Cyclic codes over $ F_2[u, v, w]/\langle u^2 = v^2, uv = 0, w^2 = w\rangle $ and its applications

Merve BULUT YILGÖR; Merve BULUT YILGÖR

doi:10.3934/math.20251249

AIMS Mathematics

2025, Volume 10, Issue 12: 28396-28406. doi: 10.3934/math.20251249

Previous Article Next Article

Research article

Cyclic codes over $ F_2[u, v, w]/\langle u^2 = v^2, uv = 0, w^2 = w\rangle $ and its applications

Merve BULUT YILGÖR ^,

Department of Basic Sciences, Faculty of Engineering and Architecture, Altınbaş University, 34218, Mahmutbey, İstanbul, Türkiye

Received: 07 October 2025 Revised: 26 November 2025 Accepted: 27 November 2025 Published: 03 December 2025
MSC : 11T71, 95B15

We investigate linear and cyclic codes over the ring $ F_2[u, v, w]/\langle u^2 = v^2, \, uv = 0, \, w^2 = w\rangle $. This is a commutative Frobenius non-chain ring, which, to the best of our knowledge, is studied here for the first time in the literature. We define a homogeneous weight on the ring and, with respect to a Gray map induced by this weight, obtain the optimal Reed-Muller code $ RM $ $ (1, 7) $. We analyze the algebraic structure of the ring in detail, determine its ideals, and present code constructions together with their Gray images.
- algebraic coding theory,
- codes over rings,
- homogeneous weight
Citation: Merve BULUT YILGÖR. Cyclic codes over $ F_2[u, v, w]/\langle u^2 = v^2, uv = 0, w^2 = w\rangle $ and its applications[J]. AIMS Mathematics, 2025, 10(12): 28396-28406. doi: 10.3934/math.20251249

Related Papers:

Abstract

We investigate linear and cyclic codes over the ring $ F_2[u, v, w]/\langle u^2 = v^2, \, uv = 0, \, w^2 = w\rangle $. This is a commutative Frobenius non-chain ring, which, to the best of our knowledge, is studied here for the first time in the literature. We define a homogeneous weight on the ring and, with respect to a Gray map induced by this weight, obtain the optimal Reed-Muller code $ RM $ $ (1, 7) $. We analyze the algebraic structure of the ring in detail, determine its ideals, and present code constructions together with their Gray images.

References

[1]	T. Abualrub, I. Siap, Cyclic codes over the rings $ Z_2+ uZ_2 $ and $ Z_2 + uZ_2 + u^2Z_2 $, Des. Codes Crypt., 42 (2007), 273–287. https://doi.org/10.1007/s10623-006-9034-5 doi: 10.1007/s10623-006-9034-5
[2]	M. Badie, A. Aliabad, F. Obeidavi, On ideals of product of commutative rings and their applications, arXiv: 2506.08537. https://doi.org/10.48550/arXiv.2506.08537
[3]	T. Alsuraiheed, E. Oztas, S. Ali, M. Yilgor, Reversible codes and applications to DNA codes over $F_4^2t[u]/(u^2-1)$, AIMS Mathematics, 8 (2023), 27762–27774. https://doi.org/10.3934/math.20231421 doi: 10.3934/math.20231421
[4]	I. Chajda, G. Eigenthaler, H. Länger, Ideals of direct products of rings, Asian-Eur. J. Math., 11 (2018), 1850094. https://doi.org/10.1142/S1793557118500948 doi: 10.1142/S1793557118500948
[5]	I. Constantinescu, W. Heise, A metric for codes over residue class rings of integers, Probl. Peredachi Inf., 33 (1997), 22–28.
[6]	H. Dinh, S. López-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50 (2004), 1728–1744. https://doi.org/10.1109/TIT.2004.831789 doi: 10.1109/TIT.2004.831789
[7]	S. Dougherty, A. Kaya, E. Salturk, Cyclic codes over local Frobenius rings of order 16, Adv. Math. Commun., 11 (2017), 99–114. https://doi.org/10.3934/amc.2017005 doi: 10.3934/amc.2017005
[8]	M. Greferath, S. Schmidt, Gray isometries for finite chain rings and a nonlinear ternary $(36, 3^12, 15)$ code, IEEE Trans. Inform. Theory, 45 (1999), 2522–2524. https://doi.org/10.1109/18.796395 doi: 10.1109/18.796395
[9]	M. Greferath, M. O'Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math., 289 (2004), 11–24. https://doi.org/10.1016/j.disc.2004.10.002 doi: 10.1016/j.disc.2004.10.002
[10]	F. Gursoy, I. Siap, B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_q+v\mathbb{F}_q$, Adv. Math. Commun., 8 (2014), 313–322. https://doi.org/10.3934/amc.2014.8.313 doi: 10.3934/amc.2014.8.313
[11]	A. Hammons, P. Kumar, A. Calderbank, N. Sloane, P. Sole, The $Z_4$-linearity of Kerdock, Preparate, Goethals, and related codes, IEEE Trans. Inform. Theory, 40 (1994), 301–319. https://doi.org/10.1109/18.312154 doi: 10.1109/18.312154
[12]	B. Yildiz, S. Karadeniz, Cyclic codes over $\mathbb F_2+u \mathbb F_2+v\mathbb F_2+uv\mathbb F_2$, Des. Codes Crypt., 58 (2011), 221–234. https://doi.org/10.1007/s10623-010-9399-3 doi: 10.1007/s10623-010-9399-3
[13]	B. Yildiz, I. Kelebek, The homogeneous weight for $ R_k $, related Gray map a new binary quasi-cyclic codes, Filomat, 31 (2017), 885–897. https://doi.org/10.2298/FIL1704885Y doi: 10.2298/FIL1704885Y
[14]	M. Yilgor, F. Gursoy, E. Oztas, F. Demirkale, Cyclic codes over $\mathbb F_2+u \mathbb F_2+v\mathbb F_2+v^2\mathbb F_2$ with respect to the homogeneous weight and their applications to DNA codes, AAECC, 32 (2021), 621–636. https://doi.org/10.1007/s00200-020-00416-0 doi: 10.1007/s00200-020-00416-0
[15]	S. Zhu, Y. Wang, M. Shi, Some results on cyclic codes over ${F}_{2}+v{F}_{2}$, IEEE Trans. Inform. Theory, 56 (2010), 1680–1684. https://doi.org/10.1109/TIT.2010.2040896 doi: 10.1109/TIT.2010.2040896

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)