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Journals / Journal of Engineering Technology and Applied Sciences / 2026 / Cilt: 7 Sayı: 1

Skew Cyclic Codes over the Non-Chain ring $\mathcal{R}_q=\mathbb{F}_q[v]/\langle v^{2}+1\rangle$

Mehmet Emin Köroğlu(Yıldız Teknik Üniversitesi)0000-0002-9173-4944
Pages
51–60
Publication Date
2022-04-30
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AI Summary & Key Takeaways

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⚡ TL;DR(Too Long; Didn't Read)

In this paper, we investigate the algebraic structure of the non-local ring $\mathcal{R}_q = \mathbb{F}_q[v]/\langle v^{2}+1\rangle$ and identify the automorphisms of this ring to study the algebraic structure of the skew cyclic codes and their duals over it.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we investigate the algebraic structure of the non-local ring $\mathcal{R}_q = \mathbb{F}_q[v]/\langle v^{2}+1\rangle$ and identify the automorphisms of this ring to study the algebraic structure of the skew cyclic codes and their duals over it.
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Key Findings & Data:In this paper, we investigate the algebraic structure of the non-local ring $\mathcal{R}_q = \mathbb{F}_q[v]/\langle v^{2}+1\rangle$ and identify the automorphisms of this ring to study the algebraic structure of the skew cyclic codes and their duals over it.
💡
Conclusion & Impact:In this paper, we investigate the algebraic structure of the non-local ring $\mathcal{R}_q = \mathbb{F}_q[v]/\langle v^{2}+1\rangle$ and identify the automorphisms of this ring to study the algebraic structure of the skew cyclic codes and their duals over it.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Non-chain ring · linear codes · skew cyclic codes

Abstract

In this paper, we investigate the algebraic structure of the non-local ring $\mathcal{R}_q = \mathbb{F}_q[v]/\langle v^{2}+1\rangle$ and identify the automorphisms of this ring to study the algebraic structure of the skew cyclic codes and their duals over it.

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Cite This Article

Köroğlu, M. E. (2022). Skew Cyclic Codes over the Non-Chain ring $\mathcal{R}_q=\mathbb{F}_q[v]/\langle v^{2}+1\rangle$. Journal of Engineering Technology and Applied Sciences, Cilt: 7 Sayı: 1, 51–60. https://doi.org/10.30931/jetas.1057395

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References (34)

