Study the Linear Equivalent of Nonlinear Sequences over Fp Where p Is Larger Than Two

Linear orthogonal binary sequences, special M-Sequences, are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form In current research trying study the construction of the linear equivalent of a product sequence on two, three, and four degrees over a linear sequences from the field Fp, where p is larger than two, and answering on the request, how is the maximum length of the linear equivalent of a product sequence (on a linear sequence {an} over the field Fp), is it less than rNh as the binary sequences?, or can reach it ?, or the length is exceed this value rNh? And is the product sequences are orthogonal? And we show that in some cases, the maximum length rNh for the binary sequences is not correct for the linear sequences in the finite field Fp for p larger than two and the result product sequences are not orthogonal, also trying study the product sequence on two different LFSRs, and how can use one shift feedback shift register LFSR as a monitor register of other p registers. In the current time, I think, there is no coders or decoders using the sequences over finite fields Fp where p is larger than 2 and from this idea this article showing very need for using in the future.


Introduction
The main obstacle to encoding and decoding is the complexity of decoding and decoding. For this reason, efforts have been made to design cryptographic and decoding methods in an easy way. The works of Hocquenghem in 1959, Reed Solomon 1960, Chaudhuri and Bose in 1960, BCH codes or Bose-Chaudhuri-Hocquenghem codes and others as Goppa, and Peterson 1961 were a new starting point for solving this issue. [1][2][3][4][5] In all stages of the encoding and the decoding, the orthogonal sequences play the main role in these processes, including the sequences with maximum period M-Sequences, Walsh sequences, Reed-Solomon sequences, and other sequences. [6][7][8][9][10][11][12] Sloane, N. J. A., discuses that the multiplication sequence {z n } on h degrees of {a n }, which has the r complexity, the complexity of {z n } can't be exceeded Orthogonal Sequences are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form, Especially in the pilot channels, the Sync channels, and the Traffic channel. [10][11][12] Shannon's classic articles, 1948-1949, were followed by many research papers on the question of finding successful ways to encode and successful decoding the media to allow it to be transmitted correctly through jammed channels. [6-8, 13, 14] The Author Al Cheikha A. H., Studied the case for p=2. [15]

M-Linear Recurring Sequences
Let k be a positive integer and 0 1 1 , , ,..., k λ λ λ λ − are elements in the field F p ={0, 1,…,(p-1)} and p >2 then the sequence 0 1 , ,... a a is called the nonhomogeneous binary linear recurring sequence of order k (or with the complexity k) iff: 1 Is called the characteristic polynomial. In this study, we are limited to 0 1 λ = .
Definition1. The ultimately sequence 0 1 , ,.... a a in F p with the smallest natural number r is called periodic with the period r iff: ; 0, 1, ... n r n a a n + = = [2][3][4][5][6] Definition2. The linear register of a linear sequence is a linear feedback shift register with only addition circuits and the number in its output in the impulse n equal to the general term of the sequence {a n } and the register denoted as LFSR. [3] Definition3. The coefficient of correlation function of x and y denoted by R x,y , is: is the nearest integer of the number 1 n p + [13]. X Y X Y G Y X Y X X Y G R ∀ ∈ ≠ ≠ + ∈ ≤ (7) [6,9] Definition 8. (Euler function φ ). ( ) n φ is the number of the all-natural numbers that are relatively prime with n. [11][12][13][14] Definition 9. The linear equivalent of a multiplication sequence {z n }, on a linear sequence {a n } which generated with the linear register LFSR1 and the sequence{z n } is a multiplication on some terms of {a n } (that is a result of multiplication circuits over the LFSR1), is a linear shift register LFSR2 generates the same sequence {z n }. [2][3]8] Definition10. The length of the linear equivalent of a multiplication sequence is the number of its complexity and equal to the degree of the characteristic polynomial which generates the same multiplication sequence, and the multiplication sequence can be generated through the linear equivalent. [8] Definition 11. The maximum length of a linear equivalent is the maximum length of the linear equivalent LFSR2 (it is the number of its complexity) which can be reached and the length of linear equivalent is always less than or equal the maximum length r h N . [

