of four points, no three of which belong to the same line.

An examination of the first two axioms shows that they are nearly identical, except that

the roles of points and lines have been interchanged. This suggests the principle of duality

for projective plane geometries, meaning that any true statement valid in all these

geometries remains true if we exchange points for lines and lines for points. The smallest

geometry satisfying all three axioms contains seven points. In this simplest of the projective

planes, there are also seven lines; each point is on three lines, and each line contains three

points. This particular projective plane is called as the projective plane of order 2,

sometimes also called the Fano plane. The projective plane of order s has s2

+ s + 1 points

and the same number of lines; each line contains s + 1 points, and each point is on s + 1

lines and the parameters of projective geometry are (s2 + s + 1, s

2 + s + 1, s +1, s+1, 1).

Therefore, incomplete block designs have close connections with finite geometrical

configurations, and many constructions of block designs are based on finite geometries if

each point is considered as a variety or treatment, as suggested by Fisher.

A balanced incomplete block design (=BIBD) is an ordinary 2-(v, k, λ) design i.e. a set

of k-subsets of an v-set such that each 2-subset is contained in exactly λ blocks and hence

balanced incomplete block design (BIBD), symmetric balanced incomplete block design

(SBIBD), finite projective and affine planes are defined by Dembowski (1968) and Hall

(1967). A finite projective plane of order s have s + 1 points on each line and the

parameters of this projective plane as a BIBD becomes (s2+s+1=v, s

2+s+1=b, s+1=r, s+1=k,

1=λ). Here symmetry of these numbers is a reflection of the fact that there is a duality

(interchangeability) between the points and lines of a finite projective plane. Many kinds of

BIBD’s are constructed from sets of points satisfying certain conditions in finite spaces.

77

In previous chapter of this study, a method of construction of BIBD for OS2 series

using MOLS has been discussed. We concentrate on the use of projective geometry for the

construction of BIB Designs for OS2 series with parameters v= s2+s+1= b, r= s+1=k and

λ=1. Here we discuss a new method of construction of BIBD for OS2 series and then

neighbour designs are constructed fulfilling circularity conditions. Further neighbours for

every treatment are observed for the design, on similar lines as in the previous chapter.

4.2 Projective Geometry

A finite projective geometry of n-dimensions, PG(n,s) over GF(s), Galois Field of

order s, s is any prime number or prime power such that s = ph, where p be a prime and h (

≥1) a positive integer; consists of the ordered set of n+1 elements ( x0, x1, x2, … , xn), xi Є

GF (s), not all xi’s simultaneously zero. Two ordered sets ( x0, x1, x2, … , xn) and ( y0, y1,

y2, … , yn) are said to represent the same points if yi= cxi for i = 0, 1, 2,…, n, where c (≠ 0 )

is an element of GF(s). The elements x0, x1, x2, … ,xn are called the co-ordinates of the

points ( x0, x1, x2, … , xn).

Clearly, the number of points in PG (n, s) is given by

Pn= (sn+1

– 1) / (s-1). …( 4.2.1)

All the points that satisfy a set of n – m independent, linear, homogeneous equations

a10 x0 + a11 x1 + … + a1nxn = 0

a20 x0 + a21 x1 + … + a2nxn = 0

.

.

.

a(n-m)0 x0 + a(n-m)1 x1 + … + a(n-m)nxn = 0 …( 4.2.2)

78

are said to form an m-flat in PG (n, s). The (n-m) equations (4.2.2) involve (n+1)

unknowns. A solution of these equations can be written as

( x0, x1, x2, … , xn)’ = b0y0 + b1y1 + … + bmym …( 4.2.3)

where the bi’s are the element of GF(s) and y0, y1, y2, … , ym are a set of (m+1) linearly

independent column vectors satisfying (4.2.2). Also, not all bi’s are simultaneously zero.

Since each of the b’s can be chosen in s ways, the total number of points satisfying (4.2.2)

is clearly (sm+1

-1). Further, since ( x0, x1, x2, … , xn) and ( cx0, cx1, cx2, … , cxn), c ≠ 0, c Є

GF(s) represent the same point in PG(n, s), the number of distinct points satisfying (4.2.2)

is

Pm = (sm+1

– 1) / (s-1) …( 4.2.4)

Thus, the number of points lying on an m-flat is Pm given by (4.2.4). The number of

m-flat in PG(n, s) is a function of n, m and s, will be denoted by φ (n, m, s). Now, each m-

flat is determined by a set of (m+1) independent points lying on it. Hence the number of m-

flats is equal to the number of ways of selecting (m+1) independent points out of the

totality of points, viz., pn, divided by the number of ways of selecting (m+1) independent

points on an m-flat. Therefore, the number of distinct m-flats in PG(n, s) is given by

φ (n, m, s) =

=

…(4.2.5)

From the expression given by (4.2.5) it is easily seen that

φ (n, m, s) = φ (n, n-m-1, s) …(4.2.6)

On the similar lines, it can be shown that the total number of distinct m-flats through

a fixed point is

79

φ (n - 1, m - 1, s) …(4.2.7)

and the number of distinct m-flats through two fixed points is

φ (n - 2, m - 2, s) …(4.2.8)

