Composed Reed Solomon Sequences Generated by ith Partial Sum of Geometrical Sequences

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*Correspondence to Author: Dr. Ahmad Hamza Al Cheikha, Math. Sci. Dep. Col. of Arts Sci. & Edu. Ahlia Uni. Manam, Bahrain

How to cite this article:

Ahmad Hamza Al Cheikha. Com-posed Reed Solomon Sequences Generated by ith Partial Sum of Geometrical Sequences. American Journal of Computer Sciences and Applications, 2017; 1:2.

eSciPub LLC, Houston, TX USA. Website: http://escipub.com/

Ahmad Hamza Al Cheikha, AJCSA, 2017; 1:2

American Journal of Computer Sciences and Applications

(ISS

N

:2

5

7 5-

77 5X

)

Research Article AJCSA, 2017, 1:2

Composed Reed Solomon Sequences Generated by i

th

_{Partial Sum}

of Geometrical Sequences

Reed–Solomon codes are an important group of error-correcting

that were introduced by Irving S. Reed and Justine Solomon in

1960. They used in the error coding control, special in systems

that have two way communication channels in two externally

ap-plications: deep telecommunications and the compact disc. They

have many important applications, the most prominent of which

include consumer technologies such as CDs, DVDs, Blue-ray

Discs, QR Codes, data transmission, technologies such as DSL

and Wi MAX, broadcast, systems such as DVB and ATSC, and

storage systems such as RAID 6 . They are also used in satellite

communication.

This research is useful to generate new Reed Solomon Codes

and their composed sequences using the ith partial sum of

geo-metrical sequences with the bigger lengths and the bigger

mini-mum distance that assists to increase secrecy of these

informa-tion and increase the possibility of correcting mistakes resulting

in the channels of communication.

Keywords:

Minimal polynomial; Minimum distance; BCH

quences; Reed-Solomon sequences; Quasi- orthogonal

Se-quences; Orthogonal seSe-quences; Code; Span.

Dr. Ahmad Hamza Al Cheikha

Math. Sci. Dep. Col. of Arts Sci. & Edu. Ahlia Uni. Manam, Bahrain

ABSTRACT

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Http://escipub.com/american-journal-of-computer-sciences-and-applications/ 00011

1. Introduction

1.1

Reed Solomon Sequences(Codes)

Reed-Solomon codes RS: The Reed-Solomon

code RS over F_qis BCH code with length Nq1

, q2, and: length, dimension and minimum

distance of the code denoted by N, K, D. This code

is a linear and cyclic. If q 2mthen we can

represent each element in F_qin row with the length

m and its components are

in F₂according to basis in F_qand the new code will

be of the form: [nmN,kmK,dD]_.

The binary representation maintains on the

linearity (but not necessary on the cyclic).

The RS codes are very important because:

• They are preferred codes when the length of

the code need to be less than the order of

field.

• They are appropriate to construct other codes

(as binary codes and concatenated codes

likes Justine codes and MDS codes).

• They are useful for correct impulsivity errors.

(Fraleigh, 1971; Yang, 1998).

2. Research Methods and Materials

• Definition 1: If F is a finite filed then any

generator of a the cyclic group F*is called

primitive element inF (Fraleigh. 1971; Jong

et al, 1998).

• Definition 2: The polynomial fF

 

x , of

degree K1is called a primitive polynomial

over Fif it irreducible for some of its primitive

elements of F (Jong at el, 1995; Lidl and

Niederreiter, 1986).

• Definition 3: Suppose G is a set of binary

vectors of the length n:





 



X;

X

x

,

x

F

0,1



G



₀ ₂



_n_₁ _i



₂



and 1*1 and 0*1, then the set G is called

Orthogonal set if it satisfies the two following

conditions :

1.



1,0,1



x

: G

X n 1

0 i

* i  



 



That is the difference between the number of “1”s

and the number of “0”s in X at most is one.

2. X,Y G and X Y: n 1x y



1,0,1



0 i

* i *

i  

 

 



(Lee and Miller, 1998; Al Cheikha, 2005)).