Study Multiplication Sequences on a Recurring M-Sequences over Fp.
Suppose the recurring M-Sequence {a n } over Fp with the complexity r and α 1 , α 2 , …, α r are its different linear independent roots of the characteristic equation of the sequence then the general term of the sequence is given through the relation; 1

The Sequence {z n } Is a Multiplication on Two
Degrees of the Sequence {a n } Suppose the multiplication sequence {z n } as multiplication on two different degrees of {a n } as the following; (1) The first degree is a n (in another case we can make a shift to the first degree).
(2) The second degree is n n b a δ + = as a shift of the first degree n a by δ .
Thus we have the following properties; P1. Each term of the first sum is not equal to zero and the number of these terms is equal to 1 P2. Also, a term of the second sum is equal to zero if and only if; Thus, it is a contradiction, then no term in the second sum is equal to zero and the number of these terms is ( 1) 2 2 r r r   − =     and the complexity of the sequence {z n }is; ( 1) Sum "one term of the first sum with one term of the second sum" is equal to zero if and only if there is different i, j, k satisfies the two conditions; 2) Thus, for the first condition each of the i, j, k can't be one, and if there is be such other values will be as; Or if i is the smallest; ( ) Thus 2 k i p − and j i p − are relatively primes and it is a contradiction then the sum of one term from the first sum with one term from the second sum can't be equal to zero and the linear equivalent reached the maximum length ( 1) Thus, the sequence {z n } is periodic with the same period of the sequence {a n }. Example 1. Suppose the following sequence {a n }, 3 ; n n N a F ∀ ∈ ∈ ;  The characteristic polynomial of the equation 3 ( ) 2 1 f x x x = + + is a prime and the characteristic equation of the sequence is; The general solution of the characteristic equation is;

The Sequence {z n } Is a Multiplication on Three
Degrees of the Sequence {a n } Suppose the new product sequence {z n } as a product of three different degrees in {a n } as following; 1) The first degree is a n (in another case we can make a shift to we arrive at the first degree) as in 3.1..
2) The second degree is n n b a δ + = (as a result of shift n by and r δ δ < ).
3) The third degree is n n c a γ + = (as a result of shift n by γ ) and and r δ γ γ < < , then: Thus, we have the following properties; P1. Each term of the first sum is not equal to zero. P2. For one term of the second sum is equal to zero is equivalent to; Or, by division on i δ γ α + ; we have the previous condition is equivalent to; P3. For one term of the third sum is equal to zero necessary and sufficient; By division on i δ γ α + we have; This equation is symmetric and can write it as follows; ( P4. Sum, one term of the first sum with one term of the second sum, is equal to zero necessary and sufficient the two conditions; 1). 3 2 The second condition can be written as; P5. Sum, one term of the first sum with one term of the third sum, is equal to zero necessary and sufficient the two conditions; 1). 3 3 Where, no two between the indexes i, j, k, and m are equal, and; 2).
( ) Or, by division on i δ γ α + , the second condition can be written as; Or; P6. Sum, one term of the second sum with one term of the third sum, is equal to zero necessary and sufficient the two conditions; 1).
By division on i δ γ α + and suppose, , , , P7. Sum, "one term of the first sum, with one term of the second sum, and with one term of the third sum" is equal to zero, necessary and sufficient the two conditions; 1).
where, no two indexes between i, j, k, h, l, and m are equal By division on i δ γ α + and suppose , , , , Each of P4 or …. or P7 properties leads to decrease the length of linear equivalent by one (for each case) relatively the maximum length r h N .
The characteristic equation of the sequence is;  1 0 the recurring formula of the sequence{z n }is;   1) i=1, j=2, k=3; 3) i=2, j=1, k=3; 5) i=3, j=1, k=2 2) i=1, j=3, k=2; 4) i=2, j=3, k=1; 6) i=3, j=2, k=1 Remembering the that the general term of the sequence is; , α α α α α α α α α α α α + + + + +  There are no other different i, j, and k as; sum one term from the first sum with one term from the second sum or sum one term from the first sum with one term from the third sum or sum one term from the first sum with one term from the second sum with one term from the third sum satisfies the first condition for P4 to P7.
P3. Is there any term of the third sum is equal to zero? Or; We can check that only for case from a), to f) there is only one case and the sum of them is 6 cases and the other properties P4,…, P7 are not satisfied.
Thus, in result the length of linear equivalent in our case is 9 only (and this length is larger than the mentioned maximum length 3 3 7 N = as in [2]) also, the expected maximum length in this case is 9 + 6=15 and it is greater than the mentioned maximum length 3 3 7 N = as in [2].