By taking the points of PG (n, s) as treatments and m-flats as blocks, the number of

treatments and blocks are respectively given by

v =φ (n, 0, s) =(sn+1

– 1) / (s-1) …(4.2.9)

and

b = φ (n, m, s) =

…(4.2.10)

where φ (n, m, s) is function defined by (4.2.5). The number of points in each m-flat is the

number of treatments contained in each block. Thus

k =φ (m, 0, s)= / …(4.2.11)

each treatment is contained in r blocks, where r is the number of m-flats passing through a

point. Thus

r = φ (n-1, m-1, s) =

…(4.2.12)

Similarly, each pair of treatments is contained in λ blocks, where λ is the number of

m-flats passing through a given pair of points, and hence

λ= φ (n-2, m-2, s) =

…(4.2.13)

Then for n = 2 and m = 1, the v, b, r, k, λ becomes (s2+s+1,s

2+s+1,s+1,s+1, 1)

respectively and the design so obtained is a 2- design i.e. Balanced Incomplete Block

Design with parameters 2- (s2 +s+1,s+1, 1; s), which is an OS2 series: {Dey (1986)}.

80

4.3 Construction of BIB Designs for OS2 Series

A BIBD of OS2 series is a finite projective plane of order s which has

s2 + s + 1 points,

s2 + s + 1 lines as blocks,

s + 1 points on each line,

s + 1 lines through each point,

where s ≥ 2 is an integer called the order of the projective plane. A projective plane of

prime order s can be obtained using the following method of construction.

Create one point P.

Create s points, which we will label P(c) :c = 0, ..., (s − 1).

Create s2 points, which we will label P(r, c) :r, c = 0, ..., (s − 1).

On these points, construct the following lines:

One line L = { P, P(0), ..., P(s − 1)}

s lines L(c) = {P, P(0,c), ..., P(s − 1,c)}, where c = 0, ..., (s − 1)

s2 lines L(r, c) = {P(c) and the points P((r + ci) mod s, i)}, where i, r, c = 0,1, ...,

(s − 1)

Note that the expression “(r + ci) mod s” will pass once through each value as i

varies from 0 to s − 1, but only if s is prime.

81

i) when s=2 i.e. a prime number

Let us consider a case when s = 2, then parameters of OS2 series becomes v=b=7,

r=k=3 and λ=1.

Fig-4.3.1 A cyclic way to represent a projective plane with seven points

For the construction of a finite projective plane of order 2, also known as Fano plane

create

a) one point P

b) s = 2 points, which we label as P(0), P(1)

c) s2 = 4 points, which we label as P(0,0), P(1,0),P(0,1), P(1,1)

On these points, construct the following lines

a) One line L = { P, P(0), P(1)}

b) s = 2 lines L(c) = {P, P(0,c), P(1,c)} : c = 0, 1 i.e.

when c=0: L(0) = {P, P(0,0), P(1,0)} &

when c=1: L(1) = {P, P(0,1), P(1,1)} .

1

2

3

4

5

6

7

82

c) s2 =4 lines L(r, c): P(c) and the points P((r + ci) mod 2, i), where i = 0,1 : r, c = 0,

1

when r,c=0,0: L(0,0) = { P(0), P((0+0.0) mod 2,0), P((0+0.1) mod 2,1) }

= { P(0), P(0,0), P(0,1) }

when r,c=1,0: L(1,0) = { P(0), P((1+0.0) mod 2,0), P((1+0.1) mod 2,1) }

= { P(0), P(1,0), P(1,1) }

when r,c=0,1: L(0,1) = { P(1), P((0+1.0) mod 2,0), P((0+1.1) mod 2,1) }

= { P(1), P(0,0), P(1,1) }

when r,c=1,1: L(1,1) = { P(1), P((1+1.0) mod 2,0), P((1+1.1) mod 2,1) }

= { P(1), P(1,0), P(0,1) }

Let us assign number to the points:

Points P(0,0) P(1,0) P(0,1) P(1,1) P P(0) P(1)

Point Number 1 2 3 4 5 6 7

after numbering and arranging in ascending or descending order (here ascending) the

resulting BIBD is:

Table 4.3.1

Blocks Contents

1

2

3

4

5 6 7

1 2 5

3 4 5

1 3 6

83

5

6

7

2 4 6

1 4 7

2 3 7

As s = 2 is the least prime number, so let us take one more prime number for the

construction of BIBD for OS2 series using projective geometry.

ii) when s=3 i.e. a prime number

Now consider a case when s = 3, then parameters of OS2 series becomes v=b=13,

r=k=4, λ=1.

