A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation

Year 2023, Volume: 16 Issue: 2, 327 - 344, 31.08.2023

Hayrullah Özimamoğlu

https://doi.org/10.18185/erzifbed.1120896

Abstract

Let R_r=Z_(2^m )+uZ_(2^m ) be a finite ring, where u^2=r for r∈Z_(2^m ), m is a positive integer, and m≥2. In this paper, we study a class of skew-cyclic codes using a skew polynomial ring over R_r with an automorphism θ_r and a derivation δ_(θ_r ). We generalize the skew-cyclic codes over Z_4+uZ_4; u^2=1 to the skew-cyclic codes over R_r, and call such codes as δ_(θ_r )-cyclic codes. We investigate the structures of a skew polynomial ring R_r [x,θ_r,δ_(θ_r ) ]. A δ_(θ_r )-cyclic code is showed to be a left R_r [x,θ_r,δ_(θ_r ) ]-submodule of (R_r [x,θ_r,δ_(θ_r ) ])/〈x^n-1〉 . We give the generator matrix of a δ_(θ_r )-cyclic code of length n over R_r. Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.

Keywords

cyclic codes, skew polynomial rings, skew-cyclic codes

References

• Blake, I. F., (1972). “Codes over certain rings”, Information and Control, 20, 396-404.
• Blake, I. F., (1975). “Codes over integer residue rings”, Information and Control, 29, 295-300.
• Boucher, D., Geiselmann, W., Ulmer, F., (2007). “Skew cyclic codes”, Appl. Algebra Engrg. Comm. Comput., 18, 379-389.
• Boucher, D., Sole, P., Ulmer, F., (2008). “Skew constacylic codes over Galois rings”, Adv. Math. Commun., 2, 273-292.
• Boucher, D., Ulmer, F., (2009). “Codes as modules over skew polynomial rings”, In Proc. of 〖12〗^th IMA International Conference, Cryptography an Coding, Cirencester, UK, LNCS, 5921, 38–55.
• Boucher, D., Ulmer, F., (2009). “Coding with skew polynomial rings”, J. of Symbolic Comput., 44, 1644–1656.
• Boucher, D., Ulmer, F., (2014). “Linear codes using skew polynomials with automorphisms and derivations”, Des. Codes Cryptogr., 70, 405–431.
• Çalışkan, B., (2022). “Skew Cyclic Codes over the Ring Z_(2^s )+uZ_(2^s ) with Derivation”, Journal of Advanced Research in Natural and Applied Sciences, 8(1), 114-123.
• Hammons, A. R., Kumar, P. V., Calderbank, A. R., Sloane, N. J. A., Sole, P., (1994). “The Z_4-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Trans. Inf. Theory, 40(2), 301-319.
• Ore, O., (1933). “Theory of Non-Commutative Polynomials”, Ann. Math., 2nd Ser, 34(3), 480–508.
• Prange, E., (1957). “Cyclic error-correcting codes in two symbols”, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, 57-103.
• Sharma, A., Bhaintwal, M., (2018). “A class of skew cyclic codes over Z_4+uZ_4 with derivation”, Adv. Math. Commun., 12(4), 723-739.
• Spiegel, E., (1977). “Codes over Z_m”, Information and Control, 35, 48-51.
• Spiegel, E., (1978). “Codes over Z_m (revisited)”, Information and Control, 37, 100-104.