* Definition 4: The set G is called quasi

orthogonal if it satisfies the two following

conditions:

1. X,Y G: n 1x k

1 i

* i 



 



(3)

2 .















1 n

0 i

* i * i

y

x

:

Y

X

G,

Y

X,

Degree of similarity could be determined by [k,] (Yang Kim,

Kumar, 2000; Yang, 1998).

• Definition 5. Minimum distance d: The

minimum distance d of a set C of binary

vectors is:

C

y

x,

y),

d(x,

min

d

C y

x,





 (Mac William and

Sloane, 1978).

• Definition 6. The code C of the form [n, k, d]

if each element (Code word) has the: length

n, rank k (the number of information

components, Message), minimum distance d

(Gong and youssef, 2002).

• Definition7. If C is a set of binary sequences

and ω is any binary vector then:



x (ω);x C



)

C(ω  _i _i  , when , we replace

each “1” in xi by ω_{and each “0” in x}i by

ω

(Al Cheikha and Ruchin, 2014; Al Cheikha,

2014).

• Definition 8. If M{x₁,x₂,...,x_n}, where:

m i F

x  ,

n



m

and F is a field , then

i

SpanM na x

1 i i

 

 (Fraleigh. 1971).

• Corollary 1

.

If in the binary vector x: the

number of ”1”s and the number of “0”s are

1

m and m₂ respectively, and in the binary

vector ω: the number of”1”s and the number

of “0”s are n₁ and n₂ respectively, then in

the binary vector

x(ω

)

: the number of”1”s

and the number of “0”s are n₁m₁n₂m₂

and n₁m₂ n₂m₁ respectively (Al Cheikha

2016; Al Cheikha 2016).

• Theorem 2. _n

2

F is isomorphic to

F

₂nand

Elements set _n

2

F have n2n1 of ones and

1 n

n2  of zeros (Lee and Miller, 1998).

• Theorem 3. If (G,+) is a additive group, A is

a subgroup of G and satisfies the following

two conditions:

A b a A b A, a 2.

A b b A b , b .

1 ₁ ₂ ₁ ₂

    

   

For eacha,b,b₁,b₂G, then: GA(bA)

(Mac William and Sloane, 1978).

3. Results and Discussion (Findings)

3.1.Generating

RS

codes

using

geometrical sequences

If F₂n is Galois field of order

n

2 ,F₂ {0,1}, αis a

prime element and we construct the geometrical

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Http://escipub.com/american-journal-of-computer-sciences-and-applications/ 00013 α, thus, we have the sequence:

) , α α, 1, , α , , α α, (1,

X 2  2n2 2  ₍₁₎

and it is a periodic sequence with the period

2

n



1

and contains all non zero elements of F₂n Al

Cheikha, 2016).

The set A



a_i,i0,1,2,...,2n 2



, where

) α ,..., α α, (1, α ) α ,...., α , (α

a_i  i i1 2n2i  i 2 2n2

when the powers are computed by mod

2

n



1

, is

linear and closed under the addition and form

Reed Solomon Code with: length N =

2

n



1

,

minimum distance D =

2

n



1

_{and dimension K= 1}

and A{a_i} where

a

_i



(

α

i

,

α

i1

,....,

α

2n2i

)

is

the binary representation of

a

_i.

The set Ais closed under the addition but not

necessary linear with: length

N



n(2

n



1 )

,

minimum

distance

D



n2

n1and dimension K2 , Thus

Ais quasi-orthogonal set and each sequence in A

contains 2n1nof “0”s_.

I. We extend

a

_i

,

i



0.1.2. 

,2

n



2

to the

sequence

~

a

_i



(a

_i

.0)



(α

i

,

α

i1

,



,

α

2n2i

,

0 )

by adding the zero of F₂n , and

A

{

a

~

}

~

i



where

II.

~

a

_i



(

α

i

,

α

i1

,....,

α

2n2i

,

0 )

, is the binary

representation of

~

a

_i .

