Placement of M-Sequences over the Field Fp in the Space R n

: Spread spectrum communication systems are widely used today in a variety of applications for different purposes such as access of same radio spectrum by multiple users (multiple access), anti-jamming capability (so that signal transmission cannot be interrupted or blocked by spurious transmission from enemy), interference rejection, secure communications, multi-path protection, etc. Several spreading codes are popular for use in practical spread spectrum systems. one of these important codes is Maximal Sequence (M-sequence) length codes, These are the longest codes that can be generated by a shift register of a specific length, The number of 1-s in the complete sequence and the number of 0-s will differ by one, Further, the auto-correlation of an m-sequence is -1, another interesting property of an M-sequence is that the sequence, when added (modulo-2) with a cyclically shifted version of itself, results in another shifted version of the original sequence. Hence, the M-sequences are also known as, pseudo-noise or PN sequences. Current article study placement M-Sequences over the finite field Fp or Mp-Sequences (where p is a prime number) in the space R n , these sequences can be generated as a closed set under the addition. These sequences form additive groups with the corresponding null sequence that was generated by the feedback shift registers. Such Mp-Sequences see a great application in the forward links of communication channels. Furthermore, they form coders and decoders that combine the information by p during the connection process with the backward links of these channels. These sequences scrutinize the transmitted information to enable it to reach the receivers in an accurate form. This study has defined eminent surfaces in the vector space ‘R’ having dimensions of ‘n’ as quadratic forms, spheres, and planes that contain these sequences.


Introduction
The sequence of order k that is homogeneously linear has been supported by recurring: With the characteristic polynomial; !"#= " " $ " $ . . " If f (x) is prime and , $ , … are not equal to zero, then the sequence , $ , & , …. is periodic with the maximum length or period 1. The calculations preformed using mod p is called Mp-Sequence over Fp, and the number of all these non-zero sequences corresponding to all cases of , $ , … is -1. These Mp-Sequences can be widely used in communication channels, and coders, and decoders by mod p as. The Mp-Sequences have zero sequence form an additive group, when the addition is pre-formed at mod p. [1,2] Example 1. The sequence over & of Linear Recurring; The prime factor with the characteristic equation " $ " 2 0 as well as the characteristic polynomial !"# " $ " 2 produce & ( . If α is a root of !"# and generates & ( , then this characteristic equation can be solved as ) , ) & . While equation (3) has a general solution that is given by; [3] 2). ) while the sequence is periodic with the period 3 $ 1 8.
The equation of the Ellipsoid in the space @ with the center o is; The equation of the sphere in the space @ with the center o and the radius r is; The equation of the plane in the space @ is;

Research Method and Materials
The Ultimately Periodic Sequence , , …. in = 0, 1, 2, … , − 1 along with r which, represents periodic or the smallest period given as; F = ; ? = 0,1, …. [1][2][3][4]. Euler function G!?#represents relative prime number of the natural numbers through n. [5][6][7][8]. The code C of the form [n, k, d] is used, if each element (Codeword) has the length n. The Rank k is the number of information components and minimum distance is d [9]. Two vectors are; " = !" , " , … , " # and H = !H , H , … , H # on having n length. The coefficient of correlations functions of x and y is indicated by @ A,I ; [4,5] @ A,I = ∑ !−1# A J I J (16) where, x i +y i is computed mod p. The periodic sequence ! # ∈K over Fp with r period comprises of the properties within "Ideal Auto Correlation" but only in the case when there is a "periodic auto Correlation". @ C !L# as given below for p > 2: When [6,7]; Any Periodic Sequence , , …. over having prime polynomial, represents an ideal autocorrelation and orthogonal cyclic code [8,9]. Theorem 1.
i. If , , …. is a sequence of order k in , which is homogeneously linear recurring and satisfying equation (1) and 0 0 ≠ λ , then it is a periodic sequence.
ii. If the current sequence is homogeneous and linear recurring then it will be periodic, and if its characteristic polynomial !"# is prime and the period of the sequence is r then |' < !"#. iii. If !"# is prime characteristic polynomial of the sequence then r the period of the sequence is given by = − 1 , known as U -Sequence over or briefly U -Sequence. [10][11] Theorem 2 (Fermat's theorem). If there is the finite field F, having q elements, then all the elements of F will satisfy this equation [12]; Theorem 3. W!x# representing the prime characteristic polynomial over Fp of the (H. L. R. S.) , , … for k degree, while the root of g !"# is ) within Fq considering any splitting field Fp, then the general term of the sequence is; And the coefficients C i determined in Fq uniquely. [13] Theorem 4. For any prime p and positive integers m and n, where m is the order of q mod e, it equals 2 in case m = e = 1, and equals to zero elsewhere. [15,16] Theorem 6. i. The necessary and sufficient condition for the square matrix j = k 9 l to be similar to the diagonal matrix  = m n jm can be shown as: = − 1 mod p. The corresponding matrix of a linear recurring sequence generated by prime polynomial or any power of the matrix satisfies the following condition; When @ 9 is the > St row. Sum of all the entries in any row (or any one period) of the Mp-Sequence is equal to zero mod p. Practical examples show that the repetition of any non-zero element in one period is p n-1 , but the repetition of zero in one period is p n-1 -1. [20]

Return a Quadratic Form to Its Regular Form
In theorem 6, if the matrix A of the quadratic form contains n independent characteristic vectors V 1 , V 2 , …, V n consequently to the characteristic values
It is known that; ∑ " $ + 2 ∑ " " 9 The Mp-Sequences lie on the surface if; Or the equation (42) will be; !"# ≡ ∑ " $ + ∑ " " 9 9,9: Or; ∑ " $ + 2 ∑ $ " " 9 ,9: The equation can be written as; [" , " $ , … , " ] The Matrix can be determined; For matrix A characteristic polynomial is represented as ‹ and given by; Subtracting the last column from the all columns results in The determinant is computed according to the first row; Computation of the determinant according to the first column; Thus; First the rule is to be proved true for n = 3 is; Therefore, the rule is proven true for n = 3. Secondly, the rule is true on n-1 and also true for n, which means: It is true and will prove that; Thus; Or; The characteristic equation becomes; The characteristic values are; The corresponding characteristic vectors were determined as; There is only one solution that is; " = " $ = ⋯ = " . While, the characteristic vector is; š ›oe = !1,1, … ,1# . The vectors: •š ›oe , š ›oe $, … . , š ›oe ž are linearly independent because their determinant not equal to zero; By adding all columns to the first column; The vectors: •š ›oe , š ›oe $ , … , š ›oe ž are linearly independent. According to the new coordinates m m $ … m corresponding of the potential vectors •š ›oe , š ›oe $ , … , š ›oe ž with the vectors unit •‹ oe , ‹ oe $ , … , ‹ oe ž; And; The sphere with the center is considered '!0,0, … ,0# and the radius r; The U -Sequences of the period − 1 lie on the surface of the sphere (The sphere has the same center for both the coordinates) but only in the following case; Or; Or; We can get the same result from equation (81)  The intersection of (81) and (85) gives; Or; Or; Also the M p -Sequences of the period − 1 lie on the plane; " + " $ + ⋯ + " = ~1 + 2 + ⋯ + ! − 1#• (89) Or; The intersection of (53) and (84) is the plan (90), and the intersection of (81) and (85) is the plan (88). And Thus: