On Some Applications Of Hadamard Matrices

14

On Some Applications Of Hadamard Matrices

Abdurzak M. Leghwel1

Abstract

Orthogonal arrays and t-designs are two of many combinatorial structures

which have been found to be very useful in the statistical area of

Experimental Design . Orthogonal arrays were introduced as suitable

designs for planned experiments by [8] . Rao also gave key statistical

properties for these arrays . On the other hand, the study of certain t-designs

pre-dates the subject of Experimental Design which began in the 1920's .

Now-a-days orthogonal arrays are used frequently in the design of quality

control experiments largely due to the efforts of [10] . Moreover, 2-designs

and 3-designs are used in all kinds of planned experiments including

agricultural experiments.

The practical utility of orthogonal arrays and t-designs has spurred

research on these topics by both mathematicians and statisticians . The main

objective of this paper is, therefore, to present constructions of certain

classes of these combinatorial structures using Hadamard matrices as a tool.

In first part of the paper we introduce the concept of an orthogonal array

and in second part of the paper we use Hadamard matrices to construct

arrays of strength 2 and strength 3 .

In the third part of the paper motivates and then introduces the definition of

t-designs . finally, we use Hadamard matrices to construct certain classes

of these designs known as Hadamard 2- and 3- designs . Examples

illustrating these constructions are also given.

Keywords : Hadamard matrix; Orthogonal arrays; t-Designs; Block designs.

1. Orthogonal Arrays

Orthogonal arrays (OA's) were first introduced in the context of statistics,

actually in the area of statistics known as experimental or statistical design .

1 University of Almergeb – Faculty of Science – Zliten - Department of Mathematics

Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya

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The definition of an OA is a combinatorial one, and, thus they have been

studied, for their properties and construction methods, by mathematicians

and statisticians . For a review of some of the basics on OA's , we refer to

the book by [6] .

The purpose of this part is to motivate and then define the concept of an

OA . To meet this objective we need some terminology from experimental

design which we summarize next . For further reading on various aspects of

experimental design we refer to [5] and [7] . We now itemize the promised

design terminology :

a) Let { } 1-s ,. . . ,2 ,1 ,0 = A ii for k ,. . . ,2 ,1=i and 2si ≥ for each ,i

be k-sets . In the context of experimental design , each set Ai is called a

factor and each element of Ai is called a level .Thus the i-th factor Ai has

si levels, which are usually chosen as the first si nonnegative integers .

b) Let

{ } Ax ; ) x ,. . . ,x ,x ( = x = A = A . . . A A = A iik21i

k

i

k21 ∈××× ∏=1

be the Cartesian product of the sets . Ai The set A is known as the

complete or full factorial based on the k-factors A ,. . . . ,A ,A k21 with the

i-th factor at si levels . Each element )x ,. . . ,x ,x( = x k21 in ,A

variously called a point (if one thinks geometrically) or a treatment (if one

thinks statistically) is a combination of levels in which one level is chosen

from each of the k factors . In an experimental design situation we may also

think of x as an experimental condition .

c) A factorial design (or simply a design) ,d based on k factors Ai each

at si levels, is a selection of treatments (points) of ,A possibly with

repetitions.

In many useful cases d is simply a subset of ,A so that d consists of

distinct points of . A Thus the full or complete factorial is an example of a

factorial design .

We now discuss the selectiondesign of problem which will ultimately

lead to definition of an OA . This discussion, including the problem

formulated below, has been taken from [4] . For further reading on this

topic we refer to [4], [12] and [11] . When an experiment is being planned,

the planner has to make certain decisions as follows :

On Some Applications Of Hadamard Matrices

16

i) One has to identify which factors (the ) sA,i are really important ; i.e.

which factors significantly effect the response . Some of these factors

would be known to the experimenter ; about others there could be some

doubt . Good planning requires that one include some factors about which

there is some doubt just in case they turn out to be really important later .

ii) Thus k factors may be selected of which perhaps ) k p ( p ≤ are really

significant in explaining the response .

iii) The next problem is to decide on the number of levels 2si ≥ of each

factor . Ai This involves a difficult balancing act, since making the si

unduly large would increase the cost of experiment without necessarily

providing more information than a choice of smaller values of the . si

iv) Finally once the sA k ,i have been identified, the number kp ≤ of

possibly significant factors being settled upon, and the si levels have been

decided, it is necessary to choose a design d* from the full factorial

A=A i

k

=1i

∏ , in order to conduct the experiment. As [4] puts it : the

pressure is now on selecting the design; we should take observations only at

points that are as ` strategically placed as possible ` .