Fig-4.3.2 A cyclic way to represent a projective plane with thirteen points

For the construction of a finite projective plane of order 3 create

a) one point P

b) s = 3 points, which we label as P(0), P(1), P(2)

c) s2 = 9 points, which we label as P(0,0), P(1,0),P(2,0),P(0,1), P(1,1), P(2,1),P(0,2),

P(1,2), P(2,2)

1

2

3

4

5

6 7

8

9

10

11

12

13

84

On these points, construct the following lines

a) One line L = { P, P(0), P(1),P(2)},

b) s = 3 lines L(c) = {P, P(0,c), P(1,c), P(2,c),} : c = 0, 1, 2 i.e.

when c=0: L(0) = {P, P(0,0), P(1,0), P(2,0)},

when c=1: L(1) = {P, P(0,1), P(1,1), P(2,1),} &

when c=2: L(2) = {P, P(0,2), P(1,2), P(2,2),} .

c) s2 = 9 lines L(r, c): P(c) and the points P((r + ci) mod 3, i), where i = 0,1,2 : r, c =

0,1,2

when r,c=0,0:

L(0,0) = {P(0),P((0+0.0)mod 3,0),P((0+0.1)mod 3,1),P((0+0.2)mod 3,2)}

= { P(0), P(0,0), P(0,1), P(0,2) }

when r,c=1,0:

L(1,0) = {P(0),P((1+0.0)mod 3,0),P((1+0.1)mod 3,1),P((1+0.2)mod 3,2)}

= { P(0), P(1,0), P(1,1), P(1,2) }

when r,c=2,0:

L(2,0) = {P(0),P((2+0.0)mod 3,0),P((2+0.1)mod 3,1),P((2+0.2)mod 3,2)}

= { P(0), P(2,0), P(2,1), P(2,2) }

when r,c=0,1:

L(0,1) = {P(1),P((0+1.0)mod 3,0),P((0+1.1)mod 3,1),P((0+1.2)mod 3,2)}

= { P(1), P(0,0), P(1,1), P(2,2) }

85

when r,c=1,1:

L(1,1) = {P(1),P((1+1.0)mod 3,0),P((1+1.1)mod 3,1),P((1+1.2)mod 3,2)}

= { P(1), P(1,0), P(2,1), P(0,2) }

when r,c=2,1:

L(2,1) = {P(1),P((2+1.0)mod 3,0),P((2+1.1)mod 3,1),P((2+1.2)mod 3,2)}

= { P(1), P(2,0), P(0,1), P(1,2) }

when r,c=0,2:

L(0,2) = {P(2),P((0+2.0)mod 3,0),P((0+2.1)mod 3,1),P((0+2.2)mod 3,2)}

= { P(2), P(0,0), P(2,1), P(1,2) }

when r,c=1,2:

L(1,2) = {P(2),P((1+2.0)mod 3,0),P((1+2.1)mod 3,1),P((1+2.2)mod 3,2)}

= { P(2), P(1,0), P(0,1), P(2,2) }

when r,c=2,2:

L(2,2) = {P(2),P((2+2.0)mod 3,0),P((2+2.1)mod 3,1),P((2+2.2)mod 3,2)}

= { P(2), P(2,0), P(1,1), P(0,2) }

Let us assign number to the points:

Points P(0,0) P(1,0) P(2,0) P(0,1) P(1,1) P(2,1) P(0,2)

Point Number 1 2 3 4 5 6 7

Points P(1,2) P(2,2) P P(0) P(1) P(2)

Point Number 8 9 10 11 12 13

86

after numbering and arranging in ascending or descending order (here ascending) the

resulting BIBD is:

Table- 4.3.2

Blocks Contents

1

2

3

4

5

6

7

8

9

10

11

12

13

10 11 12 13

1 2 3 10

4 5 6 10

7 8 9 10

1 4 7 11

2 5 8 11

3 6 9 11

1 5 9 12

2 6 7 12

3 4 8 12

1 6 8 13

2 4 9 13

3 5 7 13

So a BIBD for OS2 series using projective geometry can be constructed for any

value of s when s is a prime number. Now let us consider the case for OS2 series using

projective geometry when s is not a prime number but a prime power.

iii) when s=4 i.e. a prime power

Let us now consider a case when s = 22 = 4, then parameters of OS2 series becomes

v=b=21, r=k=5, λ=1.

87

For the construction of a finite projective plane of order 4 create

a) one point P

b) s = 4 points, which we label as P(0), P(1), P(α ), P(α2)

c) s2 = 16 points, which we label as P(0,0), P(1,0), P(α,0),…, P(α,α

2), P(α

2,α

2)

On these points, construct the following lines

a) One line L = { P, P(0), P(1),P(α), P(α2)},

b) s = 4 lines L(c) = {P, P(0,c), P(1,c), P(α,c), P(α2,c)} : c = 0, 1, α, α

2 i.e.

when c=0: L(0) = {P, P(0,0), P(1,0), P(α,0), P(α2,0)},

when c=1: L(1) = {P, P(0,1), P(1,1), P(α,1), P(α2,1)},

when c=α: L(α) = {P, P(0,α), P(1,α), P(α,α), P(α2,α)} &

when c=α2: L(α

2) = {P, P(0,α

2), P(1,α

2), P(α,α

2), P(α

2,α

2)}

c) s2 = 16 lines L(r, c): P(c) and the points P((r + ci) mod 2, i), where i = 0,1,α, α