The set

A

~

_{is linear, closed under the addition}

and form code with: length N~ 2n_{, minimum}

Distance D~ 2n 1 and dimension

K

~

=1.

The set A~is closed under the addition but not

necessary linear with: lengthN~ n2n, minimum

Distance D~ n2n1and dimension

K

~



2

.

Thus each of ~a_i,~a_j,a~_i ~a_j( if i j) , has

n

2

n1

of “1“ and n 1

2 n

 of “ 0 “ and A~ is an orthogonal

set.

III. Compose A with A or A(A)is

quasi-orthogonal sets, A(A), A(A) and A(A)are

orthogonal sets (Al Cheikha 2016).

3.2. Generating RS codes using partial

sum of geometrical sequences

First Step: We suppose

s_i

the i

th

_{partial sum of}

the Series (2):



  

  

1 α α α  1

S 2 2n 2

(2)

and

*

N i i)

s

( _

the sequence of the i

th

_{partial sums}

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,...

3 ,

2 ,

1 i

,

α

1 α

α

1 s

i 1 i

i

_











_



From

α

2n1



1

, the sequence (s_i)

is periodic with period

2

n



1 , and:

1 i

),

,

α

1 α

1 ,

α

1 α

1 0,

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 (

)

s

(

2 1 2 2 i n

















Assuming that C{c_i,i0,,2n 2} where:

0 i

),

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 (

c

i 1 2 2 i 1 i i n







 







When the power computed by

mod

2n 1

and

C{c_i}

where

c_i

is binary representation of

c_i

, we can

find:





















0 α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 c

1 2 2 0 n



                  α 1 α 1 , α 1 α 1 , , α 1 α 1 , α 1 α 1 c 1 2 3 2 1 n 

………

















 

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 ,

,

α

1 α

1 c

i 1 2 1 i i n



………

                

 ₁ _α

α 1 , , α 1 α 1 , α 1 α 1 c 2 2 1 2n 2 2 n n 

Thus each of

c

_i

,

i



0,



.,2

n



2 contains all the field elements

n

2

F

except

      α 1 1

. computing

j i c

c 

for

ji

, we can find:

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















      

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 ,

,

α

1 α

1 ,

,

α

1 α

1 ,

α

1 α

1 c

j i) (j 1 i) (j 1 2 1 2 2 j 1 j j n n



                             α 1 α α , , α 1 α α , α 1 α 1 , , α 1 α 1 , , α 1 α α , α 1 α α c c j i i) (j 1 i j i) (j 2 j 2 i 1 j 1 i j

i   

                            α 1 α 1 α , , α 1 α 1 α , α 1 α 1 , , α 1 α 1 α , , α 1 α 1 α , α 1 α 1 α i j i i j i j i j j i i j 2 i i j 1

i _ _ _ _













_







 i1 i2 ji 2 1 i i

j

i

α

,

α

,

α

,

,1

α

,

α,

,

α

1 α

1 c

c



n



And

c_i c_j

contains all elements field

F₂n

except

“0”

Where

3 0,...2

i

2,

1,...2

j

,

i

j

for

,

0 α

1 α

1

j i _n _n

















and:

a. For

c_iC

, the number of “1”s in

c_i

is





























α

1

1 w

n2

n 1

and the number of “0”s is

                        α 1 1 w n

n2n 1

and the difference

between the number of “1”s and the number

of

b.

‘0” is

_              α 1 1 2w n

.

c. For

c_i,c_jC

and

i



j

, then

c

_i



c

_j

contains

1 n

n2



of “1“s (i.e: the number of

disagreement

elements of

c_i

,

c

_j

) and (

n.2n1n

) of “0“

s (i.e: the number of agreement elements of

i

c

j

c

) and the difference between the

number of “1”s and the number of ‘0”s is

also n.

•

Thus:

C

form a quasi-orthogonal set of

degree

                    

 ,n

α 1

1 2w

n

and C,

C

are nonlinear.

•

The set C is a cyclic code but nonlinear

over

F

_q

(when

q2n

) with the length

1 q

N





and minimum distance D

=

N –

1. We can assume that C is a Reed Solomon

Code after closing eyes to the linear property.