The logic of how to implement (iv) is suggested by [4] :

since p factors are effecting the response , k p )( ≤ but whose actual identity

among the k factors is unknown, we select as the desired design d* to be

a set of points of A that project as full (and possibly repeated) factorials in

any subset of p ) kp ( ≤ factors from among the k factors . Based on this

the following combinatorial problem emerges for study and has been

suggested by [4] :

Given k sets A ,. . . ,A ,A k21 find a set d* of the Cartesian

product A . . . A A k21 ××× with as few points as possible whose (1)

projections onto any p of the k sets is equal to the Cartesian

product of those p sets .

In regard to studying problem (1) , [4] suggests the following notation

which we will use as well :

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Let the cardinality of 2s|=A| ,A iii ≥ for k ,. . . ,2 ,1=i and let kp ≤ be

given . Let d* be a design in the complete factorial A satisfying the

requirements mentioned in (1) . Then we define

= |d| = ) p ; s ,. . . ,s ,s m( *k21 the number of points in . d

* The

following observations of the function m are clear

(a) ) p ; s ,. . . ,s ,s ( m k21 is an increasing function in each of its

coordinates ,

(b) Let l = product of the p largest ss,i . Then

)( p ; s ,. . . ,s ,s m l k21≤ .

We study the function ) p ; s ,. . . ,s ,s ( m k21 for small values of p and k in the next two examples . In both examples the function m achieves its lower bound .

:1.1Example Let 3=k and let ,2|=A| i for ,3 ,2 ,1=i i.e.

{ } . 1 ,0 =A=A=A 321 Thus 321 AAA=A ×× consists of eight points . If

,2=p then d* must have at least four points as follows :

0110

1010

1100

(2)

To check this, note that in any two rows of (2) all the four points ,y ,x )'(

with { } , 1 ,0 y ,x ∈ occur . From this it follows that these four points

(columns of (2)) project as full factorials onto any two element subset of

the three sA,i . Thus . 4=) 2 ; 2 ,2 ,2 ( m

:1.2Example Let 4=k and let ,3|=A| i for ,4 ,3 ,2 ,1=i i.e.

{ } . 2 ,1 ,0 =A=A=A=A 4321 Thus 4321 AAAA=A ××× consists of 81

points . If ,3=p then d* must have at least 27 points as follows

(columns of the display in (3)) :

222222222111111111000000000

210021102102210021021212210

021210102210102021102001210

210210210210210210210120210 (3)

To check this, note that in any three rows of (3) all the 27 points

,)z y , ,x ( ′ with { } , 2 ,1 ,0 z y , ,x ∈ occur . Thus

On Some Applications Of Hadamard Matrices

18

. 27 =) 3 ; 3 ,3 ,3 ,(3 m

The design presented in Example 5.1.2 is an orthogonal array . [8]

motivated by the kind of logic which led to the problem stated in (1)

defined an orthogonal array as follows : let 2s ≥ , 1k ≥ , 1≥λ and t be

given integers with . k t 1 ≤≤ An orthogonal array (OA) is a Nk × matrix,

with entries from a set { } 1- s ,. . . ,2 ,1 ,0 of s elements such that in any t

rows of the array all possible 1t st × vectors appear with frequency . λ The

array is said to be of size ,N k ,sconstraint s levels , strength ,t and index

. λ Such an array is denoted by . ) ; t , s ,k ,N ( OA λ We refer to

λ ; t , s ,k ,N as the parameters of the array. It is clear from the

definition that s=N tλ and kt ≤ . One may check that the array in

Example 1.2 is an . ) 1 ; 3 ,3 ,4 ,27 ( OA

Since Rao's initial paper introducing OA's in 1947 much effort has been

spent in constructing OA's and studying their properties . In the next

proposition we give a construction method which depends on using a Galois

field .

:1.1nPropositio Let p=snwith 1n ≥ and p a prime . Then there exists

an ) 1 ; 1-k , s ,k ,s ( OA 1-k for each 2k ≥ .

:Proof Since p=sn with p a prime, a Galois field of order s exists .