2 : r,

c = 0,1, α, α2 i.e. i, & c are elements of G.F.(s)

when r,c=0,0: L(0,0) = { P(0), P(0,0), P(0,1), P(0,α), P(0,α2) }

when r,c=1,0: L(1,0) = { P(0), P(1,0), P(1,1), P(1,α), P(1,α2) }

when r,c= α,0: L(α,0) = { P(0), P(α,0), P(α,1), P(α,α), P(α,α2) }

when r,c= α2,0: L(α

2,0) = { P(0), P(α

2,0), P(α

2,1), P(α

2,α), P(α

2,α

2) }

when r,c=0,1: L(0,1) = { P(1), P(0,0), P(1,1), P(α,α), P(α2,α

2) }

when r,c=1,1: L(1,1) = { P(1), P(1,0), P(0,1), P(α2,α), P(α,α

2) }

when r,c= α,1: L(α,1) = { P(1), P(α,0), P(α2,1), P(0,α), P(1,α

2) }

when r,c= α2,1: L(α

2,1) = { P(1), P(α

2,0), P(α,1), P(1,α), P(0,α

2) }

88

when r,c=0, α: L(0, α) = { P(α), P(0,0), P(α,1), P(α2,α), P(1,α

2) }

when r,c=1, α: L(1, α) = { P(α), P(1,0), P(α2,1), P(α,α), P(0,α

2) }

when r,c= α, α: L(α, α) = { P(α), P(α,0), P(0,1), P(1,α), P(α2,α

2) }

when r,c= α2, α: L(α

2, α) = { P(α), P(α

2,0), P(1,1), P(0,α), P(α,α

2) }

when r,c=0, α2: L(0, α

2) = { P(α

2), P(0,0), P(α

2,1), P(1,α), P(α,α

2) }

when r,c=1, α2: L(1, α

2) = { P(α

2), P(1,0), P(α,1), P(0,α), P(α

2,α

2) }

when r,c= α, α2: L(α, α

2) = { P(α

2), P(α,0), P(1,1), P(α

2,α), P(0,α

2) }

when r,c= α2, α

2: L(α

2, α

2) = { P(α

2), P(α

2,0), P(0,1), P(α,α), P(1,α

2) }

To construct BIBD give numbering to the points:

Points P(0,0) P(1,0) P(α,0) P( α2,0) P(0,1) P(1,1) P(α,1)

Point Number 1 2 3 4 5 6 7

Points P( α2,1) P(0,α) P(1,α) P(α,α) P(α

2, α) P(0,α

2) P(1,α

2)

Point Number 8 9 10 11 12 13 14

Points P(α,α2) P(α

2,α

2) P P(0) P(1) P(α) P(α

2)

Point Number 15 16 17 18 19 20 21

[It should be noted here that the natural numbers are assigned to the points in a

sequence as P(0,0), P(1,0), … , P(r,0); P(0,1), P(1,1), … , P(r,1); … ; P(0,c), P(1,c), …,

P(r,c); P; P(0), P(1), …, P(c); where r, c = 0, ..., (s − 1); from 1, 2, …, s2 + s + 1

respectively.]

89

After numbering and arranging in ascending or descending order (here ascending)

the resulting BIBD is:

Table – 4.3.3

Blocks Contents

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

17 18 19 20 21

1 2 3 4 17

5 6 7 8 17

9 10 11 12 17

13 14 15 16 17

1 5 9 13 18

2 6 10 14 18

3 7 11 15 18

4 8 12 16 18

1 6 11 16 19

2 5 12 15 19

3 8 9 14 19

4 7 10 13 19

1 7 12 14 20

2 8 11 13 20

3 5 10 16 20

4 6 9 15 20

1 8 10 15 21

2 7 9 16 21

3 6 12 13 21

4 5 11 14 21

90

Using the above explained method of construction of BIBD through Projective

Geometry of OS2 series, it can be obtain for any value of s whether s is a prime number or

a prime power.

4.4 Construction of Neighbour Designs for OS2 Series

After the construction of BIBD, neighbour design may be constructed by using the

border plots i.e. one plot is added at each end of each block. Arrangements of treatments at

border plots at either end of the block are same as the treatment on the interior plot at the

other end of block and border plots are not used for measuring the response variable. Plots

other than border plots are described as internal plots for neighbour designs. In this design,

all the blocks shall be circular in the sense that the border treatments at either end of the

block are the same as the treatment on the interior plot at the other end of block. For a

design d, d(i, j) denotes the treatment applied to plot j of block i, particularly, d(i, 0) and

d(i, k+1) are the two treatments applied to the border plots of block i and the circularity

condition implies that d(i, 0)= d(i, k) and d(i, k+1)= d(i, 1) ; where 1≤ i≤ b &1≤ j ≤ k. These

extra parameters used as neighbour plots are needed for the effect of left and right

neighbours. Now consider the construction of neighbour design for OS2 series for any

value of s whether s is a prime number or a prime power.

