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

Yıl 2022, Cilt: 7 Sayı: 1, 51 - 60, 30.04.2022

Mehmet Emin Köroğlu

https://doi.org/10.30931/jetas.1057395

https://izlik.org/JA63TY77TC

Öz

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.

Anahtar Kelimeler

Non-chain ring , linear codes , skew cyclic codes

Kaynakça

• [1] 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.
• [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.
• [7] 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.
• [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.