The Sequence {z n } Is a Multiplication on Four
Degrees of the Sequence {a n } Suppose the sequence {z n }is a result of product four terms from the sequence {a n } as the following; First term from {a n } a n (in other case we can shift the term to the first), second term is n n b a β + = (as a result of shift the first term by β ), third term is n n c a µ + = (as a result of shift the first term by µ ), forth term is d n (as a result of shift the first term by γ ) and the all β, γ , and µ are less than or equal to r and β µ γ < < , that is;  1 0 P3. For one term of the third sum is equal to zero is necessary and sufficient; Or; (by division on i β µ γ α + + and arrange the terms as need); then for one term from the third sum is equal to zero is necessary and sufficient; ( P4. One term of the forth sum is equal to zero is necessary and sufficient; α α α α α α α α α α α α α α α α α α by division on i β µ γ α + + the latest equation can be written as; P6. For the sum of one term from the first sum with one term from the third sum is equal to zero is necessary and sufficient the two conditions; 1) Where i, j, k, and l are different.
Thus, as the same, we have the corresponding relations for the following cases; P7. Sum one term of the second sum with one term of the third sum is equal to zero.
P8. Sum one term of the second sum with one term of the forth sum is equal to zero.
P9. Sum one term of the third sum with one term of the forth sum is equal to zero.
P10. Sum three terms of the different for sums is equal to zero.
P11. Sum four terms of the different for sums is equal to zero.     Thus;   4  2  19  4  43  6  35  8  23  10  67  12  34  14  54  16  45  18  49  20   25  22  7  24  41  28  26  30  28  32  56  34  38  36  4  38  52  40  16  42 10   We have;   1  2  3  4  3  9  27  1  2  3  4   1  1,  2,  3  , , , A A A A and β µ γ α α α α α α α α First property, no term in the first sum is equal to zero. P2. Second property, one term in the second sum is equal to zero necessary and sufficient the following corresponding condition (30); Thus, no term of the second sum is equal to zero. P3. Third property, one term in the third sum is equal to zero necessary and sufficient the following corresponding condition (32);  (1 )( And we have the same result for other combination of i, j, k and l, and no term of the forth sum is equal to zero. P5. Fifths property, sum one term of the first sum with one term of the second sum is equal to zero necessary and sufficient the two conditions; (1)   Suppose the recurrent sequence {a n } on F p which has the complexity r and α is a primitive root of its characteristic equation then this sequence is periodic and its period is p r -1,{b n }is other sequence on F p which has the complexity s and β is a primitive root of its characteristic equation then this sequence is periodic and its period is p s -1, and suppose for easily, r and s are relatively prime then the roots of the characteristic equation of the sequence {z n }={a n .b n } are in the field . r s p F and this sequence has the period lcm ((p r. -1),(p .s -1)).
Example 4. Suppose, the linearly recurring sequence {a n } defining through the recurring formula a n+2 + a n+1 + 2a n =0 or a n+2 =2a n+1 + a n and the linearly recurring sequence {b n } on F p defined through the recurring formula b n+3 + 2b n+1 + b n =0 or b n+3 =b n+1 + 2b n and {z n }={a n .b n }, as in the following figure 5, which shows the feedback shift registers for the sequences {a n }, {b n }, and the product sequence {z n }.
And the general term of the sequence is; and if γ is a root of h(x) in F 2 6 then γ generates F 2 6 and; For written z n through elements in F 2 6 we need search in F 2 6 about first element e 1 which satisfies the characteristic equation 2 2 0 x x + + = and will be equal to α , and we search about second element e 2 which satisfies the characteristic equation 3 2 1 0 x x + + = and will be equal to β then after w replacing in formula of z n each α by first element e 1 and each β by the second element e 2 we have z n written through elements from the field F 3 6 , and this step is not very need in our current study. Thus the nonlinearly Property allows us construct new sequences with largest periods.