i) When s = 2: As s is a prime number i.e. s = 2, a Fano Plane, design has parameters

v=b=22+2+1=7, r=k=2+1=3 & λ=1. In the design d(i,0) and d(i,4) are the two

treatments which are applied to the border plots of block i and also d(i, 0)= d(i, 3)

and d(i, 4)= d(i, 1) ; where 1≤ i≤ 7 which fulfill the circularity conditions. Hence the

resulting design is a neighbour design:

91

Table – 4.4.1

7 5 6 7 5

5 1 2 5 1

5 3 4 5 3

6 1 3 6 1

6 2 4 6 2

7 1 4 7 1

7 2 3 7 2

In the design obtained here, no treatment is (i) immediate to itself and (ii) immediate

to any other treatment more than once, which are the conditions for a neighbour design to

be balanced either completely or partially.

ii) When s = 3: Again consider s is a prime number i.e. s = 3, then design has

parameters v=b=32+3+1 =13, r=k=3+1=4 & λ=1. In the design d(i,0) and d(i,5)

are the two treatments which are applied to the border plots of block i and also

d(i, 0)= d(i, 4) and d(i, 5)= d(i, 1) ; where 1≤ i≤ 13 which fulfill the circularity

conditions. Hence the resulting design is a neighbour design:

92

Table – 4.4.2

13 10 11 12 13 10

10 1 2 3 10 1

10 4 5 6 10 4

10 7 8 9 10 7

11 1 4 7 11 1

11 2 5 8 11 2

11 3 6 9 11 3

12 1 5 9 12 1

12 2 6 7 12 2

12 3 4 8 12 3

13 1 6 8 13 1

13 2 4 9 13 2

13 3 5 7 13 3

In the design obtained here, no treatment is (i) immediate to itself and (ii) immediate

to any other treatment more than once. These are the conditions for a neighbour design to

be balanced either completely or partially.

iii) When s = 4: Now consider s = 4 (=22) i.e. s is a prime power, design has

parameters v=b=42+4+1=21, r=k=4+1=5 & λ=1. The neighbour design

constructed by using the method as discussed earlier is:

93

Table – 4.4.3

21 17 18 19 20 21 17

17 1 2 3 4 17 1

17 5 6 7 8 17 5

17 9 10 11 12 17 9

17 13 14 15 16 17 13

18 1 5 9 13 18 1

18 2 6 10 14 18 2

18 3 7 11 15 18 3

18 4 8 12 16 18 4

19 1 6 11 16 19 1

19 2 5 12 15 19 2

19 3 8 9 14 19 3

19 4 7 10 13 19 4

20 1 7 12 14 20 1

20 2 8 11 13 20 2

20 3 5 10 16 20 3

20 4 6 9 15 20 4

21 1 8 10 15 21 1

21 2 7 9 16 21 2

21 3 6 12 13 21 3

21 4 5 11 14 21 4

In the design obtained here, again no treatment is (i) immediate to itself and

(ii) immediate to any other treatment more than once shows that the design is neighbour

balanced either completely or partially. Here again all the blocks are circular in the sense

that the border treatments at either end of the block is the same as the treatment on the

94

interior plot at the other end of block i.e. d(i, 0)= d(i, 5) and d(i, 6)= d(i, 1) ; where 1≤ i≤

21. Similarly the neighbour designs for s =5, 23=8, 3

2=9, 11 and so on, can be constructed

in the same way.

4.5Neighbours of Treatment for Neighbour Designs of OS2 Series

For the analysis of two-sided neighbour designs, left neighbours and right

neighbours of a treatment must be known. So let us consider firstly the left neighbours of a

treatment in neighbour designs of OS2 series.

4.5.1 Left-Neighbours of a Treatment

i) When s = 2: From the neighbour design given in Table-4.4.1 we can obtain left

neighbour treatments for each treatment. Now consider treatment number 1; in

second block treatment number 5 is the left neighbour, then 6 is the left

neighbour in the next block in which treatment number 1 appears. Thus a list of

left neighbours for treatment number 1in the design is: 5, 6, 7. The treatment

number 2 in second block has treatment number 1 as the left neighbour. From the

other blocks in which treatment number 2 appears in the design, a list of left

neighbours is: 1,6, 7. Similarly list of left neighbours for other treatments can be

obtained and these neighbours of design for s = 2 with parameters v=b=7, r=k=3

& λ= 1 can be summarized in the following table:

95

Table – 4.5.1

It should be noted here that each treatment has s+1 treatments i.e. 3 treatments as left

neighbours.

ii) When s = 3: From the neighbour design given in Table-4.4.2 we can obtain left

neighbour treatments for each treatment. Now consider treatment number 1; in

second block treatment number 10 is the left neighbour, then 11 is the left

neighbour in the next block in which treatment number 1 appears. Thus a list of

left neighbours for treatment number 1 in the design is: 10,11,12,13. Similarly

list of left neighbours for other treatments can be obtained and the neighbour

treatments for s = 3 with parameters v=b=13, r=k= 4 & λ= 1 can be summarized

as:

Treatment

Number (i)

Neighbours

Obtained

Common Left

Neighbour

Series

Other

Neigh-

bours

Series In Which

Treatment Number

‘i’ Lies

1

2

5, 6, 7

1, 6, 7

6, 7

6, 7

5

1

1 ≤ i ≤ s i.e.

