Mixed skew cyclic codes over rings
DOI:
https://doi.org/10.13069/jacodesmath.v12i1.339Keywords:
Skew cyclic codes, LCD skew cyclic codes, Computational resultsAbstract
This paper explores different types of skew cyclic codes by generating special subclasses with additional desirable properties. Specifically, we are interested in skew cyclic codes over mixed rings. We study some algebraic and structural properties of these codes and their constructions. We study skew cyclic codes over the mixed alphabet ring $\mathbb{F}_q(\mathbb{F}_q+v\mathbb{F}_q)$ under a mixed automorphism $(\theta,\tilde{\theta})$ and we give the structure of these codes for an arbitrary length via the non-commutative ring $\mathbb{F}_q[x,\theta](\mathbb{F}_q+v\mathbb{F}_q)[x,\tilde{\theta}]$. A condition for the existence of linear complementary dual (LCD) codes (which play an important role in practical applications such as armoring implementations against side-channel attacks and fault injection attacks) are explored specifically for skew cyclic codes.
Received: 5 July 2024 | Accepted: 13 October 2024Downloads
References
T. Abualrub, N. Aydin, and P. Seneviratne, On $\theta$-cyclic codes over $\mathbb{F}_2 + v\mathbb{F}_2$, Australas. J. Combin. 54 (2012) 115–126.
T. Abualrub, A. Ghrayeb, N. Aydin, and I. Siap, On the construction of skew quasi-cyclic codes, IEEE Trans. Inform. Theory 56(5) (2010) 2081–2090.
T. Abualrub and I. Siap, Constacyclic codes over $\mathbb{F}_2 + u\mathbb{F}_2$, J. Franklin Inst. 346(5) (2009) 520–529.
T. Abualrub, I. Siap, and N. Aydin, $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codes, IEEE Trans. Inform. Theory 60(3) (2014) 1508–1514.
N. Aydin, Some new linear codes from skew cyclic codes and computer algebra challenges, Appl. Algebra Engrg. Comm. Comput. 30(3) (2019) 185–191.
N. Benbelkacem, M. F. Ezerman, and T. Abualrub, Linear codes over $\mathbb{F}_4R$ and their MacWilliams identity, Discrete Math. Algorithms Appl. 12(6) (2020) 2050085.
N. Benbelkacem, M. F. Ezerman, T. Abualrub, and A. Batoul, Skew cyclic codes over $\mathbb{F}_4R$, J. Algebra Appl. 21(4) (2022) 2250065.
S. Biswas and M. Bhaintwal, Quantum codes from $\mathbb{Z}_2\mathbb{Z}_2[u]/\langle u^4\rangle$-cyclic codes, Des. Codes Cryptogr. 90(2) (2022) 343–366.
J. Borges, C. Fernandez-Cordoba, and R. Ten-Valls, On $\mathbb{Z}_{p^r}\mathbb{Z}_{p^s}$-additive cyclic codes, Adv. Math. Commun. 12(1) (2018) 169–179.
D. Boucher, W. Geiselmann, and F. Ulmer, Skew-cyclic codes, Appl. Algebra Engrg. Comm. Comput. 18(4) (2007) 379–389.
D. Boucher and F. Ulmer, Linear codes using skew polynomials with automorphisms and derivations, Des. Codes Cryptogr. 70(3) (2014) 405–431.
C. Carlet and S. Guilley, Complementary dual codes for counter-measures to side-channel attacks, Adv. Math. Commun. 10(1) (2016) 131–150.
P. M. Cohn, Free rings and their relations, Bull. Amer. Math. Soc. (N.S.) 21(1) (1989) 139–142.
G. D. Forney Jr., N. J. A. Sloane, and M. D. Trott, The Nordstrom-Robinson code is the binary image of the octacode, in Coding and Quantization, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 14, Amer. Math. Soc., Providence, RI (1993) 19–26.
M. Grassl, Bounds on the minimum distance of linear codes and quantum codes (online), http://www.codetables.de.
F. Gursoy, I. Siap, and B. Yildiz, Construction of skew cyclic codes over $\mathbb{F}_q + v\mathbb{F}_q$, Adv. Math. Commun. 8(3) (2014) 313–322.
F. Gursoy and I. Aydogdu, On $\mathbb{Z}_2\mathbb{Z}_4[\xi]$-skew cyclic codes, J. Appl. Math. Comput. 68(3) (2022) 1613–1633.
A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301–319.
J. L. Massey, Linear codes with complementary duals, Discrete Math. 106–107 (1992) 337–342.
E. S. Oztas, B. Yildiz, and I. Siap, A novel approach for constructing reversible codes and applications to DNA codes over the ring $\mathbb{F}_2[u]/(u^{2k}-1)$, Finite Fields Appl. 46 (2017) 217–234.
H. Rifà-Pous, J. Rifà, and L. Ronquillo, $\mathbb{Z}_2\mathbb{Z}_4$-additive perfect codes in steganography, Adv. Math. Commun. 5(3) (2011) 425–433.
A. Sharma and M. Bhaintwal, $\mathbb{F}_3R$-skew cyclic codes, Int. J. Inf. Coding Theory 3(3) (2016) 234–251.
I. Siap, T. Abualrub, N. Aydin, and P. Seneviratne, Skew cyclic codes of arbitrary length, Int. J. Inf. Coding Theory 2(1) (2011) 10–20.