Using Shift Feedback Register LFSR as Monitor Register
Suppose there is a set S of p linear feedback shift registers that is; S={LFSR(0), LFSR(1), ….., LFSR(p-1)} and the shift feedback shift register LFSR and we will use it as an monitor register as following: a) If the output of the register LFSR is 0 then the output of the system S is the output of LFSR(0). b) If the output of the register LFSR is 1 then the output of the system S is the output of LFSR(1).
……………….. c) If the output of the register LFSR is p-1 then the output of the system S is the output of LFSR(p-1).
Suppose, output of LFSR(0) is I 0 , output of LFSR(1) is I 1 , …, output of LFSR(p-1) is Ip-1, and through Solving the system of equations for 0 1 x p ≤ ≤ − ; We can find the coefficients a 0 , a 1 , …, a p-1 , where "x" is the output of the monitor register LFSR and "s" is the output of the system S which is equal to the output of the register LFSR(x) for 0 1 x p ≤ ≤ − and x k computed by mod p, thus we get the solution of our problem.
Exampe 5. 1) For p=2 that is the all registers are binary registers and the all arithmetic operations are the operation on F 2 , we can find the coefficients a 0 , a 1 in the equation (49) as following; The formula (49) will be; 2) For p=3, the all arithmetic operations are the operations on F 3 , we can find the coefficients a 0 , a 1 , a 2 in the equation (43) as following; The formula (43) will be; For example, for r=3 and h=2 the length of the linear equivalent of {z n } is 6.
2) If the sequence {z n } is multiplication on only two degrees of the recurring sequence {a n } over Fp and the characteristic polynomial of the sequence {a n } is prime of degree r then the sequence {z n } is also periodic and its period is equal to that is the half period of {a n } but the set of the cyclic permutations of one period of {z n }is don't form an orthogonal set as the sequence {a n }.
3) If the sequence {z n } is multiplication on h degrees of the recurring sequence {a n }over Fp and the characteristic polynomial of the sequence {a n }is prime of degree r and h >2 then the length of equivalent LFSR of {z n } is larger than of the maximum length in the binary sequences r h N , where; ( For example, for r=3 and h=3 the length of the linear equivalent is 9 and larger than 3 3 7 N = and for r=4, h=4 the length of the linear equivalent of {z n } is 29 4) Length of the linear equivalent of a multiplication sequence {z n } on h degrees of the linear recurring sequence {a n } over Fp is pending not only with the roots of the characteristic equation of the sequence {a n }but also pending with the coefficients of the terms in the general solution of the sequence{a n } and with the shifts of the terms of the sequence {a n } which on them occur the multiplication and any where is larger than (or equal for h=2) the maximum length r h N in the binary sequences. 5) we suggestion that the maximum length of the linearly shift register equivalent of multiplication sequence {z n } on h degrees of a linear sequence {a n }over Fp which has the complexity r is ... − , and has the complexity r.s. If r and s are not relatively prime then the period of the sequence {z n }is (p lcm(r,s) -1) and its complexity is lcm(r,s). 7) Using multiplication operation on different sequences operation leads to getting sequences with high complexity and with a high period but not orthogonal. 8) We can use one shift feedback register LFSR over Fp as monitor register for other p registers over Fp. Limitation: The method for reading a page which has a block will be according to the following direction as in figure 10.