1 ≤ i ≤ 2

3

4

5, 1, 2

3, 2, 1

1, 2

1, 2

5

3

s+1 ≤ i ≤ 2si.e.

3 ≤ i ≤ 4

5 7, 2, 4 2, 7 4 i = s2+1 i.e. i=5

6

7

5, 3, 4

6, 4, 3

3, 4

3, 4

5

6

s2+2 ≤ i ≤ s

2+s+1

i.e. 6 ≤ i ≤ 7

96

Table – 4.5.2

It should be noted here that each treatment has s+1 treatments i.e. 4 treatments as left

neighbours. Here it should be further noticed that the left neighbours for each treatment are

same as have been obtained in chapter-III for s = 3 in Table-3.5.1, difference is only in their

order of occurrence as a left neighbour.

iii) When s = 4: For s = 4 i.e. a prime power, the neighbour design so obtained is

given in Table-4.4.3. From the table we observed left neighbour treatments for

each treatment. Consider the treatment number 1; in second block of the design

has 17 as the left neighbour, then 18 is the left neighbour in the next block in

Treatment

Number (i)

Neighbours

Obtained

Common Left

Neighbour

Series

Other

Neigh-

bours

Series In Which

Treatment Number

‘i’ Lies

1

2

3

10,11,12,13

1,11,12,13

2,11,12,13

11, 12, 13

11, 12, 13

11, 12, 13

10

1

2

1 ≤ i ≤ s

i.e.

1 ≤ i ≤ 3

4

5

6

10,1,2,3

4,2,1,3

5,3,2,1

1, 2, 3

1, 2, 3

1, 2, 3

10

4

5

s+1 ≤ i ≤ 2s

i.e.

4 ≤ i ≤ 6

7

8

9

10,4,6,5

7,5,4,6

8,6,5,4

4, 5, 6

4, 5, 6

4, 5, 6

10

7

8

2s+1≤ i ≤ s2

i.e.

7 ≤ i ≤ 9

10 13,3,6,9 3,6,13 9 i = s2+1 i.e. i=10

11

12

13

10,7,8,9

11,8,9,7

12,9,8,7

7, 8, 9

7, 8, 9

7, 8, 9

10

11

12

s2+2 ≤ i ≤ s

2+s+1

i.e.

11≤ i ≤13

97

which treatment number 1 appears. Similarly, we get neighbours from the other

blocks in which treatment number 1 appears. Thus a list of neighbours for

treatment number 1 in the design is: 17,18,19,20,21. The treatment number 2 in

second block has 1 as the left neighbour. From the other blocks in which

treatment number 2 appears a list of neighbours in the design is: 1,18,19,20,21.

Similarly list of neighbours for other treatments is obtained and is given in the

following table:

Table – 4.5.3

Treatment

Number (i)

Neighbours

Obtained

Common Left

Neighbour

Series

Other

Neigh-

bours

Series In Which

Treatment

Number ‘i’ Lies

1

2

3

4

17,18,19,20,21

1,18,19,20,21

2,18,19,20,21

3,18,19,20,21

18,19,20,21

18,19,20,21

18,19,20,21

18,19,20,21

17

1

2

3

1 ≤ i ≤ s

i.e.

1 ≤ i≤ 4

5

6

7

8

17,1,2,4,3

5,2,1,4,3

6,3,4,1,2

7,4,3,2,1

1,2,3,4

1,2,3,4

1,2,3,4

1,2,3,4

17

5

6

7

s+1 ≤ i ≤ 2s

i.e.

5 ≤ i ≤ 8

9

10

11

12

17,5,8,6,7

9,6,7,5,8

10,7,6,8,5

11,8,5,7,6

5,6,7,8

5,6,7,8

5,6,7,8

5,6,7,8

17

9

10

11

2s+1≤ i ≤ 3s

i.e.

9 ≤ i≤ 12

13

14

15

16

17,9,10,11,12

13,10,9,12,11

14,11,12,9,10

15,12,11,10,9

9,10,11,12

9,10,11,12

9,10,11,12

9,10,11,12

17

13

14

15

3s+1 ≤ i ≤ s2

i.e.

13 ≤ i≤ 16

98

17 21,4,8,12,16 4,8,12,21 16 i = s2+1 i.e.i = 17

18

19

20

21

17,13,14,15,16

18,16,15,14,13

19,15,14,16,13

20,14,16,15,13

13,14,15,16

13,14,15,16

13,14,15,16

13,14,15,16

17

18

19

20

s2+2 ≤ i ≤ s

2+s+1

i.e

18 ≤ i≤ 21

It should be noted here that each treatment has s+1 (5 in this case) treatment as the

left neighbours. Here it should be further noticed that the left neighbours for each treatment

are same as have been obtained in chapter-III for s = 4 in Table-3.5.2, difference is only in

their order of occurrence as a left neighbour. Therefore, we can obtain the left neighbours

for any treatment of OS2 series for any value of s whether s is a prime number or prime

power.