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C

Number of “1”s

Number of “0”s

Number of “1”s

Number of “0”s





























α

1

1 w

n2

n 1 _

                      α 1 1 w n

n2n 1

_



























α

1

1 w

n2

n 1 _

                      α 1 1 w n n2n 1

* For

c_k C

we define the set:

P_k C

 

c_k {p_i c_i

 

c_k ,c_iC}

then:

a. The number of “1”s in

p_i

is:

                                                                               α 1 1 w n n2 α 1 1 w n n2 α 1 1 w n2 α 1 1 w n2 1 n 1 n 1 n 1 n                           α 1 1 2nw ) 2 (1 n α 1 1 w 2 2

2n 2 n

2 1)

2(n 2

b. The number of “0”s in

p_i

is:

                                       α 1 1 w n n2 α 1 1 w n2

2 n 1 n 1

2 n 2 1) 2(n 2 α 1 1 w 2 α 1 1 w 2n 2 n 2 2n _                        

c. The difference between the number of “1“s and the number of “0“s is:

2 2 2 α 1 1 2w n n α 1 1 w 4n α 1 1 w 4                                      

d. for

p

_i

,

p

_j



P

_k

and

i j

the

p

_i



p

_j



(

c

_i



c

_j

)(

c

_k

)

then:

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



_

  

  

   

 

     

   

    

 

     



  

 

α 1

1 w n n2

n n2 α

1 1 w n2

n2n 1 n 1 n 1 n 1





_

    

 

 

 

α 1

1 w n 2 1 n 2

2n2 2(n 1) 2 n

.

•

The number of “0” s in

p

_i



p

_j

is:





_

  

 

     

  

     

  

   

 

     

 

  

 

α 1

1 w n2

n n2 α

1 1 w n n2

n2n 1 n 1 n 1 n 1

     

 



 

α 1

1 w n 2 n 2

2n2 2(n 1) 2 n

of “0”s.

And the difference between the number of “1”s

and the number of “0”s is:

_     

 

α 1

1 w 2n

n2

.

Thus

P_k

is a quasi-orthogonal set of degree

[

2

α 1

1 2w

n _

  

 

     



 ,

[

_

    

 

α 1

1 w 2n

n2

]]

.

Second Step: Extending C

By extending the sequences (

c_iC

) to the

sequences

















α

1

1 ,

c

~

i

by adding the term

* 2n

F α 1

1





, and assuming

C {~c}

~

i



and

} c ~ {

C~  _i

where

~c

is the binary representation

of

~c

.

For (

i



j

), each of

~c_i,~c_j,~c_i ~c_j

contains all

the field elements of

n

2

F

, also each of

j i j

i,~c ,~c ~c

c

~ _

_contain

n 1

n2



of “1“s and the

same number of “0“s , and

C~

form orthogonal

set .

The sum of two different elements of

C~

does

not belong to

C~

, i.e:

C~

and

C~

are nonlinear.

Compose

C~

with

C~

or

C

~

(

C

~

)

C

~

C

~

Number of “1”s

Number of “0”s

Number of “1”s

Number of “0”s

1 n

(9)

* For

c_kC

we define the set:

 

~c {~p ~c

 

~c ,c C}

C~ P ~

i k i i k

k    

then:

a. The number of “1”s in

~p_i

is:

2 n

2

2(n1)

of “

1”s.

b. The number of “0”s in

~p_i

is:

2 n

2

2(n1)

of “

0”s.

c. The difference between the number of “ 1 “s

and the number of “ 0 “s in

~p_i

is zero.

d.

For

~p_i,~p_j~P_k

and

i j

thus

)

c

~

)(

c

~

c

~

(

p

~

p

~

k j i j

i







has the same number of

“1“s and the same number of “0“s and the

same difference.

Thus:

P~_k

is an orthogonal set.

