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

Ahmad Hamza Al Cheikha

doi:doi:10.11648/j.wcmc.20200801.12

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

Ahmad Hamza Al Cheikha

Published in International Journal of Wireless Communications and Mobile Computing (Volume 8, Issue 1)

Received: 17 July 2020 Accepted: 27 September 2020 Published: 14 October 2020

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Abstract

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.

Published in	International Journal of Wireless Communications and Mobile Computing (Volume 8, Issue 1)
DOI	10.11648/j.wcmc.20200801.12
Page(s)	9-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Walsh Sequences, M-sequences, Additive Group, Coefficient of Correlation, Orthogonal Sequences, Quotient Ring

References

[1]	Sakrison D. J., (1968), Communication Theory: Transmission of Waveforms and Digital information, Publisher: John Wiley & Sons Inc.
[2]	Al Cheikha A. H. (July, 2017). Compose M-Sequences. Australian Journal of Business, Social Science and Information Technology. AJBSSIT. Vol. 3, Issue 3. Pp. 119-126.
[3]	Al Cheikha A. H. (2017). Compose Binary Matrices. American Journal of Computer sciences and Applications. AJCSA. Vol. 1, Issue 2. Pp. 0001-0017.
[4]	Al Cheikha A. H. (September, 2014). Some Properties of M-Sequences Over Finite Field Fp. International Journal of Computer Engineering & Technology. IJCER. Vol. 5, Issue 9. Pp. 61-72.
[5]	Al Cheikha A. H. (September, 2014). Composed Walsh Sequences and M-Sequences. International Journal of Computers & Technology. IJCT. Vol. 15, Issue 7. Pp. 6933-6939.
[6]	Al Cheikha A. H. (2017). Composed Reed Solomon Sequences Generated by ith Partial Sum of Geometrical Sequences. American Journal of Computer sciences and Applications. AJCSA. Vol. 1, Issue 1. Pp. 0001-000116.
[7]	Byrnes, J. S.; Swick. (1970), “Instant Walsh Functions”, SIAM Review., Vol. 12, pp. 131.
[8]	David, J., “Introductory Modern Algebra,” Clark University, USA, 2008.
[9]	Jong-Seon No, Solomon W. & Golomb, (1998), “Binary Pseudorandom Sequences For period 2ⁿ-1 with Ideal Autocorrelation. IEEE Trans. Information Theory”, Vol. 44 No 2, PP 814-817.
[10]	Lee J. S & Miller L. E, (1998), “CDMA System Engineering Hand Book”, Artech House. Boston, London.
[11]	Lidl, R. & Pilz, G., (1984), ”Applied Abstract Algebra”, Springer–Verlage New York.
[12]	Lidl, R. & Nidereiter, H., (1994), “Introduction to Finite Fields and Their Application”, Cambridge University USA.
[13]	Al Cheikha A. H. (2018). Generating New Binary Sequences Using Quotient Rings Z/p^mZ, Research Journal of Mathematics and Computer Science, RJMCS, ISSN: 2576-3989, Vol. 2 Issue 11. Pp. 1-13.
[14]	Mac Williams, F. G & Sloane, N. G. A., (2006), “The Theory of Error-Correcting Codes”, North-Holland, Amsterdam.
[15]	Sloane, N. J. A., (1976), “An Analysis Of The Stricture and Complexity Of Nonlinear Binary Sequence Generators, IEEE Trans. Information Theory” Vol. It 22 No 6, PP 732-736.
[16]	Thomson W. Judson, (2013), “Abstract Algebra: Theory and Applications”, Free Software Foundation.
[17]	Yang S. C, (1998), “CDMA RF System Engineering”, ArtechHouse. Boston-London.

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APA Style

Ahmad Hamza Al Cheikha. (2020). New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. International Journal of Wireless Communications and Mobile Computing, 8(1), 9-17. https://doi.org/10.11648/j.wcmc.20200801.12

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ACS Style

Ahmad Hamza Al Cheikha. New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. Int. J. Wirel. Commun. Mobile Comput. 2020, 8(1), 9-17. doi: 10.11648/j.wcmc.20200801.12

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AMA Style

Ahmad Hamza Al Cheikha. New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. Int J Wirel Commun Mobile Comput. 2020;8(1):9-17. doi: 10.11648/j.wcmc.20200801.12

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@article{10.11648/j.wcmc.20200801.12,
  author = {Ahmad Hamza Al Cheikha},
  title = {New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers},
  journal = {International Journal of Wireless Communications and Mobile Computing},
  volume = {8},
  number = {1},
  pages = {9-17},
  doi = {10.11648/j.wcmc.20200801.12},
  url = {https://doi.org/10.11648/j.wcmc.20200801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wcmc.20200801.12},
  abstract = {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 pm, 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.},
 year = {2020}
}

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TY  - JOUR
T1  - New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers
AU  - Ahmad Hamza Al Cheikha
Y1  - 2020/10/14
PY  - 2020
N1  - https://doi.org/10.11648/j.wcmc.20200801.12
DO  - 10.11648/j.wcmc.20200801.12
T2  - International Journal of Wireless Communications and Mobile Computing
JF  - International Journal of Wireless Communications and Mobile Computing
JO  - International Journal of Wireless Communications and Mobile Computing
SP  - 9
EP  - 17
PB  - Science Publishing Group
SN  - 2330-1015
UR  - https://doi.org/10.11648/j.wcmc.20200801.12
AB  - 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 pm, 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.
VL  - 8
IS  - 1
ER  -

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Author Information

Ahmad Hamza Al Cheikha

Department of Mathematical Science, College of Arts-science and Education, Ahlia University, Manama, Bahrain

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APA Style

Ahmad Hamza Al Cheikha. (2020). New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. International Journal of Wireless Communications and Mobile Computing, 8(1), 9-17. https://doi.org/10.11648/j.wcmc.20200801.12

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ACS Style

Ahmad Hamza Al Cheikha. New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. Int. J. Wirel. Commun. Mobile Comput. 2020, 8(1), 9-17. doi: 10.11648/j.wcmc.20200801.12

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AMA Style

Ahmad Hamza Al Cheikha. New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers. Int J Wirel Commun Mobile Comput. 2020;8(1):9-17. doi: 10.11648/j.wcmc.20200801.12

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@article{10.11648/j.wcmc.20200801.12,
  author = {Ahmad Hamza Al Cheikha},
  title = {New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers},
  journal = {International Journal of Wireless Communications and Mobile Computing},
  volume = {8},
  number = {1},
  pages = {9-17},
  doi = {10.11648/j.wcmc.20200801.12},
  url = {https://doi.org/10.11648/j.wcmc.20200801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.wcmc.20200801.12},
  abstract = {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 pm, 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.},
 year = {2020}
}

Copy | Download

TY  - JOUR
T1  - New Orthogonal Binary Sequences Using Quotient Rings Z/nZ Where n Is a Multiple of Some Prime Numbers
AU  - Ahmad Hamza Al Cheikha
Y1  - 2020/10/14
PY  - 2020
N1  - https://doi.org/10.11648/j.wcmc.20200801.12
DO  - 10.11648/j.wcmc.20200801.12
T2  - International Journal of Wireless Communications and Mobile Computing
JF  - International Journal of Wireless Communications and Mobile Computing
JO  - International Journal of Wireless Communications and Mobile Computing
SP  - 9
EP  - 17
PB  - Science Publishing Group
SN  - 2330-1015
UR  - https://doi.org/10.11648/j.wcmc.20200801.12
AB  - 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 pm, 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.
VL  - 8
IS  - 1
ER  -

Copy | Download