{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,20]],"date-time":"2026-02-20T18:32:57Z","timestamp":1771612377204,"version":"3.50.1"},"reference-count":36,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T00:00:00Z","timestamp":1742774400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>An even Eisenstein integer is a multiple of an Eisenstein prime of the least norm. Otherwise, an Eisenstein integer is called odd. An Eisenstein integer that is not an integer multiple of another one is said to be primitive. Such integers can be used to construct signal constellations and complex-valued codes over Eisenstein integers via a carefully designed modulo function. In this work, we establish algebraic properties of even, odd, and primitive Eisenstein integers. We investigate conditions for the set of all units in a given quotient ring of Eisenstein integers to form a cyclic group. We perform set partitioning based on the multiplicative group of the set. This generalizes the known partitioning of size a prime number congruent to 1 modulo 3 based on the multiplicative group of the Eisenstein field in the literature.<\/jats:p>","DOI":"10.3390\/e27040337","type":"journal-article","created":{"date-parts":[[2025,3,24]],"date-time":"2025-03-24T12:01:50Z","timestamp":1742817710000},"page":"337","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On Algebraic Properties of Primitive Eisenstein Integers with Applications in Coding Theory"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6536-9894","authenticated-orcid":false,"given":"Abdul","family":"Hadi","sequence":"first","affiliation":[{"name":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia"},{"name":"Department of Mathematics, Universitas Riau, Tampan, Pekanbaru 28293, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5313-813X","authenticated-orcid":false,"given":"Uha","family":"Isnaini","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0390-8682","authenticated-orcid":false,"given":"Indah Emilia","family":"Wijayanti","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Universitas Gadjah Mada, Sekip Utara Bulaksumur 21, Yogyakarta 55281, Indonesia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5851-2717","authenticated-orcid":false,"given":"Martianus Frederic","family":"Ezerman","sequence":"additional","affiliation":[{"name":"School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Ireland, K., Rosen, M.I., and Rosen, M. (1990). A Classical Introduction to Modern Number Theory, Springer Science & Business Media. [2nd ed.].","DOI":"10.1007\/978-1-4757-2103-4"},{"key":"ref_2","first-page":"518","article-title":"The Euler \u03c6-function in the Gaussian Integers","volume":"90","author":"Cross","year":"1983","journal-title":"Am. Math. Mon."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"602","DOI":"10.1080\/00029890.2005.11920231","article-title":"Finding factors of factor rings over the Gaussian integers","volume":"112","author":"Dresden","year":"2005","journal-title":"Am. Math. Mon."},{"key":"ref_4","first-page":"165","article-title":"On the factor rings of Eisenstein integers","volume":"6","author":"Ozkan","year":"2013","journal-title":"Erzincan Univ. J. Sci. Technol."},{"key":"ref_5","first-page":"58","article-title":"Factor rings and their decompositions in the Eisenstein integers ring Z[\u03c9]","volume":"5","author":"Misaghian","year":"2013","journal-title":"Armen. J. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.12988\/imf.2014.111121","article-title":"Finding factors of factor rings over Eisenstein integers","volume":"9","author":"Bucaj","year":"2013","journal-title":"Int. Math. Forum"},{"key":"ref_7","unstructured":"Gullerud, E., and Mbirika, A. (2020). An Euler phi function for the Eisenstein integers and some applications. Integers, Available online: https:\/\/math.colgate.edu\/~integers\/u20\/u20.pdf."},{"key":"ref_8","unstructured":"Hadi, A., Isnaini, U., Wijayanti, I.E., and Ezerman, M.F. (2024). On codes over Eisenstein integers. arXiv."},{"key":"ref_9","unstructured":"Stern, S., Rohweder, D., Freudenberger, J., and Fischer, R.F.H. (2019, January 24\u201326). Binary multilevel coding over Eisenstein integers for MIMO broadcast transmission. Proceedings of the International ITG Workshop Smart Antennas (WSA 2019), Vienna, Austria."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"7576","DOI":"10.1109\/TIT.2013.2274264","article-title":"An algebraic approach to physical-layer network coding","volume":"59","author":"Feng","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2713","DOI":"10.1109\/TCOMM.2013.050813.120759","article-title":"Lattice network codes based on Eisenstein integers","volume":"61","author":"Sun","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1988","DOI":"10.1109\/LCOMM.2013.081313.131410","article-title":"On lattice-partition-based physical-layer network coding over GF(4)","volume":"17","author":"Sun","year":"2013","journal-title":"IEEE Commun. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Fang, D., Burr, A., and Wang, Y. (2014, January 23\u201326). Eisenstein integer based multi-dimensional coded modulation for physical-layer network coding over F4 in the two-way relay channels. Proceedings of the European Conference on Networks and Communications (EuCNC 2014), Bologna, Italy.","DOI":"10.1109\/EuCNC.2014.6882643"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5306","DOI":"10.1109\/TIT.2015.2451623","article-title":"Lattices over Eisenstein integers for compute-and-forward","volume":"61","author":"Tunali","year":"2015","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1848","DOI":"10.1109\/18.705563","article-title":"Groups of algebraic integers used for coding QAM signals","volume":"44","author":"Dong","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1093\/imamci\/19.3.229","article-title":"Multiplicative groups used for coding QAM signals","volume":"19","author":"Dong","year":"2002","journal-title":"IMA J. Math. Control Inform."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"546","DOI":"10.1016\/j.phycom.2017.10.012","article-title":"Signal constellations employing multiplicative groups of Gaussian and Eisenstein integers for enhanced spatial modulation","volume":"25","author":"Raghavendra","year":"2017","journal-title":"Phys. Commun."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"556","DOI":"10.1109\/LCOMM.2016.2630701","article-title":"Signal constellations based on Eisenstein integers for generalized spatial modulation","volume":"21","author":"Freudenberger","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"3114","DOI":"10.1109\/TCOMM.2013.061913.120742","article-title":"New coding techniques for codes over Gaussian integers","volume":"61","author":"Freudenberger","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"2420","DOI":"10.1109\/TCOMM.2015.2441691","article-title":"New four-dimensional signal constellations from Lipschitz integers for transmission over the Gaussian channel","volume":"63","author":"Freudenberger","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"5079","DOI":"10.1109\/TCOMM.2021.3083323","article-title":"Four-dimensional Hurwitz signal constellations, set partitioning, detection, and multilevel coding","volume":"69","author":"Rohweder","year":"2021","journal-title":"IEEE Trans. Commun."},{"key":"ref_22","first-page":"39","article-title":"On some perfect codes over Hurwitz integers","volume":"1","year":"2018","journal-title":"Math. Adv. Pure Appl. Sci."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"652","DOI":"10.15672\/hujms.1137425","article-title":"An algebraic construction technique for codes over Hurwitz integers","volume":"52","author":"Duran","year":"2023","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s12215-023-00996-z","article-title":"Encoder Lipschitz integers: The Lipschitz integers that have the \u201cdivision with small remainder\u201d property","volume":"Volume 73","author":"Duran","year":"2024","journal-title":"Rendiconti del Circolo Matematico di Palermo Series 2"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2911","DOI":"10.1109\/TSP.2019.2910480","article-title":"Coprime sensing via Chinese remaindering over quadratic fields\u2014Part II: Generalizations and applications","volume":"67","author":"Li","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1828","DOI":"10.1109\/LSP.2021.3108526","article-title":"Multi-channel modulo samplers constructed from Gaussian integers","volume":"28","author":"Gong","year":"2021","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1007\/s10623-013-9850-3","article-title":"ETRU: NTRU over the Eisenstein integers","volume":"74","author":"Jarvis","year":"2013","journal-title":"Des. Codes Cryptog."},{"key":"ref_28","unstructured":"Conway, J.H., and Sloane, N.J.A. (2019). Sphere Packings, Lattices and Groups, Springer. [3rd ed.]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1090\/conm\/168\/01696","article-title":"Codes over Eisenstein-Jacobi integers","volume":"168","author":"Huber","year":"1994","journal-title":"AMS. Contemp. Math."},{"key":"ref_30","unstructured":"L\u00f6fgren, S. (2022). The Eisenstein Integers and Cubic Reciprocity, Uppsala University. Technical Report."},{"key":"ref_31","unstructured":"Vargas, J.D., and Recillas, H.T. (1989). La functi\u00f3n de Euler en Los Enteros de Eisenstein-Jacobi, Departamento de Matem\u00e1ticas, Universidad Aut\u00f3noma Metropolitana-Iztapalapa. Reportes de Investigaci\u00f3n."},{"key":"ref_32","first-page":"17","article-title":"The Euler totient function on quadratic fields","volume":"52","author":"Vargas","year":"2021","journal-title":"JP J. Algebra Number Theory Appl."},{"key":"ref_33","unstructured":"Hadi, A., Isnaini, U., Wahyudi, E.E., Wijayanti, I.E., and Ezerman, M.F. (2024). RSA-like schemes over Eisenstein integers. under review."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2898","DOI":"10.1109\/TSP.2019.2910498","article-title":"Coprime sensing via Chinese remaindering over quadratic fields\u2014Part I: Array designs","volume":"67","author":"Li","year":"2019","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Awad, Y., El-Kassar, A.N., and Kadri, T. (2018, January 25\u201326). Rabin public-key cryptosystem in the domain of Gaussian integers. Proceedings of the 2018 International Conference on Computer and Applications (ICCA), Beirut, Lebanon.","DOI":"10.1109\/COMAPP.2018.8460338"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1514","DOI":"10.1109\/18.923731","article-title":"Lattice constellations and codes from quadratic number fields","volume":"47","author":"Interlando","year":"2001","journal-title":"IEEE Trans. Inf. 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