On double cyclic, constacyclic and linear complementary dual codes over $\mathbf{R_{q}^{n}}$

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Arjun College of Technology, Coimbatore, Tamilnadu, India.

2 Department of Mathematics, University of Science and Technology Houari Boumediene (USTHB) of Algiers, Algeria.

3 Department of Mathematics, Vellore Institute of Technology, Chennai, Tamilnadu, India.

Abstract

In this paper, we study double cyclic and double constacyclic code over the field $\mathbf{R_{q}^{n}}$ and a non-chain ring $\mathbf{R_{q}^{n}}=\mathbb{F}_{q}^{n}+u\mathbb{F}_{q}^{n}$, $u^{2}=u.$ We determine the generator polynomials and minimum spanning sets for $\mathbf{R_{q}^{n}}$ and the image of double cyclic code over $\mathbf{R_{q}}$ is a double cyclic code over $\mathbf{R_{q}^{n}}.$ Also we discuss the linear complementary dual codes(LCD) over $\mathbf{R_{q}^{n}}$ has the application in a multi secret-sharing scheme.

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References

[1] A. Hammons, P. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Sole, The Z 4 -linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40 (1994) 301-319.
[2] T. Abualrub, I. Siap and N. Aydin, Z2Z4-Additive cyclic codes, IEEE Trans. Inform. Theory, 60 (2014) 1508-1514.
[3] I. Aydogdu and I. Siap, The structure of Z2Z2s -additive codes: Bounds on the minimum distance, Appl. Math. Inf. Sci., 7 No. 6 (2013) 2271-2278.
[4] I. Aydogdu, T. Abualrub and I. Siap, On Z2Z2[u]-additive codes, Int. J. Comput. Math., 92 (2015) 1806-1814.
[5] I. Aydogdu, T. Abualrub and I. Siap, The Z2Z2[u]-cyclic and constacyclic codes, IEEE Trans. Inform. Theory, 63 No. 8 (2016) 4883-4893.
[6] J. Borges, C. Fernandez-Cordoba, J. Pujol and J. Rifa, Z2Z4-linear codes: generator matrices and duality, Des. Codes Cryptogr., 54 (2010) 167-179.
[7] J. Borges, C. Fernandez-Cordoba and R. Ten-Valls, Z2Z4-additive cyclic codes, generator polynomials and dual codes, IEEE Trans. Inform. Theory, 62 (2016) 6348-6354.
[8] J. Borges, C. F. Cordoba and R. T. Valls, Z2-cyclic codes, Des. Codes Cryptogr., 86 (2018) 463-479.
[9] S. Bathala and P. Seneviratne, Some results on F4[v]-double cyclic codes, Comput. Appl. Math., 40 No. 2 (2021) 64.
[10] C. Carlet, S. Mesnager, C. M. Tang, Y. F. Qi, New characterization and parametrization of LCD codes, IEEE Trans. Inf. Theory, 65 No. 1 (2019) 39-49.
[11] C. Carlet, S. Mesnager, C. M. Tang, Y. F. Qi, On σ-LCD codes, IEEE Trans. Inf. Theory, 65 No. 3 (2019) 1694-1704.
[12] C. S. Ding, C. J. Li and S. X. Li, LCD cyclic codes over finite fields, IEEE Trans. Inf. Theory, 63 No. 7 (2017) 4344-4356.
[13] C. Fernandez-Cordoba, J. Pujol and M. Villanueva, Z2Z4-linear codes: Rank and kernel, Des. Codes Cryptogr., 56 No. 1 (2010) 43-59.
[14] J. Gao, M. Shi, T. Wu and F. W. Fu, On double cyclic codes over Z4, Finite Fields Their App., 39 (2016) 233-250.
[15] M. Harada and K., Saito, Binary linear complementary dual codes, Cryptogr. Commun., 11 No. 4 (2019) 677-696.
[16] J. L. Massey, Linear codes with complementary duals, Discrete Math., 106 No. 107 (1992) 337-342.
[17] X. Yang and J. L. Massey, The condition for a cyclic code to have a complementary dual, Discrete Math., 126 No. (1-3) (1994) 391-393.
[18] X. S. Liu, Y. Fan and H. L. Liu, Galois LCD codes over finite fields, Finite Fields Their App., 49 (2018) 227-242.
[19] S. X. Li, C. J. Li, C. S. Ding and H. Liu, Two families of LCD BCH codes, IEEE Trans. Inf. Theory, 63 No. 9 (2017) 5699-5717.
[20] B. B. Pang and S. X. Zhu and Z. H. Sun, On LCD negacyclic codes over finite fields, J. Syst. Sci. Complex, 31 No. 4 (2018) 1065-1077.
[21] H. Rifa-Pous, J. Rifa and L. Ronquillo, Z2Z4-additive perfect codes in steganography, Adv. Math. Commun., 5 No. 3 (2011) 425-433.
[22] N. Sendrier, Linear codes with complementary duals meet the Gilbert-Varshamov bound, Discrete Math., 285 (2004) 345-347.
[23] B. Srinivasulu and B. Maheshanand, The Z2(Z2+uZ2)-additive cyclic codes and their duals, Discrete Math. Algorithm. Appl., 8 (2016) 1793-8317.
[24] T. Yao and S. Zhu, ZpZps -additive cyclic codes are asymptotically good, Cryptogr. Commun., 12 (2020) 253-264.
Algebraic Structures and Their Applications

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