Let (s)GF=F be such a Galois field . Then

{ } F x : ) x ,. . . ,x ,x ( = x = F . . . F F = F ik21k ∈′×××

is a vector space over F under componentwise addition and scalar

multiplication )( x ,. . . ,x ,x = x k21 αααα for . F∈α Let d* be the

following subset of Fk :

Then d

* is the set of vectors in Fk which are orthogonal to the vector

) 1 ,. . . ,1 ,1 ( = 1′ in Fk . Thus d

* is a subspace of Fk of dimension

. 1-k Hence s|=d| 1-k* . If we write each element of d* as a column vector

and display all the elements of d* in this way we claim that the result is an

, = ) 1 ; 1-k , s ,k ,s ( OA 1-k Γ say . We have already noted that d* has

s1-k elements and hence, displayed as mentioned, is a sk 1-k× matrix with

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entries from F (i.e. the s elements of F form the levels of the array) . It

remains to check that Γ has strength . 1-k Due to symmetry in the

argument, it is enough to consider the subarray Γ′ of Γ consisting of the

first 1-k rows in checking that Γ has strength . 1-k Take any vector of

the form )( 0 ,y ,. . . ,y ,y = Y1-k21

′ in Fk . Then let Fy ∈ be the

unique element such that 0 =y + y i

1-k

=1i

. Then

d y ,y ,. . . ,y ,y = z *

1-k21∈′ )( and since y is unique, Y ′ appears

exactly once in the subarray Γ′ of Γ obtained by deleting the last row in

.Γ Hence Γ has strength 1-k and index 1. �

We illustrate the proposition in our next example .

:1.3Example Let ,4=2=s 2 and . 3=k We now consider

{ } 1+x ,x ,1 ,0 = GF(4) = F under mod )( 1+x+x ,2 2 arithmetic, where

1+x+x2 is the primitive irreducible polynomial used to generate . GF(4)

Then we list the 13 × column vectors of the vector space F3 orthogonal to1′

(by Proposition 1.1 this gives us the desired OA) :

1+x1+x1+x1+xxxxx11110000

1x1+x0x011+xx1+x011+xx10

x101+x0x1+x11+xx101+xx10

If one wishes the entries of the OA to be in the set { } 3 ,2 ,1 ,0 this is

easily accomplished by the following recoding : replace x by 2 , and 1+x

by 3 in the above array . We then get :

3333222211110000

1230201323013210

2103023132103210

which is an . ) 1 ; 2 ,4 ,3 ,16 ( OA

Many different construction methods for OA's have been developed

(see e.g. [6] and the references therein) . In the next part we discuss one of

∑V k

k-1

i=1

On Some Applications Of Hadamard Matrices

20

these methods which involves the use of Hadamard matrices for

constructing OA's involving two levels .

2. Construction Of Orthogonal Arrays Via Hadamard Matrices

The purpose of this part is to present a method of constructing two level

orthogonal arrays using Hadamard matrices . We begin with the observation

that if an OA is of strength t then the same array is of strength t′ for

t<t1 ′≤ .

:2.1nPropositio If ) ; t , s ,k ,N ( OA= λΓ then Γ is also an

orthogonal array of strength ,t<t′ specifically

. ) s ; t , s ,k ,N ( OA= t-t ′′Γ λ

:Proof Let Γ be an orthogonal array )( ; t , s ,k ,N λ and let

. t < t 1 ′≤

Consider any t rows of -Γ for convenience we choose the first t rows, and

note that since Γ is given to be of strength t and index ,λ any t-plet,

, v

u

will repeat λ times, where u has t′ entries and v has t-t ′ entries .

With u fixed, there exist st-t ′ choices for . v Now since ,s =N tλ then

. s ) s ( = ) s . s ( =N tt-ttt-t ′′′′ λλ

�

We are now ready to state our main result, namely that an OA with two

levels and strength 3 and n constraints exists if and only if a Hadamard

matrix of order n exists .

:2.1Theorem An orthogonal array )( 4

n ; 3 ,2 ,n ,2n OA exists if

and only if a Hadamard matrix of order n exists .