4.5.2Right-Neighbours of a Treatment for OS2 Series

i) When s = 2: From the neighbour design given in Table-4.4.1 we can obtain right

neighbour treatments for each treatment. Consider treatment number 1; in second

block treatment number 2 is the right neighbour, then 3 is the right neighbour in

the next block in which treatment number 1 appears in the design. Thus a list of

right neighbours for treatment number 1 in the design is: 2,3,4. Similarly the

neighbour for other treatments is obtained and the following table for s = 2 with

parameters v=b=7, r=k= 3 & λ= 1 is obtained:

99

Table – 4.5.4

It should be noted here that each treatment has s+1 treatments i.e. 3 treatments as

right neighbours.

ii) When s = 3: From the neighbour design given in Table-4.4.2, consider treatment

number 1; in second block treatment number 2 is the right neighbour, then 4 is

the right neighbour in the next block in which treatment number 1 appears. Thus

a list of right neighbours for treatment number 1 in the design is: 2,4,5,6.

Similarly the neighbour for other treatments is obtained and then the following

table for s = 3 with parameters v=b=13, r=k= 4 & λ= 1 can be written as:

Treatment

Number (i)

Neighbours

Obtained

Common Right

Neighbour Series

Other

Neighbours

Series In Which

Treatment Number

‘i’ Lies

1

2

2,3,4

5,4,3

3, 4

3, 4

2

5

1 ≤ i ≤ si.e.

1 ≤ i ≤ 2

3

4

4,6,7

5,6,7

6, 7

6, 7

4

5

s+1 ≤ i ≤ 2si.e.

3 ≤ i ≤ 4

5 6,1,3 1, 3 6 i = s2+1 i.e. i=5

6

7

7,1,2

5,1,2

1, 2

1, 2

7

5

s2+2 ≤ i ≤ s

2+s+1

i.e. 6 ≤ i ≤ 7

100

Table – 4.5.5

It should be noted here that each treatment has s+1 treatments i.e. 4 treatments as

right neighbours. Here it should be further noticed that the right neighbours for each

treatment are same as have been obtained in chapter-III for s = 3 in Table-3.5.4, difference

is only in their order of occurrence as right neighbour.

iii) When s = 4: For s = 4 i.e. when s is a prime power, the neighbour design so

obtained is given in Table-4.4.3 from which we observed right neighbour

treatments of each treatment. Consider the treatment number 1; in second block it

has 2 as the right neighbour, then 5 is the right neighbour in the next block in

Treatment

Number (i)

Neighbours

Obtained

Common Right

Neighbour Series

Other

Neighbours

Series In Which

Treatment Number

‘i’ Lies

1

2

3

2,4, 5, 6

3, 5, 6, 4

10,6, 4, 5

4, 5, 6

4, 5, 6

4, 5, 6

2

3

10

1 ≤ i ≤ s

i.e.

1 ≤ i ≤ 3

4

5

6

5,7,8,9

6,8,9,7

10,9,7,8

7, 8, 9

7, 8, 9

7, 8, 9

5

6

10

s+1 ≤ i ≤ 2s

i.e.

4 ≤ i ≤ 6

7

8

9

8,11,12,13

9,11,12,13

10,11,12,13

11, 12, 13

11, 12, 13

11, 12, 13

8

9

10

2s+1≤ i ≤ s2

i.e.

7 ≤ i ≤ 9

10 11,1,4,7 1,4,7 11 i = s2+1 i.e. i=10

11

12

13

12,1,2,3

13,1,2,3

10,1,3,2

1, 2, 3

1, 2, 3

1, 2, 3

12

13

10

s2+2 ≤ i ≤ s

2+s+1

i.e.

11≤ i ≤13

101

which treatment number 1 appears in the design. Similarly, we get neighbours

from the other blocks in which treatment number 1 appears. Thus a list of

neighbours for treatment number 1 in the design is: 2,5,6,8,7. The list of

neighbours for other treatments has been obtained and the following table is

constructed:

Table – 4.5.6

Treatment

Number (i)

Neighbours

Obtained

Common Right

Neighbour

Series

Other

Neighbours

Series In Which

Treatment Number

‘i’ Lies

1

2

3

4

2,5,6,7,8

3,6,5,8,7

4,7,8,5,6

17,8,7,6,5

5,6,7,8

5,6,7,8

5,6,7,8

5,6,7,8

2

3

4

17

1 ≤ i ≤ s

i.e.

1 ≤ i≤ 4

5

6

7

8

6,9,12,10,11

7,10,11,9,12

8,11,10,12,9

17,12,9,11,10

9,10,11,12

9,10,11,12

9,10,11,12

9,10,11,12

6

7

8

17

s+1 ≤ i ≤ 2s

i.e.

5 ≤ i ≤ 8

9

10

11

12

10,13,14,15,16

11,14,13,16,15

12,15,16,13,14

17,16,15,14,13

13,14,15,16

13,14,15,16

13,14,15,16

13,14,15,16

10

11

12

17

2s+1≤ i ≤ 3s

i.e.