Third Step: Compose

C

with

C~

or

C(C~)

C

_C

~

Number of “1”s

Number of “0”s

Number

of

“1”s

Number of “0”s

   

 

     

 



α 1

1 w

n2n 1 _

  

  

   

 

     

  



α 1

1 w n n2n 1

1 n

n2



n2

n1

* For

c_k C

we define the set:

Z_k C

 

~c_k {z_i c_i

 

~c_k ,c_i C}

then:

a. The number of “1”s in

z_i

is:

    

  

    

  

   

 

     

   

   

 

     

 

   

α 1

1 w n n2

α 1

1 w n2

n2n 1 n 1 n 1

1 n 2 1) 2(n 2

2 n 2

2n   



b. The number of “0”s in

z_i

is:

_   

  

    

  

   

 

     

   

   

 

     

 

   

α 1

1 w n n2

α 1

1 w n2

n2n 1 n 1 n 1

2n222(n1)n22n1

(10)

d. for

z

_i

,

z

_j



Z

_k

the

z

_i



z

_j



(

c

_i



c

_j

)(

~

c

_k

)

contains:

2 2(n1) 2 n 1

2 n

2 2n



of “1“s

and the same number of “0“ and the

difference between the number of “1”s and the

number of “0”s is zero.

Thus:

Z_k

is an orthogonal set

.

Forth Step: Compos

C~

with

C

or

C

~

(

C

)

C

~

C

Number of “1”s Number

of

“0”s

Number of “1”s

Number of “0”s

1 n

n2



n2

n1





























α

1

1 w

n2

n 1 _

  

  

   

 

     

  



α 1

1 w n n2n 1

* For

c_k C

we define the set:

~Z_k C~

 

c_k {~z_i ~c_i

 

c_k ,c_i C}

then:

a. The number of “1”s in

~z_i

is:

    

  

    

  

   

 

     

   

   

 

     

 

   

α 1

1 w n n2

α 1

1 w n2

n2n 1 n 1 n 1

1 n 2 1) 2(n 2

2 n 2

2n   



b. The number of “0”s in

~z_i

is:

    

  

    

  

   

 

     

   

   

 

     

 

   

α 1

1 w n n2

α 1

1 w n2

n2n 1 n 1 n 1

1 n 2 1) 2(n 2

2 n 2

2n   



c. The difference between the number of

“1“s and the number of “0“s is zero.

d. for

~

z

_i

,

~

z

_j



Z

_k

the

~

z

_i



~

z

_j



(

~

c

_i



~

c

_j

)(

c

_k

)

contains the same number of “1“s, the same

number of “0“s and the same difference

between the number of “1”s and the number of

“0”s.

(11)

Http://escipub.com/american-journal-of-computer-sciences-and-applications/ 000110 Example 1. Using the prime polynomial

1 x x

f(x)  3   over F2 , extending F2 to

F

₂3and if

α

is a root of f(x) and prime element in

F

₂3then

} α α, 1,

{ 2 is a basis of

F

₂3and the following

Table 1: A binary representation of ₃

2

F

Binary Representation

3

2

F

Binary Representation

3

2

F

011

1 α α3 _ _

000 0

110

α α α4 _ 2 _

001 1

111

1 α α

α5 _ 2 _ _

010



101

1 α α6 _ 2 _

100

2

α

Table 2: Elements A and A~after adding the null column

0

i

a

i a ~

0

6

α

5

α

4

α

3

α

2

α 

1

0

a ~

0 1

6

α

5

α

4

α

3

α

2

α 

1

a ~

0



1

6

α

5

α

4

α

3

α

2

α

2

a ~

0

2

α 

1

6

α

5

α

4

α

3

α

3

a ~

0

3

α

2

α 

1

6

α

5

α

4

α

4

a ~

0

4

α

3

α

2

α 

1

6

α

5

α

5

a ~

0

5

α

4

α

3

α

2

α 

1

6

α

6

a ~







1 α

α

1 α

S

2 3 6

2 3 5 4 5 2 3

3 2

1

α α α s ; α α α 1 s

; α α 1 s ; 1 s

   

  

   

1 s ; 0 α 1 α s

; 1 α α α s ; α α α s

8 6

2 7

2 5 6

4 2 5

 

  

    

(12)

Noting that

α 3 α4

α 1

1 _ _





.

Table 3: Elements

C

and

C~

in

F

₂3

4

α

i

c

i

c ~

4

α

0

6

α

2

α

5

α

3

α

1

0

c

~

4

α

1

0

6

α

2

α

5

α

3

α

1

c ~

4

α

3

α

1

0

6

α

2

α

5

α

2

c ~

4

α

5

α

3

α

1

0

6

α

2

α

3

c

~

4

α

2

α

5

α

3

α

1

0

6

α

4

c ~

4

α

2

α

5

α

3

α

1

0

6

α

5

c

~

4

α

6

α

2

α

5

α

3

α

1

0

6

c

~

Table 4: Elements

C

and

C~

in

F

₂3

110

i

c

i

c ~

110

000

101

010

100

111

011

001

0

c

~

110

001

000

101

010

100

111

011

1

c

~

110

011

001

000

101

010

100

111

2

c

~

110

111

011

001

000

101

010

100

3

c ~

110

100

111

011

001

000

101

010

4

c

~

110

010

100

111

011

001

000

101

5

c ~

110

101

010

100

111

011

001

000

6

(13)

Noting that

w

 

α 2 α

1 1

w _ 4 

    



, we can find:

1. For

c_i C

, the number of ones is

10

2 )

3(2

31





and the number of

zeros i

3(231)



32



11

and

C~ c ~

i 

, contains

1 3

2 (

3



) of ones and

the same number of zeros .

2. For

c_i,c_jC,i j

then

c

_i



c

_j

contains

3(231)12

of ones ( which

is the number of

3. disagreement numbers between

j i

,

c

) and

3 (

2

31

)



3 

9 of zeros (

which is the

4. number of agreement numbers

between

c

_i

,

c

_j

) , also

C~

form a

nonlinear orthogonal set.

5. Compose

C~

with

c

₀

or

C~(c0)



0

c

001011111100010101000,

c

₀



110100000011101010111



)

c

(

c

~

0

0 110100000011101010111110100000011101010111001011111100010101000

110100000011101010111001011111100010101000001011111100010101000

001011111100010101000001011111100010101000001011111100010101000

001011111100010101000110100000011101010111110100000011101010111

110100000011101010111001011111100010101000110100000011101010111

001011111100010101000110100000011101010111001011111100010101000

110100000011101010111110100000011101010111110100000011101010111

001011111100010101000001011111100010101000110100000011101010111



)

c

(

c

~

0

1 110100000011101010111001011111100010101000001011111100010101000

001011111100010101000001011111100010101000001011111100010101000

001011111100010101000110100000011101010111110100000011101010111

110100000011101010111001011111100010101000110100000011101010111

001011111100010101000110100000011101010111001011111100010101000

110100000011101010111110100000011101010111110100000011101010111

110100000011101010111110100000011101010111001011111100010101000

001011111100010101000001011111100010101000110100000011101010111



)

c

(

c

~

0

2 001011111100010101000001011111100010101000001011111100010101000

001011111100010101000110100000011101010111110100000011101010111

(14)

001011111100010101000110100000011101010111 001011111100010101000

110100000011101010111110100000011101010111 110100000011101010111

110100000011101010111110100000011101010111 001011111100010101000

110100000011101010111001011111100010101000 001011111100010101000

001011111100010101000001011111100010101000 110100000011101010111



)

c

(

c

~

0

3 001011111100010101000110100000011101010111110100000011101010111

110100000011101010111001011111100010101000110100000011101010111

001011111100010101000110100000011101010111001011111100010101000

110100000011101010111110100000011101010111 110100000011101010111

110100000011101010111110100000011101010111 001011111100010101000

110100000011101010111001011111100010101000 001011111100010101000

001011111100010101000001011111100010101000 001011111100010101000

001011111100010101000001011111100010101000 110100000011101010111



)