:Proof Assume that there exists an orthogonal array

4

n ; 3 ,2 ,n ,2n OA= )(∆ . Let ∆

* be obtained from the matrix ∆ by

replacing the 0 symbol in ∆ by -1 so that the only entries in the matrix are

now -1 and 1 . Also observe that each row of ∆* has precisely n plus ones

and n minus ones, since ∆* is also (by Proposition 2.1) an array of strength

Majalat Al-Ulum Al-Insaniya wa

1=t and index . n=λ Then without loss of generality, we can arrange the

first three rows R ,R ,R 321 ′′′ of ∆

where H ,H 21 are matrices of order

( ,1- ,1- ,1 ,. . . ,1 ,1 = R1′

n minus ones, and ( ,u = R 12′

rows of ∆* apart from . R1′ Note that only the 3

(1,-1,1) , (1,-1,-1) will appear among the first three rows of

these four 3- plets must appear with frequency

∆* has strength 3=t and index

. 0 = v . 1 = v . u = u . 11111

′′

. 0 = v . 1- = v . u = u .1-2222

′′

of ,*∆ it follows that either of H

. n Conversely, let H be a Hadamard matrix of order

( ) . H- | H = *∆ Note that ∆

*

and -1. In order to prove that

parameters, it is only necessary to show t

arbitrary 3-plet will appear 4

n=λ

any two rows of ∆* all possible 2

)( k j , ,i f denote the frequency of the +/

. H Thus if ; x = ) + ,+ ,+ ( f 1

; x = ) + ,+ ,- ( f 3 ,- ,- ( f

orthogonal in pairs, the ix satisfy the following equations :

Insaniya wat - Tatbiqiya

21

Then without loss of generality, we can arrange the

∆* in the partitioned form as follows:

are matrices of order n,

)() 1- ,1 = 1- ,. . . , ′′ has n plus ones and

)() v ,v = R , u 2132 ′ are any two distinct

Note that only the 3-plets (1,1,1) , (1,1,-1) ,

1) will appear among the first three rows of . H 1 Each of

plets must appear with frequency 4

n( i.e. ))(mod 4 0n ≡ since

and index . 4

n=λ From this it follows that

A similar argument shows that

Since )( 3 ,2 ,1=i ,Ri are arbitrary rows

H 1 and H 2 is a Hadamard matrix of order

be a Hadamard matrix of order n and let * has 2n columns, n rows, and 2 levels, +1

∆* is an orthogonal array with the given

parameters, it is only necessary to show that in any 3 rows of ∆* an

4

n times . Since H- is the negative of ,H in

all possible 2-plets will appear 2

n times . Let

denote the frequency of the +/- triplet k j , ,i in any 3 rows of

; x = ) + ,- ,+ ( f 2

,x = ) + 4 then since the rows of H are

satisfy the following equations :

On Some Applications Of Hadamard Matrices

22

0=x+x-x-x 4321 , 0=x-x-x+x 4321 , 0=x-x+x-x 4321 ,

n=x+x+x+x 4321 . From these equations, 4

n=x=x=x=x 4321 , and the

theorem is established. �

:2.1Corollary Let n be a Hadamard number . Let ( ) H- | H = nx2n

∆ ,

where H is a Hadamard matrix of order . n Let ∆* be obtained from the

matrix ∆ by replacing each -1 by 0 . Then

. ) 4

n ; 3 ,2 ,n ,2n ( OA = *

∆

:Proof This follows immediately from the proof of Theorem 2.1 . �

:2.2Corollary If n is a Hadamard number, then one can construct an

orthogonal array ,Γ such that . ) 2

n ; 2 ,2 ,n ,2n ( OA = Γ

:Proof This follows immediately by using Proposition 2.1 and Theorem

2.1. �

:2.1Example Consider the following Hadamard matrix H of order 4 :

11-1-1

1-1-11

1-11-1

1111

=H .

Then ( ) ,

1-111-

111-1-

11-11-

1-1-1-1-

11-1-1

1-1-11

1-11-1

1111

= H- | H =

∆ and

Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya

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thus

∆

01101001

11000011

10100101

00001111

=* is an orthogonal array

. ) 1 ; 3 ,2 ,4 ,8 ( OA

:2.2Example Consider the Hadamard matrix H of order 12 given in

[1], [2] and [3] . Then ) H- | H ( = ∆

and thus ∆* below is an orthogonal array ) 3 ; 3 ,2 ,12 ,24 ( OA :

On Some Applications Of Hadamard Matrices

24

3. Block Designs

In this part we introduce the concept of a t-design . For the basics of the

theory of such designs and their properties we refer to [4] and [9].

An appropriate starting point for defining the notion of a t-design is to

first define a cover of a set . LetV be any nonempty set . If I is a nonempty

set then

{ } VB,Ii:B = ii ⊆∈ζ is called a Cover of V if and only if

BV i

Ii

U∈

⊆ . In design theory and in statistics the following terminology is

customary and we adopt it here:

(a) the set V is called the setunderlying and the elements of V are

called points ,

(b) the set ζ is also called a coveringof V and the elements Bi of ζ are

called blocks .

