Compose Quotient Ring Sequences with Walsh’s Sequences and M-Sequences

Quotient ring sequences are completely new orthogonal sets without coders and decoders to the moment but Walsh sequences of the order 2 k , k positive integer, and M-Sequences with zero sequence form additive groups, Except the zero sequences, Walsh sequences, and M-Sequences formed orthogonal sets and used widely in the forward links and inverse links of communication channels for mixing and sifting information as in the systems CDMA and other channels. The current paper studied the orthogonal sets (which are also with the corresponding null sequence additive groups) generated through compose quotient ring sequences with self, Compose quotient ring sequences with the best and very important sequences Walsh sequences and M-sequences and by inverse for getting these new orthogonal sets or sequences with longer lengths and longer minimum distances in order to increase the confidentiality of information and increase the possibility of correcting mistakes in the communication channels.


Orthogonal Quotient Ring Sequences
We can get quotient ring sequences from the . [1,2] For p = 2 we get Walsh sequences. [3]

Walsh Sequence
Walsh sequences are binary sets with 2 k of rows (or sequences), except the zero row, each set is orthogonal, the length of each row is 2 k and contains 2 k-1 of "0.s" and the same number of "1.s", and forms an additive group with the zero row where the addition performed by mod 2, also they are known under the name Walsh functions.
The Walsh functions can be generated by any of the following methods: (1) Using Rademacher functions.

Binary M-Sequences: M-Linear Recurring Sequences
Let k be a positive integer and 0 1 1 , , ,..., k λ λ λ λ − are elements in the field 2 F then the sequence 0 1 , ,... z z is called non homogeneous linear recurring sequence of order k iff: Is called the characteristic polynomial. In this study, we are limited to 0 1 λ = . [8,9].
Orthogonal quotient rings sequences are good and important sequences absolutely new, published one month ago, linear, suitable and sufficient lengths and minimum distances, to this moment there is no coders and decoders for them.
Thus, sequence generated showed increased secrecy and increased possibility of correcting error in communication channel because it exhibited bigger length and the bigger minimum distance.  Denoted by R x,y , is:

Research Method and Materials
Definition 4. Any Periodic Sequence 0 1 , ,.... z z over 2 F with prime characteristic polynomial is an orthogonal cyclic code and ideal auto correlation. [13,14] Definition 5. Suppose G is a set of binary vectors of length n: Let 1 * = -1 and 0 * =1. The set G is said to be orthogonal if the following two conditions are satisfied: That is, the absolute value of "the number of agreements minus the number of disagreements" is equal to or less than 1. [15] Definition 6. Hamming distance ( , ) d x y : The Hamming distance between the binary vectors is the number of the disagreements of the corresponding components of x and y. [16] Definition 7. If C is a set of binary sequences and ω is any binary vector then each "1" in x i by ω and each "0" in x i by ω .
Corollary 1. If in the binary vector x: the number of "1.s" and the number of "0.s" are 1 m and 2 m respectively, and in the Binary vector w the number of"1.s" and the number of "0.s" are 1 n and 2 n respectively then in the binary vector ( ) x w . The number of "1.s" and the number of "0.s" are 1  primitive then the period of the Sequence is 2 1 k − , and this sequence is called M-sequence and each of these sequences contains 1 2 k − of "1"s and 1 2 1 k − − of "0"s. [18,19]

Compose Quotient Ring Sequences with Other Quotient Ring Sequences
(1) The number of "1.s" in i a is: The number of "0.s" in i a is: The difference between the number of "1"s and the number of "0"s is one b) For , The difference between the number of "0.s" and the number of "1.s" is 2 n p Thus, k A is not an orthogonal set and ( ) The difference between the number of "0.s" and the number of "1.s" is 2 n p . Thus, ( )   1 1 ( ) Q q ɶ ɶ are not orthogonal sets.

Compose Quotient Ring Sequences with Walsh
Sequences and Increase

Compose Quotient Ring Sequences with Walsh
The difference between the number of "1.s" and the number of "0.s" is zero b) For , The number of "1.s" and the number of "0.s" in i b ɶ is 1 2 n m p − , the difference between them is zero.
The difference between the number of "1.s"and the number of "0.s" in b i and i j b b ⊕ is zero.

Thus ( , )
k B ⊕ is an orthogonal set and ( ), ) Q W ⊕ are orthogonal sets.

Compose Walsh Sequences with Quotient Ring
Sequence Suppose   For ,

Compose Quotient Ring Sequences and M-Sequences
The difference between the number of "0.s" and the number of "1.s" is one

Compose M-Sequences and Quotient Ring Sequences and Finding ( )
(2) The difference between the number of "0.s" and the number of "1.s" is one b) For ,  Table 1. Contains the multiplication on 5 Z and their binary representation.  Each of 1 r and 2 r contains 5 1 3 2