Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property

Year 2023, , 792 - 812, 25.08.2023

Ramazan Duran

https://doi.org/10.16984/saufenbilder.1248060

Abstract

Considering error-correcting codes over Hurwitz integers, prime Hurwitz integers are considered. On the other hand, considering transmission over Gaussian channel, Hurwitz integers, whose the norm is either a prime integer or not a prime integer, are considered. In this study, we consider Hurwitz integers, the greatest common divisor of components of which is one, i.e., primitive Hurwitz integers. We show, with the help of a proposition, that some primitive Hurwitz integers accompanied by a related modulo function are not suitable for constructing Hurwitz signal constellations. To solve this problem, we show, with the help of a proposition, the existence of primitive Hurwitz integers that have the "division with small remainder" property used to construct the Hurwitz constellations. We also call the set of these integers named as "Encoder Hurwitz Integers" set. Moreover, we examine some properties of the mentioned set. In addition, we investigate the performances of Hurwitz signal constellations, which are constructed accompanied by a related modulo function using Hurwitz integers, each component of which is in half-integers, for transmission over the additive white Gaussian noise (AWGN) channel by means of the constellation figure of merit (CFM), average energy, and signal-to-noise ratio (SNR).

Keywords

Quaternion integers, Hurwitz integers, residual class, signal constellations, code constructions

Kodlayıcı Hurwitz Tamsayıları: Küçük Kalanlı Bölme Özelliğine Sahip Hurwitz Tamsayıları

Abstract

Keywords

Kuaterniyon tamsayıları, Hurwitz tamsayıları, kalan sınıfı, sinyal yıldızkümeleri, kod yapıları

References

• K. Huber, "Codes over Gaussian integers," IEEE Transactions on Information Theory, vol. 40, no. 1, pp. 207–216, 1994.
• S. Bouyuklieva, "Applications of the Gaussian integers in coding theory," Proceedings of the 3rd International 39 Colloquium on Differential Geometry and its Related Fields, Veliko Tarnovo, Bulgaria, 2012, pp. 39–49. J. Freudenberger, F. Ghaboussi, S. Shavgulidze, "New coding techniques for codes over Gaussian integers," IEEE Transactions on Communications, vol. 61, no. 8, pp. 3114–3124, 2013.
• J. Freudenberger, F. Ghaboussi, S. Shavgulidze, "Set Partitioning and Multilevel Coding for Codes Over Gaussian Integer Rings," SCC 2013; 9th International ITG Conference on Systems, Communication, and Coding, Munich, Germany, pp. 1–5, 2013.
• M. Özen, M. Güzeltepe, “Quantum codes from codes over Gaussian integers with respect to the Mannheim metric,” Quantum Information & Computation, vol. 12, pp. 813–819, 2012.
• M. Özen, M. Güzeltepe, "Codes over quaternion integers," European Journal of Pure and Applied Mathematics, vol. 3, no. 4, pp. 670–677, 2010.
• M. Özen, M. Güzeltepe, "Cyclic codes over some finite rings," Selcuk Journal of Applied Mathematics, vol. 11, no. 2, pp. 71–76, 2010.
• Y. J. Choie, S. T. Dougherty, "Codes over and Jacobi forms over the quaternions," Applicable Algebra in Engineering, Communication and Computing, vol. 15, no. 2, pp. 129–147, 2004.
• M. Özen, M. Güzeltepe, "Cyclic codes over some finite quaternion integer rings," Journal of the Franklin Institute, vol. 348, no. 7, pp. 1312–1317, 2011.
• T. Shah, S. S. Rasool, "On codes over quaternion integers," Applicable Algebra in Engineering, Communication and Computing, vol. 24, no. 6, pp. 477–496, 2013.
• M. Güzeltepe, G. Çetinel, N. Sazak, “Constacyclic codes over Lipschitz integers,” 2023. (Early access)
• O. Heden, M. Güzeltepe, “Perfect 1–error–correcting Lipschitz weight codes,” Mathematical Communication, vol. 21, no. 1, pp. 23–30, 2016.
• M. Güzeltepe, “The MacWilliams identity for Lipschitz weight enumerators,” Gazi University Journal of Science, vol. 29, no. 4, pp. 869–877, 2016.
• M. Özen, M. Güzeltepe, “Quantum codes from codes over Lipschitz integers,” Global Journal of Pure and Applied Mathematics, vol. 7, pp. 201–206, 2011.
• M. Güzeltepe, "Codes over Hurwitz integers," Discrete Mathematics, vol. 313, no. 5, pp. 704–714, 2013.
• D. Rohweder, S. Stern, R. F. H. Fischer, S. Shavgulidze, J. Freudenberger, "Four-Dimensional Hurwitz Signal Constellations, Set Partitioning, Detection, and Multilevel Coding," in IEEE Transactions on Communications, vol. 69, no. 8, pp. 5079–5090, 2021.
• M. Güzeltepe, "On some perfect codes over Hurwitz integers," Mathematical Advances in Pure and Applied Sciences, vol. 1, no. 1, pp. 39–45, 2018.
• M. Güzeltepe, A. Altınel, "Perfect 1–error–correcting Hurwitz weight codes," Mathematical Communications, vol. 22, no. 2, pp. 265–272, 2017. M. Güzeltepe, O. Heden, "Perfect Mannheim, Lipschitz and Hurwitz weight codes," Mathematical Communications, vol. 19, no. 2, pp. 253–276, 2014.
• O. Heden, M. Güzeltepe, "On perfect –error-correcting codes," Mathematical Communications, vol. 20, no. 1, pp. 23–25, 2015.
• M. Güzeltepe, G. Güner, "Perfect codes over Hurwitz integers induced by circulant graphs," Journal of Universal Mathematics, vol. 5, no. 1, pp. 24–35, Mar. 2022.
• K. Abdelmoumen, H. Ben Azza, M. Najmeddine, "About Euclidean codes in rings," British Journal of Mathematics and Computer Science, vol. 4, no. 10, pp. 1356–1364, 2014.
• InetDaemon Enterprises. (2023). Signal constellation [Online]. Available: https://www.inetdaemon.com/tutorials/basic_concepts/communication/signals/constellation.shtml G. Davidoff, P. Sarnak, A. Valette, Elementary Number Theory, Group Theory, and Ramanujan Graphs, Cambridge University Press, 2003.
• J. H. Conway, D. A. Smith, On Quaternions and Octonions, A.K. Peters, 2003.
• Wikipedia. (2023). Parity (mathematics) [Online]. Available: https://en.wikipedia.org/wiki/Parity_(mathematics)
• J. Freudenberger, S. Shavgulidze, "New four-dimensional signal constellations from Lipschitz integers for transmission over the Gaussian channel," IEEE Transactions on Communications, vol. 63, no. 7, pp. 2420–2427, 2015.
• G. Forney, L. F. Wei, "Multidimensional constellations. I. Introduction, figures of merit, and generalized cross constellations," IEEE Journal on Selected Areas in Communications, vol. 7, no. 6, pp. 877–892, 1989.
• Wikipedia. (2023). Signal-to-noise ratio [Online]. Available: https://en.wikipedia.org/wiki/Signal-to-noise_ratio