Skew cyclic codes over a finite non-chain ring and an application
DOI:
https://doi.org/10.13069/jacodesmath.v13i2.453Keywords:
Linear codes, $\Theta_t$-cyclic codes, Non-chain rings, Gray maps, $(\Theta_t,\lambda)$-constacyclic codes, DNA-codesAbstract
This article studies \( \Theta_t \)-cyclic and \( (\Theta_t,\lambda) \)-constacyclic codes over the finite commutative non-chain Frobenius ring
\[ R = \mathbb{F}_q[u,v,w]/\langle u^2-u,\ v^2-v,\ w^2-1,\ uv,\ uw-wu,\ wv-vw \rangle .\]
Gray maps, structural decompositions, and generator descriptions are developed for both odd- and even-characteristic cases. The paper further determines principal generators in the associated skew polynomial rings, dual codes, idempotent generators, and conditions for self-duality. It also presents explicit examples over specific finite fields and extends the framework to DNA codes in the even-characteristic setting through reversibility and complement constraints. Spanning sets, cardinality formulas, and optimal DNA-code constructions meeting the Griesmer bound are also obtained.
Received: 12 September 2025 | Accepted: 05 April 2026
Downloads
References
T. Abualrub, A. Ghrayeb and X. Nian Zeng, Construction of cyclic codes over GF(4) for DNA computing, J. Franklin Inst. 343(4-5) (2006) 448–457.
T. Abualrub, N. Aydin and P. Seneviratne, On θ-cyclic codes over F2 + vF2, The Australasian Journal of Combinatorics 54 (2012) 115–126.
T. Abualrub, M. F. Ezerman, P. Seneviratne, P. Solé, Skew Generalized Quasi-Cyclic Codes, TWMS J. Pure Appl. Math. 9(2) (2018) 123–134.
L. Adleman, Molecular computation of solutions to combinatorial problems, Science 266 (1994) 1021–1024.
M. Ashraf and G. Mohammad, Skew cyclic codes F3 + vF3, Int. J. Inf. Coding Theory 2(4) (2014) 218–225.
M. Ashraf and G. Mohammad, Skew cyclic codes Fq + uFq + vFq, Asian-European Journal of Mathematics 11(5) (2018) 1850072.
N. Bennenni, K. Guenda and S. Mesnager, DNA cyclic codes over rings, Advances in Mathematics of Communications 11(1) (2017) 83–98.
S. Bhardwaj and M. Raka, Skew constacyclic codes over a non-chain ring Fq[u, v]/<f(u), g(v), uv−vu>, arXiv:1905.12933 [cs.IT] (2019).
D. Boucher, W. Geiselmann and F. Ulmer, Skew-cyclic codes, Appl. Algebra Eng. Comm. 18 (2007) 379–389.
D. Boucher, P. Solé and F. Ulmer, Skew constacyclic codes over Galois rings, Adv. Math. Commun. 2(3) (2008) 273–292.
Y. Cengellenmis, A. Dertli, On the Skew Cyclic Codes and the Reversibility Problem for DNA 4-Bases, Mathematics in Computer Science 14 (2020) 431–435.
A. Dertli and Y. Cengellenmis, Skew cyclic codes over Fq + uFq + vFq + uvFq, Journal of Science and Arts. 2(39) (2017) 215–222.
H. Q. Dinh, Bac T. Nguyen, Songsak Sriboonchitta, Skew Constacyclic Codes over Finite Fields and Finite Chain Rings, Mathematical Problems in Engineering (2016) 1–17.
F. Gursoy, E. S. Oztas and I. Siap, Reversible DNA Codes Using Skew Polynomial Rings, AAECC 28 (2017) 311–320.
P. Gaborit, O. D. King, Linear construction for DNA codes, Theoretical Computer Science 334(1-3) (2005) 99–113.
J. Gao, F. Ma and F. Fu, Skew constacyclic codes over the ring Fq + vFq, Appl. Comput. Math. 6(3) (2017) 286–295.
K. Gowdhaman, T. A. Gulliver, C. Mohan, D. Chinnapillai, Constacyclic codes over the non-chain finite commutative ring Z4 + uZ4 + vZ4, J. Discrete Math. Sci. Cryptogr. 27(6) (2024) 1867–1885.
F. Gursoy, I. Siap and B. Yildiz, Construction of skew cyclic codes over Fq + vFq, Adv. Math. Commun. 8(3) (2014) 313–322.
H. Islam and O. Prakash, Skew cyclic and skew (α1 + uα2 + vα3 + uvα4)-constacyclic codes over Fq + uFq + vFq + uvFq, Int. J. Inf. Coding Theory 5(2) (2018) 101–116.
H. Islam and O. Prakash, A note on skew constacyclic codes over Fq + uFq + vFq, Discrete Mathematics, Algorithms and Applications 11(3) (2019) 1950030.
S. Jitman, S. Ling and P. Udomkavanich, Skew constacyclic codes over finite chain rings, Adv. Math. Commun. 6 (2012) 39–63.
O. King, Bounds for DNA Codes with Constant GC-Content, Journal of Combinatorics 10 (2003) 1–13.
T. Y. Lam and A. Leroy, Vandermonde and Wronskian matrices over division rings, Journal of Algebra 119(2) (1988) 308–336.
A. Melakhessou, N. Aydin, Z. Hebbache, K. Guenda, Zq(Zq + uZq)-Linear Skew Constacyclic Codes, Journal of Algebra Combinatorics Discrete Structures and Applications 7(1) (2019) 85–101.
C. Mohan, C. Durairajan, Skew-constacyclic codes over R, Malaya J. Mat. 8(4) (2020) 1502–1508.
R. Mohammadi, S. Rahimi, H. Mousavi, On Skew Cyclic Codes over A Finite Ring, Iranian Journal of Mathematical Sciences and Informatics 14(1) (2019) 135–145.
E. S. Oztas, I. Siap, On a generalization of lifted polynomials over finite fields and their applications to DNA codes, International Journal of Computer Mathematics 92(9) (2015) 1976–1988.
V. Rykov, A. J. Macula, D. Torney, P. White, DNA sequences and quaternary cyclic codes, IEEE International Symposium on Information Theory (ISIT 2001) Washington DC (2001) 248.
I. Siap, T. Abulraub and A. Ghrayeb, Cyclic DNA codes over the ring F2[u]/(u2 − 1) based on the deletion distance, J. Franklin Inst. 346(8) (2009) 731–740.
I. Siap, T. Abualrub, N. Aydin and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2 (2011) 10–20.
K. Shiromotoa, L. Storme, A Griesmer bound for linear codes over finite quasi-Frobenius rings, Discrete Applied Mathematics 128 (2003) 263–274.
T. Yao, M. Shi and P. Sole, On Skew cyclic codes over Fq + uFq + vFq + uvFq, J. Algebra Comb. Discrete Struct. Appl. 2(3) (2015) 163–168.
B. Yildiz, I. Siap, Cyclic codes over F2[u]/(u4 − 1) and applications to DNA codes, Computers and Mathematics with Application 63 (2012) 1169–1176.