The concept of a covering for a set is an important one and has led to the

development of important mathematical ideas . For example, the use of

certain types of coverings has led to the notion of compactness in set

topology, and the assumption of compactness is quite often indispensable in

the development of theorems in real analysis or more broadly in functional

analysis .

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In the present topic as well coverings of a certain kind will play an

important role. First we require that the underlying set of points V be a

. setfinite It is convenient to choose V to be the set of the first n positive

integers, namely { }. n ,. . . ,2 ,1 =V Second, we consider coverings

)B(=Iii ∈ζ of V where certain demands are placed on the blocks . These

are

(C1) for each Ii ∈ , k|=B| i , where k is a fixed integer ,n k 1 ≤≤

and

(C2) let t be a nonnegative integer satisfying the restriction k t 0 ≤≤

be given in advance, where k is the cardinality of each block . Then the

number of blocks Bi in ζ which contain any t element subset of V is a

fixed number λ t say depending on t .

We illustrate these two requirements (C1) and (C2) in the following

example :

:3.1Example Let 2n ≥ be any integer . Select a value k such that

. n < k 1≤ Let { } n ,. . . ,3 ,2 ,1 =V be the underlying set of n points .

Let ∑V k be the set of all subsets of V of cardinality ,k that is

{ VB:B = k V ⊆∑ and } k|=B| .

Clearly ∑V k is a covering of V and by definition ∑V k satisfies (C1) .

Moreover, ∑V k satisfies (C2) in an abundant manner, that is, for every

choice of ,k t 0 ,t ≤≤ any t-element subset of V is included in exactly

t-k

t-n=tλ blocks in . k V∑

Example 3.1 prompts one to put another restriction on the kind of

coverings of V we wish to study . This restriction is

(C3) once k is specified, not every k element subset is a block .

We are now ready to formally define a t-design :

Let { } n ,. . . ,3 ,2 ,1 =V be a given underlying set of n points . A

subset B of V with k elements )( n k ≤ will be called a subset-k .

Suppose that a number t is given such that . n < k t 0 ≤≤

A ) ,k ,n (-t tλ design is a pair ,) ,V ( ζ where ζ is a collection of

k subsets of V with the property that each t-subset of V occurs in exactly

λ t blocks . If =ζ ∑V k the ) ,k ,n (-t tλ design is called complete ;

On Some Applications Of Hadamard Matrices

26

otherwise it is called .incomplete With less stringent emphasis we refer to

such a pair ) ,(V ζ as a t-design .

It is clear from the definition that the interesting t-designs involve

coverings ζ of V satisfying (C1), (C2) and (C3) .

If ) ,V ( ζ is a ) ,k ,n (-t tλ design the numbers k ,n ,t and λ t are

collectively called the parametersof the t-design . The number k is usually

referred to as the sizeblock and the number n as the number of treatments

in the design .

The following example illustrates the definition and may be verified

manually . From the discussion in the next section it will be clear how it is

constructed and in the next section we will see that it is a Hadamard 2-

design.

:.2Example 3 Let { } . 8 ,7 ,6 ,5 ,4 ,3 ,2 ,1 =V Thus the

number of treatments is . 8=n We choose ,4=k so that the block size of

each block in the desired covering of V will be 4 . Finally, we take 3=t

and . 1=tλ It may now be verified that the table that follows is a

1) ,4 ,(8-3 design in which the columns are blocks :

1 2 3 4 5 6 7 3 4 5 6 7 1 2

2 3 4 5 6 7 1 5 6 7 1 2 3 4

4 5 6 7 1 2 3 6 7 1 2 3 4 5

8 8 8 8 8 8 8 7 1 2 3 4 5 6

Many interesting questions can be asked about t-designs . We list three

such questions . For answers someto or partial answers we refer to the

books referred to earlier :

1Question : What conditions must be satisfied by the parameters

λ t ,k ,n ,t in order that a t-design with the given

λ t ,k ,n ,t exists ?

An answer to Question 1 would provide necessary conditions for the

existence of t-designs .

2Question : Given a t-design or several t-designs how can they be

used to produce a new t-design ?

An answer (or answers) to Question 2 would provide recursive

construction methods for t-designs .