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  2. Boucher, D., Geiselmann, W., Ulmer, F., "Skew-cyclic codes", Applicable Algebra in Engineering, Communication and Computing 18(4) (2007): 379-389.
  3. Boucher, D., Solé, P., Ulmer, F., "Skew constacyclic codes over Galois rings", Advances in mathematics of communications 2(3) (2008): 273.
  4. Boucher, D. and Ulmer, F., "Codes as modules over skew polynomial rings", In IMA International Conference on Cryptography and Coding (2009): 38-55.
  5. Boucher, D. and Ulmer, F., "Coding with skew polynomial rings", Journal of Symbolic Computation 44(12) (2009): 1644-1656.
  6. Boucher, D., Ulmer, F., "Self-dual skew codes and factorization of skew polynomials", Journal of Symbolic Computation 60 (2014): 47-61.
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  8. Dinh, H. Q., López-Permouth, S. R., "Cyclic and negacyclic codes over finite chain rings", IEEE Transactions on Information Theory 50(8) (2004): 1728-1744.
  9. Dinh, H. Q., "Constacyclic codes of length $p^{s}$ over $\mathbb{F}_{p^{m}} +u\mathbb{F}_{p^{m}}$", Journal of Algebra 324(5) (2010): 940-950.
  10. Gao, J., Ma, F., Fu, F., "Skew constacyclic codes over the ring $\mathbb{F}_{q} +v\mathbb{F}_{q}$ ", Applied and Computational Mathematics 6(3) (2017): 286-295.
  11. Gao, J., "Skew cyclic codes over $\mathbb{F}_{q} +v\mathbb{F}_{q}$", Journal of Applied Mathematics and Informatics 31(3-4) (2013): 337-342.
  12. Gursoy, F., Siap, I., Yildiz, B., "Construction of skew cyclic codes over $\mathbb{F}_q+ v\mathbb{F}_q$", Advances in Mathematics of Communications 8(3) (2014): 313-322.
  13. Jitman, S., Ling, S., Udomkavanich, P., "Skew constacyclic codes over finite chain rings", Advances in Mathematics of Communications 6(1) (2012): 39-63.
  14. Martìnez-Moro, E., Rùa, I. F., "Multivariable codes over finite chain rings: serial codes", SIAM Journal on Discrete Mathematics 20(4) (2006): 947-959.
  15. Shi, M., Yao, T., Solè, P., "Skew cyclic codes over a non-chain ring", Chin. J. Electron. 26(3) (2017): 544-547.
  16. Norton, G. H., Sâlâgean, A., "Strong Gröbner bases and cyclic codes over a finite-chain ring", Electron. Notes Discrete Math. 6(2001): 240-250.
  17. Siap, I., Abualrub, T., Aydin, N., Seneviratne, P., "Skew Cyclic codes of arbitrary length", Int. J. Information and Coding Theory 2(1) (2011): 10-20.
  18. Amarra, M. C. V., Nemenzo, F. R., "On $(1-u)$-cyclic codes over $\mathbb{F}_{p^{k}} +u\mathbb{F}_{p^{k}}$ ", Applied Mathematics Letters 21(11) (2008): 1129-1133.
  19. Boucher, D., Geiselmann, W., Ulmer, F., "Skew-cyclic codes", Applicable Algebra in Engineering, Communication and Computing 18(4) (2007): 379-389.
  20. Boucher, D., Solé, P., Ulmer, F., "Skew constacyclic codes over Galois rings", Advances in mathematics of communications 2(3) (2008): 273.
  21. Boucher, D. and Ulmer, F., "Codes as modules over skew polynomial rings", In IMA International Conference on Cryptography and Coding (2009): 38-55.
  22. Boucher, D. and Ulmer, F., "Coding with skew polynomial rings", Journal of Symbolic Computation 44(12) (2009): 1644-1656.
  23. Boucher, D., Ulmer, F., "Self-dual skew codes and factorization of skew polynomials", Journal of Symbolic Computation 60 (2014): 47-61.
  24. Bonnecaze, A., Udaya, P., "Cyclic codes and self-dual codes over $\mathbb{F}_{2} +u\mathbb{F}_{2}$", IEEE Transactions on Information Theory 45(4) (1999): 1250-1255.
  25. Dinh, H. Q., López-Permouth, S. R., "Cyclic and negacyclic codes over finite chain rings", IEEE Transactions on Information Theory 50(8) (2004): 1728-1744.
  26. Dinh, H. Q., "Constacyclic codes of length $p^{s}$ over $\mathbb{F}_{p^{m}} +u\mathbb{F}_{p^{m}}$", Journal of Algebra 324(5) (2010): 940-950.
  27. Gao, J., Ma, F., Fu, F., "Skew constacyclic codes over the ring $\mathbb{F}_{q} +v\mathbb{F}_{q}$ ", Applied and Computational Mathematics 6(3) (2017): 286-295.
  28. Gao, J., "Skew cyclic codes over $\mathbb{F}_{q} +v\mathbb{F}_{q}$", Journal of Applied Mathematics and Informatics 31(3-4) (2013): 337-342.
  29. Gursoy, F., Siap, I., Yildiz, B., "Construction of skew cyclic codes over $\mathbb{F}_q+ v\mathbb{F}_q$", Advances in Mathematics of Communications 8(3) (2014): 313-322.
  30. Jitman, S., Ling, S., Udomkavanich, P., "Skew constacyclic codes over finite chain rings", Advances in Mathematics of Communications 6(1) (2012): 39-63.
  31. Martìnez-Moro, E., Rùa, I. F., "Multivariable codes over finite chain rings: serial codes", SIAM Journal on Discrete Mathematics 20(4) (2006): 947-959.
  32. Shi, M., Yao, T., Solè, P., "Skew cyclic codes over a non-chain ring", Chin. J. Electron. 26(3) (2017): 544-547.
  33. Norton, G. H., Sâlâgean, A., "Strong Gröbner bases and cyclic codes over a finite-chain ring", Electron. Notes Discrete Math. 6(2001): 240-250.
  34. Siap, I., Abualrub, T., Aydin, N., Seneviratne, P., "Skew Cyclic codes of arbitrary length", Int. J. Information and Coding Theory 2(1) (2011): 10-20.