9 ≤ i≤ 12

13

14

15

16

14,18,19,20,21

15,18,19,20,21

16,18,19,20,21

17,18,19,20,21

18,19,20,21

18,19,20,21

18,19,20,21

18,19,20,21

14

15

16

17

3s+1 ≤ i ≤ s2

i.e.

13 ≤ i≤ 16

102

It is to be noted here that each treatment has s+1 (5 in this case) treatment as the

right neighbours. Here it should be further noticed that the right neighbours for each

treatment are same as have been obtained in chapter-III for s = 4 in Table-3.5.5, difference

is only in their order of occurrence as right neighbour. Therefore, we can obtain the right

neighbours for any treatment of OS2 series for any value of s whether s is a prime number

or prime power.

4.6 Two-Sided Neighbours for OS2 Series

From the tables of neighbours for different values of s, we conclude that the left

neighbours and the right neighbours for a treatment is same, whether the neighbour design

for OS2 series is obtained using method of MOLS or method of Projective Geometry. Only

difference is in their order of occurrence, which may be due to the shifting of the blocks in

the neighbour designs for OS2 series when constructed using Projective Geometry.

Therefore, two-sided neighbours can be easily summarized in the manner given in the

following table:

17 18,1,5,9,13 1,5,9,13 18 i = s2+1 i.e. i = 17

18

19

20

21

19,1,2,3,4

20,1,2,3,4

21,1,4,2,3

17,1,3,4,2

1,2,3,4

1,2,3,4

1,2,3,4

1,2,3,4

19

20

21

17

s2+2 ≤ i ≤ s

2+s+1

i.e

18 ≤ i≤ 21

103

Table-4.6.1

Other

Left

Neigh-

bour

Common Left

Neighbour Series

Treat-

ment

Number

(i)

Common Right

Neighbour Series

Other

Right

Neigh-

bour

Series In

Which

Treatment

Number ‘i’

Lies

s2+1

i-1

.

.

.

i-1

s2+2,…, s

2+s+1

s2+2,…, s

2+s+1

.

.

.

s2+2 ,…, s

2+s+1

1

2

.

.

.

s

s+1,…,2s

s+1,…,2s

.

.

.

s+1,…,2s

i+1

i+1

.

.

.

s2+1

1 ≤ i ≤ s

s2+1

i-1

.

.

.

i-1

1,…,s

1,…,s

.

.

.

1,…,s

s+1

s+2

.

.

.

2s

2s+1,…,3s

2s+1,…,3s

.

.

.

2s+1,…,3s

i+1

i+1

.

.

.

s2+1

s+1 ≤ i ≤ 2s

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

s2+1

i-1

(s-2)s+1,…,(s-2)s+s

(s-2)s+1,…,(s-2)s+s

(s-1)s+1

(s-1)s+2

s2+2 ,…, s

2+s+1

s2+2 ,…, s

2+s+1

i+1

i+1

104

.

.

.

i-1

.

.

.

(s-2)s+1,…,(s-2)s+s

.

.

.

(s-1)s+s

=s2

.

.

.

s2+2 ,…, s

2+s+1

.

.

.

i+1=

s2+1

(s-1)s+1 ≤ i

≤ s2

i-1 = s2 s,2s,…,(s-1)s&

s2+s+1

s2+1 1,s+1,2s+1,…, (s-

1)s+1

i+1=

s2+2

i = s2+1

s2+1

i-1

.

.

.

i-1

(s-1)s+1,…, s2

(s-1)s+1,…, s2

.

.

.

(s-1)s+1,…, s2

s2+2

s2+3

.

.

.

s2+s+1

1,…,s

1,…,s

.

.

.

1,…,s

i+1

i+1

.

.

.

s2+1

s2+2 ≤ i ≤

s2+s+1

The above obtained table is exactly same as that of table 3.6.1 obtained in chapter

III, where the neighbour design is constructed using the method of MOLS and has been

discussed in detail there. Hence, we conclude here that the pattern of neighbours for any

treatment is exactly same whether a neighbour designs for OS2 series is constructed using

method of MOLS or using method of Projective Geometry. Therefore, one can find out the

neighbours of any treatment of neighbour design for OS2 series according to the steps

given in section 3.6 of chapter III. After finding the neighbours of a treatment, one shall be

interested to analyze the data for the significance. Therefore, we shall discuss the analysis

of data, obtained from neighbour design of OS2 series in the succeeding chapter.

References

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k, λ) – designs. Designs, Codes and Cryptography, 34(1), 55-70.

Dembowski, P. (1968): Finite geometries. Springer-Verlag, New York.

105

Dey, A. (1986): Theory of Block Designs. Wiley Eastern Limited, New Delhi.

M. Hall, Jr. (1967): Combinatory Theory. Blaisdell Publishing Company, Waltham, Mass.

Miyakawa, M., Munemasa, A. and Yoshiara, S.(1995):On a Class of Small 2-Designs over

GF(q). Journal of Combinatorial Designs 3,pages 61-77.

Suzuki, H.,(1990): 2-Designs over GF(2m). Graphs and Combinatorics 6, pages 293-296.

Thomas, S. (1987): Designs over Finite Fields. Geometriae Dedicata 24 (1987), pages 237-

242.