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3Question : Given parameters , ,k ,n ,t tλ satisfying the necessary

conditions obtained as an answer to Question 1 , can one

construct a t-design with these specified parameters ?

An answer (or answers) to Question 3 would provide ab initio

construction methods for t-designs .

Prior to the 1920's combinatorial lists interested in t-designs were busy

answering questions of the kind posed above . This led to the development

of the theory of t-designs but there were no real practical applications for

this topic . The only applications appeared to be " brain teaser " problems

based on t-designs circulated for mathematical fun ; an area of mathematics

known as recreational mathematics . Post 1920 the situation changed

dramatically. Sir Ronald A. Fisher had his first job posting at an

Agricultural Experimentation station in Rothamsted, England . He soon

realized that the experimental data gathered there had to be carefully

obtained if it was to be meaningfully interpreted for prediction purposes .

He began developing a subject which was named the Design of

Experiments or Experimental Design for short and to go along with it the

statistical theory of Analysis of Variance . One aspect of Experimental

Design in the beginning years required the use of 2-designs . Statisticians

refer to 2-designs as balanced incomplete block ) B I B ( designs .

The knowledge that t-designs had a very practical use gave a huge

boost to the people working in this area and intensive study has been going

on in this area since the 1920's . Returning to Question 3 mentioned above a

vague answer may be given as follows : the demands on the cover are of a

regulated nature ; the blocks are uniform in size (condition C1) , and every

t-subset must be contained in precisely λ t (condition C2) . Thus one might

conjecture that finite algebraic structures such as finite abelian groups and

finite fields may help in the construction of t-designs . The latter algebraic

structures have regulated addition and multiplication tables . This in fact is

the case and many t-design constructions are based on such structures .

In this context there is another regulated structure that comes to mind .

This is, of course, a Hadamard matrix . It's entries are regulated ; they must

be drawn from the set { -1, 1} . Further, the columns and rows must be

mutually orthogonal . Thus one might conjecture that Hadamard matrices

could serve as useful tools in the construction of t-designs for certain

values of . t This in fact turns out to be the case and will be discussed in the

next part .

On Some Applications Of Hadamard Matrices

28

4. Constructions of Block Designs Via Hadamard Matrices

Finally, we give two construction methods which utilize Hadamard

matrices to produce certain 2- and 3-designs . The 2- and 3-designs that we

get are usually called Hadamard 2- and 3- designs .

To make the presentation clearer we begin with two lemmas which will

aid in the justification of our construction procedures . To formulate the

lemmas the following notation and terminology will be useful .

A matrix C of order nm × will be called orthogonal columnwise if

and only if ,D=CC n′ where = Dn diagonal nn × matrix . Thus by

definition the set of columns of a columnwise orthogonal matrix is a

mutually orthogonal set of vectors . In this terminology a Hadamard matrix

is a square (-1,1)- columnwise orthogonal matrix.

:4.1Remark Let C be any nm × columnwise orthogonal matrix . Let

C* be obtained from C by either multiplying a row of C by -1 or by

multiplying a column of C by -1 . Then C* is columnwise orthogonal . To

justify this let E be a diagonal matrix whose i-th entry is -1 and all other

diagonal entries are 1 . If the i-th row of ,C for example, is multiplied by -1

then . C E =C* Hence ,D = C C = C E E C = C C n

** ′′′′ a diagonal matrix .

A similar argument show that C* is columnwise orthogonal if it results

from C when the i-th column of C is multiplied by -1 .

Let A and B be 2n × and 3n × (-1,1)-matrices respectively . We

define the functionsfrequency fA

and fB

on the rows of A and ,B

respectively, as follows:

for { } , 1 ,1- k j , ,i ∈ let

i) = )j i (fA

frequency with which the 21 × vector )j i ( occurs as a

row of ,A

ii) = ) kj i (fB

frequency with which the 31 × vector ) kj i ( occurs as a

row of . B

Note that each of the functions fA

, fB

assume nonnegative integer values .

fA

has for its domain the four element set of 21 × vectors D1 where

{ } . - - ,+ - ,- + ,+ +=D1 )()()()( Similarly fB

has for its domain

the eight element set of 31 × vectors

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{ } . - - - ,+ - - ,- + - ,+ + - ,- - + ,+ - + ,- + + ,+ + + =D2 )()()()()()()()(

:4.1Lemma Suppose that A is a columnwise orthogonal (-1,1)- matrix

of order . 2n × Assume that for each column of ,A the number plus ones in

that column equals the number of minus ones . Then

4

n = -) (-f = +) (-f = -) (+f = +) (+f

AAAA and . 4) ( 0n mod≡

:Proof Since the two columns of A are orthogonal we must have

(1) . +) (-f+-) (+f=-) (-f++) (+fAAAA

Further, since each column has by assumption2

n plus ones and

2

n minus

ones we have

(2)

2

n= -) (-f++) (-f=-) (+f++) (+fAAAA

.

