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).
Quaternion integers, Hurwitz integers, residual class, signal constellations, code constructions
Kuaterniyon tamsayıları, Hurwitz tamsayıları, kalan sınıfı, sinyal yıldızkümeleri, kod yapıları
Primary Language | English |
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Subjects | Mathematical Sciences |
Journal Section | Research Articles |
Authors | |
Early Pub Date | August 19, 2023 |
Publication Date | August 25, 2023 |
Submission Date | February 5, 2023 |
Acceptance Date | April 26, 2023 |
Published in Issue | Year 2023 Volume: 27 Issue: 4 |
Bibtex | @research article { saufenbilder1248060, journal = {Sakarya University Journal of Science}, eissn = {2147-835X}, address = {}, publisher = {Sakarya University}, year = {2023}, volume = {27}, number = {4}, pages = {792 - 812}, doi = {10.16984/saufenbilder.1248060}, title = {Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property}, key = {cite}, author = {Duran, Ramazan} } |
APA | Duran, R. (2023). Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property . Sakarya University Journal of Science , 27 (4) , 792-812 . DOI: 10.16984/saufenbilder.1248060 |
MLA | Duran, R. "Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property" . Sakarya University Journal of Science 27 (2023 ): 792-812 <https://dergipark.org.tr/en/pub/saufenbilder/issue/79486/1248060> |
Chicago | Duran, R. "Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property". Sakarya University Journal of Science 27 (2023 ): 792-812 |
RIS | TY - JOUR T1 - Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property AU - RamazanDuran Y1 - 2023 PY - 2023 N1 - doi: 10.16984/saufenbilder.1248060 DO - 10.16984/saufenbilder.1248060 T2 - Sakarya University Journal of Science JF - Journal JO - JOR SP - 792 EP - 812 VL - 27 IS - 4 SN - -2147-835X M3 - doi: 10.16984/saufenbilder.1248060 UR - https://doi.org/10.16984/saufenbilder.1248060 Y2 - 2023 ER - |
EndNote | %0 Sakarya University Journal of Science Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property %A Ramazan Duran %T Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property %D 2023 %J Sakarya University Journal of Science %P -2147-835X %V 27 %N 4 %R doi: 10.16984/saufenbilder.1248060 %U 10.16984/saufenbilder.1248060 |
ISNAD | Duran, Ramazan . "Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property". Sakarya University Journal of Science 27 / 4 (August 2023): 792-812 . https://doi.org/10.16984/saufenbilder.1248060 |
AMA | Duran R. Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property. SAUJS. 2023; 27(4): 792-812. |
Vancouver | Duran R. Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property. Sakarya University Journal of Science. 2023; 27(4): 792-812. |
IEEE | R. Duran , "Encoder Hurwitz Integers: Hurwitz Integers that have the “Division with Small Remainder” Property", Sakarya University Journal of Science, vol. 27, no. 4, pp. 792-812, Aug. 2023, doi:10.16984/saufenbilder.1248060 |
Sakarya University Journal of Science (SAUJS)