Some properties of m sequences over finite field fp

1. INTERNATIONAL JOURNAL OF COMPUTER ENGINEERING &
International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
SOME PROPERTIES OF M-SEQUENCES OVER FINITE FIELD Fp
Dr. Ahmad Hamza Al Cheikha
Department of Mathematical Science, College of Arts-science and Education/
Ahlia University, Exhibition Street, Manama, Bahrain
l l l l l
+ + + + + = Î = −
... 0, , 0,1,..., 1
a a a a F i k
+ − + − − + −
1 1 2 2 0
n k k n k k n k n i q
61
ABSTRACT
M-Sequences over Fp, when p is odd prime, are compatible with binary M-Sequences by
orthogonal property but their even period gives them some other properties that are not exist in
binary M-Sequences.
This research shows or clarifies some of these properties that: the set of cyclic permutations
of elements one non zero period is not closed under the addition, the matrix of these permutations is
symmetric for the second diagonal, the sum of any two rows, one of them is translated by index
equal to half of the periodis equal to zero sequences, the repetitions of the non-zero elements in one
period are equal and sum of the squares of all entries in any row or any column by mod p is equal to
zero.
Keywords: M-Sequences, Cyclic Permutation, Orthogonal Sequences, Repetition, Translation by
Index.
I. INTRODUCTION
M-Sequences: M- Linear Recurring Sequences
Let k be a positive integer and l ,l0,l1,...,lk −1 are elements in the field Fq , then the sequence
a0,a1,...is called non homogeneous linear recurring sequence of order k iff:
0 (1)
−
1
l l
+ + =
or a a
+ +
n k i n i
=
1
k
i
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2. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
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The elements a0 ,a1,...,ak −1 are called the initial values (or the vector (a0 ,a1,...,ak −1) is called
the initial vector). If l = 0 then the sequence a0 ,a1,... is called homogeneous linear recurring
sequence (H. L. R. S. ), except the zero initial vector, and the polynomial
k (2)
r for r
62
( ) = +l 1 +...+l +l −
1 0
1
f x x k
k
− x x is called the characteristic polynomial. In this study, we are limited tol0 =1. [1]-[3]
II. RESEARCH METHODS AND MATERIALS
Definition 1.
The Ultimately Periodic Sequence a0, a1,.... in Fp with the smallest period r is called periodiciff:
an+r = an ; n = 0,1, ... [1]-[4]
Definition 2.
The complement of the vector: X = (x1, x2 ,..., xn ) , when xi ÎFp , is the vector X = (x1, x2,...,xn ) ,when
xi = p −1− xi , and − X = (−x1,−x2 ,...,−xn ) when − xi = p − xi mod p . [1]-[4]
Definition3. (Euler functionj ).
j (n) is the number of the natural numbers that are relatively prime with n.[5]-[8]
Definition 4.
Hamming distance d(x, y) : The Hamming distance between the binary vectors
x = ( x0 , x1 ,..., xn −1 ) and y = (y0, y1,..., yn−1) , when xi , yi in F2 , is the number of the
disagreements of the corresponding components of x and y.[9]
Definition 5.
The code C of the form [n, k, d] if each element (Codeword) has the: length n, the rank k is the
number of information components (Message), minimum distance d.[9]
Definition6.
Suppose x = ( x 0 , x1 ,..., x n −1 ) and y = ( y0 , y1,..., yn−1) are vectors of length n on
GF(p)={0,1, …,P-1}. The coefficient of correlations function of x and y, denoted by Rx,y, is:
−
+ = −
R i i
, ( 1)
=
1
0
n
i
x y
x y
Where xi +yiis computed mod p.. [13]
Definition7.
The periodic sequence (ai )iÎN over FP with the period r has the property of “IdealAutoCorrelation”
if and only if its periodic autoCorrelation Ra (t )of the form:
( )
º
=
otherwise
Ra 0 ;
; t 0 mod
t

3. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
63
When: ( ) ( + ) + ( ) = −
−
Ra ( t ) 1
t [1],[2]
=
1
0
r
t
a t a t
Definition8.
Any Periodic Sequence a0,a1,....over Fp with prime characteristic polynomial is anorthogonal
cyclic code and ideal autocorrelation [1]-[10].
Theorem1.
i. If a0,a1,.... is a homogeneous linear recurring sequence of order kin Fp , satisfies (1) then this
sequence is periodic.
ii. If this sequence is homogeneous linear recurring sequence, periodic with the period r, and its
characteristic polynomial f (x) then r ord f (x) .
iii. If the polynomial f (x) is primitive then the period of the sequence is −1 k p , this sequence is
called M – sequence over Fp = GF( p) , or briefly M-Sequences.[6]
Lemma 2.( Fermat’s theorem ).
If F is a finite field and has q elements then every element a of F satisfies the equation:
x x q = . [6],[9]
Theorem 3.
If g(x) is a characteristic prime polynomial of the (H. L. R. S.) a0,a1,....of degree k, and a is a
root of g(x) in any splitting field of Fp then the general bound of the sequence is:
n
p
k
n i
i
i
C a
=
−
=
1
1
a
Theorem 4.
i. q q m n m n ( −1) ( −1)Û (4)
ii. If Fq is a field of order n q = p then any subfield of them of the order m p and
m n
and
m n
and by inverse if
m n
then in the field Fq there is a subfield of order m p .
[6],[9],[11]
Theorem 5.
The number of irreducible polynomials in Fq (x) of degree m and order e is j (e) /m,
if e ³ 2 , When m is the order of q by mod e, and equal to 2if m = e = 1 and equal to zero
elsewhere. [6]-[9]
* The study here, is limited to the fields Galois ( ) n GF p , and p 2 , then the period = −1 k r p is
even.
III. RESULTS AND DISCUSSION

4. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
A. First step
Theorem 7:
Suppose a0 , a1,.... is a non zero homogeneous linear recurring sequence of order k over
Fp ={0,1,..., p −1} and f (x) is its prime characteristic polynomial then the first = −1 k r p bounds
with all its cyclic shifts and the zero sequence forming an additive group.[13]
Theorem 8:
Suppose a1,a2 ,.....is a non zero homogeneous linear recurring sequence of order k in Fp and f(x)
is their primitive characteristic polynomial, S1 is the initial bounds where = −1 k r p and
{ } $ = S1, S2 ,..., Sr are the all cyclic shifts. Let Ais
a matrix which its rows are elements $ respectively, then by{A i r} i , =1,..., , or by powers of its
permutations of Awe can represent all multiplicative subgroups in k p
−
1
= = = − −
( ) ( ) ( ) . ( ) −
64
F relatively to product and
addition of matrices, having the period of S1(x) and rows of i A are the shifts to rows of A. [13]
B. Second Step
Theorem 9.
In the field ( ) n GF p where p 2:
i. The square roots of unit in the field ( ) n GF p are 1 and p-1.
− n p
x and the elements
1
=
ii. Half of the nonzero elements in ( ) n GF p are roots to the equation 2 1
−
1
= −
n p
of the other second half are roots of the equation x 2 p
1
.
iii. The additional inverse to the number
2
1
1
−
£ £
n p
i where i is: -i = (p– 1)i.
Proof.
n − p
x and
1
=
i. The all elements of the field ( ) n GF p satisfies the equation 1 1 = − n p x then: 2 1
(p-1)2 =1 modp then the roots of order two of the unit in the field ( ) n GF p are 1 and (p-1).
ii. The non zero elements in the field ( ) n GF p are multiplicative cyclic group of the order −1 n p
they are: : = 0,1,..., − 2 i n a i p when a is a root of the prime polynomial generating field ( ) n GF p ,
and we get:
n
• If i is even then i = 2k and: ( ) 2 ( ) k ( p 1) ( p 1
) k
1. p
i n n
a a a
• If i is odd then i = 2k+1 and :
1
2 1
2
1
1
2 1
1
2 2 1
−
−
−
+
−
= = =
n
n
n
n n
p
p
k p
p
k
p
i a a a a a

5. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
= n − p a then all elements of the field ( ) n GF p satisfies the equation
−
¹ − n p a and fori is odd ( ) 2 1
p
R x R
M
l l
L M L
65
1
If ( ) 1 2 1
− n p
x and the order of ( ) * n GF p is
1
=
2 1
p −1
2
and not −1 n p , and that is
1
Contradiction then ( ) 1 2 1
1
= p
−
n p
i a
iii. For all i that are:
2
1
1
−
n p
i we see that:
( p −1)i = pi − i( p −1) i = −i (mod p)
p
n − 1
x =
, or the equation
Result 1. All non zero elements in ( ) n GF p satisfies the equation 2 1
−
1
= −
n p
2 1
x p
.
Result 2. The corresponding matrix “of a linear recurring sequence generating by prime
polynomial” or any power of the matrix satisfies the following condition:
2
1
0 : 1,2,3,...,
2
1
−
+ = =
−
+
n
p
i
i
p
R R n i
When R j is the th j row, because, i
p
i
n
n
2
1
2
1
−
−
+
R n ¹ R
= and i p
i
−
+
2
1
that is
i
R n
p
i
−1
2
+
permutation of Ri then
n i R p Ri R
i
p
i
= − = −
−
+
( 1)
2
1
.
Result 3.The rows of the corresponding matrix or any power to it are cyclic permutations of the
first row then are symmetric respectively to the second diagonal of these matrices and
2
1
0 : 1,2,3,...,
2
1
−
+ = =
−
+
n
p
j
j
p
C C n j ,
whenCj is the th j column in these matrices.
Result 4. From the Results 2 andResult 3 we can write the corresponding matrix of any power of it
as a partitioned matrix of the form:
M
l l

8. L M L
−
−
=

11. −
−
=
l l
l l
M
M
( 1)
( 1)
p
p
A

12. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
k k
M
l l
And 2 ; 1,2,..., 1 1 = −
−
1
2 =
66

16. L M L
−
−
= − A k p
k k
k k
M
l l
when the calculations are performed by mod p.
Result 5. The results not pending on the initial vector of the period of the sequence then are true for
any cyclic permutations of the rows of the corresponding matrix.
Result 6.Sum all entries in any row (or any column) of the matrix A is equal to zero mod p.
Result 7. By practical examples showing the repetition in one period of any non zero Element in Fp
is pk-1, but the repetition of zero in one period is pk-1-1.
Result 8. From Result 1 and Result 2 we get 0
=
1
n p
i
ai , that is:
p
( 1) 0
−
2
1
−
1 1 1
= − (1)
−
2
1
1
2 = −
+
−
=
p
p
p
a p n n
p
i
i
n
Then the sum of the squares of all entries in any row (or any column) is equal to zero.
C. Third Step
Example 1: If a is a root of the prime polynomial ( ) 2 2 f x = x + x + and generates (3 ) 2 GF then
the representation of the elements of (3 ) 2 GF in F3is:
Where (i), i = 0,1, 2, ..., 8is the symbol of the sequence i.
Suppose the Linear Recurring Sequence be
an+2 + an+1 + 2an = 0 or an+2 = 2an+1 + an (5)
Figure(1): Linear feedback register of degree2 generates sequence (5)

17. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
With the characteristic equation 2 0 2 x + x + = and the characteristic polynomial ( ) 2 2 f x = x + x +
,which is a prime and generates 2 3
F and if ( 2 ) x =a ÎGF 3 is a root of f (x) then the solutions of
characteristic equation are { , } n 3n a a . The general solution of equation (1) is given by
an 3 = 2a ×a + (1+a ) ×a , and the sequence is periodic with the period3 1 8 2 − = .
For the initial position: a1 = 1 , a2 = 0 , then S1 = (1 01 2 2 0 21) and by the cyclic permutations on
S1 we have $ = {S1, S2 , S3, S4 , S5, S6 , S7 , S8}where:
= = = =
(11 0 1 2 2 0 2) ; (2 11 0 1 2 2 0) ; (0 2 11 0 1 2 2) ; (2 0 2 11 0 1 2) ;
S S S S
2 3 4 5
= = =
67
n n
(2 2 0 2 11 0 1) ; (1 2 2 0 2 11 0) ; (0 1 2 2 0 2 11)
S S S
6 7 8
The first two digits in each sequence are the initial position of the feedback register. In this
example the resulting sequences is:
1 0 1 2 2 0 2 1 1 0 1 2 2 0 …, and the corresponding matrix is:
Thus:

18. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
consequently, $is a representation of the multiplicative group (3 ) * 2 GF .
68
We denote in this example that:
• Ri + Ri+4 = 0 , i = 1,...,4 and C j + C j+4 = 0 , j =1,...,4
•
The matrix is symmetric for the second diagonal.
•
Sum all entries in any row or any column is equals to zero.
•
The repetition of any 1, 2, in one period is 32-1= 3, but the repetition of the zero is 3-1 = 2.
• Sum of the squares of all entries in any row (or any column) is equal to zero.
The field (3 ) 2 GF contains (3 1) 2 2 2 j − = third degree irreducible polynomials are
( ) 2 2 f x = x + x + and ( ) 2 2 2 g x = x + x + .
Example 2.Suppose the Linear Recurring Sequence over F5 be:
an+2 + an+1 + 2an = 0 or an+2 = 4an+1 + 3an (6)
that we can generate it by the following Linear Feedback Register
Figure(2): Linear feedback register of degree2 generates sequence (6)

19. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
2 With the characteristic equation x + x + 2 = 0 and the prime characteristic polynomial
g ( x ) = x 2 + x + 2 , which is generates F 2 . In this example the resulting sequence is:
5
0 1 4434022331304 1 12 1 0 3 2 2 401 4 ……
69
And the permutations corresponding matrix is:
We denote that in this example that is:
• Ri + Ri+12 = 0 , i =1,...,12and C j + C j+12 = 0 , j = 1,...,12
•
The matrix is symmetric for the second diagonal.
•
Sum all entries in any row (or any column)is equals to zero.
•
The repetition of any 1, 2, 3, 4in one period is 52-1 = 5, but the repetition of zero is 5-1 = 4.
• Sum of the squares of all entries in any row (or any column) is equal to zero.
• There are (5 1) 2 4 2 j − = irreducible polynomials of second degree that are:
( ) 2 3 2 f x = x + x + , ( ) 2 2 g x = x + x + and their conjugate ( ) 4 2 2
f1 x = x + x + ,
and ( ) 3 3 2
g1 x = x + x + , and the corresponding matrices have the same behavior.
Example 3.Suppose the Linear Recurring Sequence over F3 be:
an+3 + 2an+1 + an = 0 or an+3 = an+1 + 2an (7)
then we can generate it by the following Linear Feedback Register

20. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
Figure (3): Linear feedback register of degree3 generates sequence (7)
3 With the characteristic equation x + 2 x + 1 = 0 and the prime characteristic polynomial
g ( x ) = x 3 + 2 x + 1 , which is generates F 3 . In this example the resulting sequence is:
3
1 0 0 2 0 2 1 2 2 1 0 2 2 2 0 0 1 0 1 2 1 1 2 0 1 1 1 0 0 2 0 2 1
70
And the permutations corresponding matrix is:
We denote that in this example that is:
• Ri + Ri+13 = 0 , i = 1,...,13 and C j + C j+13 = 0 , j = 1,...,13
•
The matrix is symmetric for the second diagonal.
•
Sum all entries in any row or any column is equals to zero.
•
The repetition any of 1, 2 in one period is 33-1 = 9, and the repetition of zero is 8
• There are (5 1) 2 4 2 j − = irreducible polynomials of second degree that are:

21. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
• We denote that the repetition of any non zero element of Fp in one period of M-sequence
over Fp is n−1 p while the repetition zero element is n−1 −1 p .
• Sum of the squares of all entries in any row or any column is equal to zero.
k k
M
l l
L M L
71
IV. CONCLUSION
p
n − 1
x =
, or the equation
1. All non zero elements in ( ) n GF p satisfies the equation 2 1
−
1
= −
n p
2 1
x p
.
2. The corresponding matrix “of a linear recurring sequence generating by prime polynomial”
or any power of the matrix satisfies the following condition:
2
1
0, 1,2,3,...,
2
1
−
+ = =
−
+
n
p
i
i
p
R R n i and
2
1
0, 1,2,3,...,
2
1
−
+ = =
−
+
n
p
j
j
p
C C n j
3. The rows of the corresponding matrix or any power to it are cyclic permutations of the first
row, are found to be symmetric respectively to the second diagonal of these matrices.
4. From the Result 2 and Result 3, we can write the corresponding matrix or any power of it as
a partitioned matrix of the form:

25. 2 ; 1,2,..., 1 1 = −
−
−
= − A k p
k k
k k
M
l l
when the calculations are performed by mod p.
5. Sum all entries in any row or any column of the matrix A is equal to zero mod p.
6. By practical examples showing the repetition in one period of any no zero element in Fp is
pk-1, but the repletion of zero in one period is pk-1-1.
7. The Sum of the squares of all entries in any row or any column is equal to zero.
8. The results are not pending on the initial vector of the period of the sequence.
Hence, are true for any cyclic permutations of the rows of the corresponding matrix.
V. ACKNOLEDGMENT
The author expresses his gratitude to Prof. Abdulla Y Al Hawaj, President of Ahlia
University for all the support provided.
VI. REFERENCE
[1] Yang K , Kg Kim y Kumar l. d ,“Quasi – orthogonal Sequences for code – Division
Multiple Access Systems, “IEEE Trans .information theory, Vol. 46, No3, 200, PP 982-993.
[2] Jong-Seon No, Solomon W. Golomb, “Binary Pseudorandom Sequences For period 2n-1
with Ideal Autocorrelation,” IEEE Trans. Information Theory, Vol. 44 No 2, 1998,
PP 814-817.

26. International Journal of Computer Engineering and Technology (IJCET), ISSN 0976-6367(Print),
ISSN 0976 - 6375(Online), Volume 5, Issue 9, September (2014), pp. 61-72 © IAEME
[3] Lee J.S Miller L.E, “CDMA System Engineering Hand Book,” Artech House. Boston,
72
London, 1998.
[4] Yang S.C, “CDMA RF System Engineering,” ArtechHouse.Boston-London, 1998.
[5] Lidl,R. Pilz,G.,” Applied Abstract Algebra,” Springer – VerlageNew York, 1984.
[6] Lidl,R. Niderreiter,H., “Introduction to Finite Fields and Their Application,” Cambridge
university USA,1994.
[7] Thomson W. Judson, “Abstract Algebra: Theory and Applications,” Free Software
Foundation, 2013.
[8] Fraleigh,J.B., “A First course in Abstract Algebra, Fourth printing. Addison-
Wesleypublishing company USA,1971.
[9] MacWiliams,F.G Sloane,N.G.A., “The Theory of Error-Correcting Codes,” North-
Holland, Amsterdam, 2006.
[10] Kasami,T.Tokora, H., “TeoriaKodirovania,”Mir(Moscow), 1978.
[11] David, J., “Introductory Modern Algebra,”ClarkUniversityUSA, 2008.
[12] Sloane, N.J.A., “An Analysis of the Stricture and Complexity of Nonlinear Binary Sequence
Generators,” IEEE Trans. Information Theory Vol. It 22 No 6, 1976, PP 732-736.
[13] Al Cheikha A. H. “Matrix Representation of Groups in the finite Fields GF(pn),”
International Journal of Soft Computing and Engineering, Vol. 4, Issue 2, May 2014,
PP 118-125.
[14] Dr. Ahmad Hamza Al Cheikha and Dr. Ruchin Jain, “Composed Short Walsh’s Sequences
and M-Sequences”, International Journal of Computer Engineering Technology (IJCET),
Volume 5, Issue 8, 2014, pp. 144 - 158, ISSN Print: 0976 – 6367, ISSN Online:
0976 – 6375.