Since each column has n entries , we have

(3) n= -) (-f++) (-f+-) (+f++) (+fAAAA

.

Solving these three equations gives the solution 4

n = j) (if

A for any

,Dj) (i 1∈ and the lemma is established . �

:4.2Lemma Let B be any columnwise orthogonal 3n × (-1,1)-matrix .

Then 4

n = -) - (-f + ) + + (+f

BB and . 4) ( 0n mod≡

:Proof Multiply each row of B which begins with a -1 by -1 and call the

resulting matrix B* . Then the first column of B

* has all its entries +1 .

Moreover, by Remark 4.1 B* is also columnwise orthogonal . Since each of

the second and third columns of B* are orthogonal to the first column of

B* this implies that each of these columns has

2

n plus ones and

2

n minus

ones . From these observations and Lemma 4.1 we have

,4

n = +) (+f = +) + (+f = -) - (-f + +) + (+f

BBBB *** where B** is the

2n × submatrix of B* consisting of the second and third columns of . B

*

Hence the Lemma is established . �

On Some Applications Of Hadamard Matrices

30

I . Construction Of a Hadamard 2-Design

We give the construction procedure in steps :

1Step . Begin with any Hadamard matrix H of order . 8n ≥ Using

Remark 4.1, we can convert ,H if necessary, into a new

Hadamard matrix H* such that each entry in the first row and

first column of H* is plus one .

2Step . Delete the first row and first column of H* and call the resulting

1)-(n1)-(n × (-1,1)-matrix H** .

3Step . Let

{ } 1-n ,. . . ,3 ,2 ,1 =V (4)

and we use V to index each column and each row of H** in

numerical order from left to right and from top to bottom

respectively .

4Step . We will use the set V as the points of setunderlying of the block

design we are trying to construct

5Step . Let ,1-n j 1 ,1-n i 1 ,)h(=H ij** ≤≤≤≤ where { } . 1- ,1+ hij ∈

for each ,1-n i 1 ,i ≤≤ we define the subset Bi of V as follows

:

{ } 1=h,Vk:k =B iki ∈ .

Let { } 1-n i 1:B = i ≤≤ζ . (5)

Steps 1 to 5 , therefore yield the pair ) ,V ( ζ where

{ } 1-n ,. . . ,2 ,1 =V is a set of the first 1-n positive integers and ζ is a

collection of 1-n subsets of . V We note that the value of n must be

congruent to zero mod 4 since it is a Hadamard number . The important

feature of ) ,V ( ζ is recorded in the next theorem (we refer to the setting

described in steps 1 to 5 in the proof of this theorem) :

:4.1Theorem The pair ) ,V ( ζ where V is defined in (4) and ζ is

defined in (5) is a ) 1-4

n ,1-

2

n ,1-n (-2 design .

:Proof By definition the number of elements in . 1-n|=V| ,V Since

H* has for its first row a vector each of whose entries is 1 , every other row

Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya

31

of H* being orthogonal to this row has

2

n plus ones and

2

n minus ones .

Since the blocks Bi are obtained from ,H** it follows that 1-

2

n|=B| i for

any . 1-n i 1 ,i ≤≤ Take any two element subset { } . V j ,i ⊆ Then we

have

(*) { } B j ,i ⊆ for some ζ∈B if and only if there exists some

,1-n r 1 ,r ≤≤ such that . 1 = h = h j ri r

From (*) it follows that the numbers of blocks in ζ which include the set

{ }j ,i as a subset is equal to the frequency ,+) (+fK

where K is the

21)-(n × submatrix of H** consisting of its i-th and j-th columns . But by

Lemma 4.1 , we know that 4

n = +) (+ f

K where K is the 2n × submatrix

of H* consisting of the i-th and j-th columns of .H

* Hence

. 1-4

n = 1-+) (+ f = +) (+f

KK The theorem is established. �

We illustrate Theorem 4.1 in the next Example .

