New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers

Orthogonal Sequences (as M-Sequences, Walsh Sequences,...) are used widely at the forward links of communication channels to mix the information on connecting to and at the backward links of these channels to sift through this information is transmitted to reach the receivers this information in a correct form, especially in the pilot channels, the Sync channels, and the Traffic channel. This research is useful to generate new sets of orthogonal sequences (with the bigger lengths and the bigger minimum distance that assists to increase secrecy of these information and increase the possibility of correcting mistakes resulting in the channels of communication) from quotient rings Z/nZ, where Z is the integers and n is not of the form p m, where p is prime, replacing each event number by zero and each odd number by one, also, the increase in the natural number does not necessarily lead to an increase in the size of the biggest orthogonal set in the corresponding quotient ring. The length of any sequence in a biggest orthogonal set in the quotient ring Z/nZ is n and the minimum distance is between (n-3)/2 and (n-1)/2 and the sequences can be used as keywords or passwords for secret messages.


Introduction
Shannon's classic articles, 1948-1949, were followed by many research papers on the question of finding successful ways to encode a successful encoding of the media to allow it to be transmitted correctly through jammed channels. [1] 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 Petrson 1961 were a new starting point for solving this issue. [2][3][4][5][6][7][8] In all stages of encoding and decoding the orthogonal sequences play the main role in these processes n all stages of encoding and decoding, the orthogonal sequences play the main role in these processes, including: the sequences with maximum period M-Sequences, the Walsh sequences, the Reed-Solomon sequences, and the other. [9][10][11][12] In 2018 Al Cheikha A. H. publish an article "Generating New Binary Sequences Using Quotient Rings / m Z p Z "and current article is extending to this article. [13] Orthogonal Sequences are used widely at the forward links of communication channels to sift through this information is transmitted to reach the receivers this information in a correct form, especially in the pilot channels, the Sync channels, and the Traffic channel. [14][15][16][17]  , [1,2,[14][15][16][17] (2)

Research Method and Material
). [6][7][8][12][13][14][15][16] Definition 5. Suppose G is a set of binary vectors of length n: Let's 1 * =-1 and 0 * =1, The set G is said to be orthogonal if the following two conditions are Satisfied: Or; the difference between the number of "0.s" and the number of "1.s" is at most one.
where d is all divisors of m including 1 and m. [11,12,16] Result. if n larger than 1 to ) (n ϕ is even except

Results and Discussion
In this study we restrict our self m p n ≠ , p is prime.

n Is Odd
The best method for getting the binary representation of the multiplication table of the quotient ring ) /(nZ Z , where Z is the integers and n is natural number larger than 1, is replacing each event number by "0" and replacing each odd number by "1', by this way each row with the index i relatively prime with n contains 2 / ) 1 ( + n of "0.s' and 2 / ) 1 ( − n of "1.s" and the row ri is the conjugate of r(n-i) that is the entries in ) ( i n r ri − + are equal to zero by mod n. We searching between these rows about a comfortable subset of rows which with the null row form additional subgroups achieve the number of "0.s" and the number of "1.s" or orthogonal conditions in the vector space n 2 , where the addition is performed by mod 2.
6) The following table showing the addition between some of the rows in the set where Ri + Rj denoted by Ri+j. Table 8 showing the sum of the row R1 with the row R2: Table 8. Sum R1 with R2.
sets with the dimension 2 and the size 3 of each a set, and minimum distance 10, while the expected dimensions and sizes for each of set are at least 4 and 17 respectively.

n = 5 (7) = 35
The following table 9 showing the multiple in the quotient ring Z Z 35 with restriction over the basic useful numbers which are relatively prime with 35 (are half of the numbers which are relatively prime with 35).  The binary representation of table 9 showing in the following table 10:  contains (35+1)/2 of "0.s" and (35-1)/2 of "1.s" and the first condition of orthogonal is verified. 4) Ri and its conjugate R(n-i) can't be in one Span, each of R1 + 3, R1+4, R1+9, R1+12, R1+13, R1+17, R2+6, R2+8, and R2+11 does not meet the first conditions of orthogonal and can't be in one Span. Table 11 showing the sum of R1 and R2 with the some other rows.   Table 12 showing the of Span R1, R2 and R16.
sets with the dimension 3 and the size 7 of each a set, and minimum distance 17, while the expected dimension and size for each set are at least 5 and 31 respectively.

n Is Even
1) If the index i is even then the entries in the row ri in the table Z/nZ contains only even numbers and its corresponding binary representation Ri is zero row and does not satisfy the first condition of orthogonal. 2) If the index i is odd then the entries in the row ri in the table Z/nZ contains one even entry and after one odd entry periodically and its corresponding binary representation Ri the entries are one o and after 1 periodically and satisfies the first condition of orthogonal.

3)
If the indexes i, j is odd numbers the distribution of odd and even numbers the same in ri and rj and the distribution of "0.s" and "1.s" in Ri and Rj also the same and Ri+j is the zero row. Thus if n is even to Z/nZ don't have orthogonal sets and the following tables of representation of Z/10Z illustrated the ideas. The following table 17 showing the multiplication on  quotient ring Z/10Z:  6  5  4  3  2  1   Table 18 showing the binary representation of multiplication on quotient ring Z/10Z

Conclusions
When studying the quotient rings Z/15Z, Z/21Z, Z/35Z, Z/45Z and Z/10Z and their binary representation we found the following results:

For n Is Odd, and m n p ≠ Where p Is Prime
(1) In binary representation of nZ Z / , the length of each row is n, started by zero, each row with index relatively prime with n has 2 / ) 1 ( + n of "0.s", 2 / ) 1 ( − n of "1.s", satisfy the first condition of orthogonal, the number of these rows is ) (n ϕ , the first half of them is basic and the second half is their conjugates where the indexes computed mod n. (2) If i is prime then Ri+ R(n-i) = [0 1 1 1…1] n that is ) ( i n R Ri − = except the first entry is zero in both of them (3) In Z/15Z; the number of the biggest binary orthogonal closed sets (in the space 2 15 ) which we can get them from Z/15Z is at most 56 3

For n Is Even
(a) The number of the biggest binary orthogonal closed sets (in the space 2 n ) which we can get them from Z/nZ is only n/2 sets with; dimension 1, length n, and size or capacity 1 of each set, and this case is very trivial.
(b) From above in the quotient ring Z/nZ, and n is odd, the dimension of orthogonal set is don't increase or increase very slaw with increasing n and consequently their capacity but we can get orthogonal sets with biggest lengths and biggest minimum distances (n-1)/2.
(c) The increase in the natural number does not necessarily lead to an increase in the size of the biggest orthogonal set in the corresponding quotient ring (see 3.1.1 and 3.1.2).
(d) In Z/nZ the length any sequence in orthogonal set is n and the minimum distance is between (n-3)/2 and (n-1)/2.
Limitation: This method of compose sequences is useful for only binary sequences and the addition on the sequences computed by "mod 2 " also used Microsoft Word 2010 and the Microsoft equation 3.0 for written the math equations.
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