c

(

c

~

0

4 110100000011101010111001011111100010101000110100000011101010111

001011111100010101000110100000011101010111 001011111100010101000

110100000011101010111110100000011101010111 110100000011101010111

110100000011101010111110100000011101010111 001011111100010101000

110100000011101010111001011111100010101000 001011111100010101000

001011111100010101000001011111100010101000 001011111100010101000

001011111100010101000 110100000011101010111 110100000011101010111

001011111100010101000001011111100010101000 110100000011101010111



)

c

(

c

~

0

5 001011111100010101000110100000011101010111 001011111100010101000

110100000011101010111110100000011101010111 110100000011101010111

110100000011101010111110100000011101010111 001011111100010101000

110100000011101010111001011111100010101000 001011111100010101000

001011111100010101000001011111100010101000 001011111100010101000

001011111100010101000110100000011101010111 110100000011101010111

(15)

001011111100010101000001011111100010101000110100000011101010111



)

c

(

c

~

0

6 110100000011101010111110100000011101010111110100000011101010111

110100000011101010111110100000011101010111001011111100010101000

110100000011101010111001011111100010101000001011111100010101000

001011111100010101000001011111100010101000001011111100010101000

001011111100010101000 110100000011101010111110100000011101010111

110100000011101010111001011111100010101000 110100000011101010111

001011111100010101000110100000011101010111 001011111100010101000

001011111100010101000001011111100010101000 110100000011101010111

We can look that: Length C~(c₀)is



2

1



2 n

N

~_z



2 n n



= 3(23)(23 -1) = 168,

Minimum distance is

d

_z~



2n

2

2(n1)



n

2

n1=

1 3 2 1) 2(3 2

2

3 )2

2(3



 = 252.

Dimension



2

4. Conclusion

If αis a primitive element in the field _n

2

F then the

geometrical sequence X(1,α,α2...,αn,...)

and S , where S is the sequence of ith_{partial sum}

of X , are periodic with period 2n 1, and: if C is

the all permutations of one period of S, C~ is

extending of C by adding

α 1

1

 to the end of each

period in C and C~&C~ the binary representation of

C~ &

C respectively, then:

1. The set C is cyclic sequences and form Reed

Solomon code of: length N2n 1, minimum

distance

D



2

n



2

and dimension of Span C,

K



2

_.

2. The set C_{is nonlinear, not cyclic and}Span Cis

Quasi-Orthogonal set (code) of order

    

  

   

 

     



 ,n

α 1

1 2w

n , with:length Nn(2n 1)

, minimum distance _

    

   

α 1

1 w 2

n

d n 1 and

dimension K 2.

3. The set C~_{, extending C by adding} α 1

1

 , is

nonlinear, not cyclic sequence and form codes

with: length N~ 2n, minimum distance

1

2 d

~

_

n

_

and Span C~of dimension

k

~



2

.

4. The set C~_{is nonlinear, not cyclic sequences and}

(16)

Http://escipub.com/american-journal-of-computer-sciences-and-applications/ 000115 length

N

~



n2

n, minimum distance

d

~



n2

n1,

and dimension

~

k



2

.

5. The sets PC(C) are nonlinear, not cyclic and

quasi-orthogonal sets with degrees:

[

2

α 1

1 2w

n _

  

 

     



 ,

[

_

    

 

α 1

1 w 2n

n2

]]

6. The sets P~C~(C~) are orthogonal sequences

with: length N~_P~ n222n, minimum distance

1) 2(n 2 P

~ 2n 2

d

~ _

 and dimension 2_.

7. The sets Z C(C~) are orthogonal sequences

with: length

N

_z



n

2

n



2

n



1



, minimum

distance

d

_z



2n

2

2(n1)



n

2

n1and dimension

2 k_z  _.

8. The sets Z~C~(C) are orthogonal sequences

with: length

N

_z~



n

2

n



2

n



1



, minimum

distance

d

_z~



2n

2

2(n1)



n

2

n1 and

dimension

k

~_z



2

5.

Acknowledgment

The author express their gratitude to Prof. Abdulla Y

Al Hawaj, President of Ahlia University for all the

support

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