:4.1Example Let

H-H

HH=H

88

88

, where

H-H

HH=H

11

11

8 and

+--+

-+-+

--++

++++

=H 1 . Then H is a Hadamard matrix of order 16 and has

for its first row and first column, vectors each of whose entries is +1 .

Deleting the first row and first column of H we arrive at a 1515 × matrix

H** displayed below ; we index the rows and columns as outlined in steps 1

to 5 with the elements of the set { } : 15 ,. . . ,3 ,2 ,1 =V

On Some Applications Of Hadamard Matrices

32

From H

** we may now derive the 2-(15, 7, 3) Hadamard design using the

construction method outlined in steps 1 to 5 ; in the table below the 15

columns are the 15 blocks B ,. . . . ,B ,B 1521 based on the underlying set

V :

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33

II . Construction Of a Hadamard 3-Design

We give the construction procedure for a Hadamard 3-design in steps :

1Step . Let H be any Hadamard matrix of order . n If necessary, using

Remark 4.1 , convert H into a new Hadamard matrix H* so that

the first row of H has each entry equal to +1 . This will require

one to multiply certain columns of H by -1 .

2Step . Delete the first row of H* to obtain the n1)-(n × matrix H

** .

3Step . Let

{ }n ,. . . 3, ,2 ,1 =V and { } . 1-n ,. . . ,3 ,2 ,1 =V* (6)

Index the columns of H** by the elements of the set V in order

from left to right . Index the rows of H** by the elements of V

*

in order from top to bottom .

4Step . Let 1-n i 1 ,)h(=H ij** ≤≤ and . n j 1 ≤≤ Then { } 1 ,1- h j i ∈ .For

,1-n i 1 ≤≤ we define subsets Bi and Bci of V as follows :

{ Vk:k =Bi ∈ and } , 1=h k i

{ Vk:k =Bci ∈ and } . -1=h k i

5Step . Let

{ } { } , 1-n i 1:B 1-n i 1:B = cii ≤≤≤≤ Uζ (7)

where the Bi , Bci are defined in step 4 .

The important feature of the pair ) ,V ( ζ is recorded in the next

theorem .

:4.2Theorem The pair ) ,V ( ζ where the set V is defined in (6) and

the set ζ is defined in (7) is a ) 1-4

n ,

2

n ,n (-3 design .

:Proof By definition the number of points in the underlying set V is . n

Since the first row of H* is a vector of plus ones, and this is orthogonal to

each row of ,H** it follows that each row of H

** has 2

n plus ones and

2

n

minus ones . Hence by the definition of the blocks Bi and Bci in step 4 ,

we have that for each ,i ,1-n i 1 ≤≤ the block size is 2

n|=B|=|B| c

ii ,

On Some Applications Of Hadamard Matrices

34

Finally, let { } k j , ,i be any three element subset of . V Then by the

definition of the blocks given in step 4 we have

(**) { } B k j , ,i ⊆ for some ζ∈B if and only if there exists some

Vr *∈ such that 1=h=h=h k rj ri r or -1=h=h=h k rj ri r .

From (**) we get that the number of blocks ζ∈B which contain{ } k j , ,i

as a subset is equal to ,-) - (-f + +) + (+fMM

where M is the 31)-(n ×

submatrix of H** consisting of the i-th , j-th and k-th columns of . H

**

Consider the 3n × matrix

M

+++=M

* . This matrix is columnwise

orthogonal and hence by Lemma 4.2 . 4

n = -) - (- f + +) + (+ f

MM**

Hence 1-4

n = -) - (-f + +) + (+f

MM. This completes the proof . �

We conclude this part by illustrating Theorem 4.2 in the following

example :

:4.2Example

Let

+--+

-+-+

--++

++++

=H 1 and

H-H

HH=H

11

11

8 . Finally take

. Then H is a Hadamard matrix of order 16 and has a

first row and first column each of whose entries is +1 . Deleting the first

row of H we arrive at a 1615 × matrix H** displayed below ; we index the

rows and columns as outlined in steps 1 to 5 with the elements of the set

{ } : 16 ,. . . ,3 ,2 ,1 =V

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35

From H

** we may now derive the 3-(16, 8, 3) Hadamard design using the

construction method outlined in steps 1 to 5 ; in the table below the 30

columns are 30 blocks B ,B ,. . . ,B ,B ,B ,B ,B ,Bc1515

c33

c22

c11 based

on the underlying set V :

On Some Applications Of Hadamard Matrices

36

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