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Dergiler / Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi / 2026 / Cilt: 16 Sayı: 2

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

Hayrullah Özimamoğlu(Nevşehir Hacı Bektaş Veli Üniversitesi)0000-0001-7844-1840
Sayfa
327–344
Yayın Tarihi
2023-08-31
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.

📌 3 Maddede Bu Makale

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Amaç & Hipotez: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.
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Temel Bulgular & Veri: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.
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Sonuç & Katkı:Also, we present the generator matrix of the dual of a free δ_(θ_r )-cyclic code of even length n over R_r.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:cyclic codes · skew polynomial rings · skew-cyclic codes

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.

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Bu Makaleye Atıf Yap

Özimamoğlu, H. (2023). A Class of Skew-Cyclic Codes over (Z_(2^m ) [u])/〈u^2-r〉 with Derivation. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, Cilt: 16 Sayı: 2, 327–344. https://doi.org/10.18185/erzifbed.1120896

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Kaynakça (14)

  1. Blake, I. F., (1972). “Codes over certain rings”, Information and Control, 20, 396-404.
  2. Blake, I. F., (1975). “Codes over integer residue rings”, Information and Control, 29, 295-300.
  3. Boucher, D., Geiselmann, W., Ulmer, F., (2007). “Skew cyclic codes”, Appl. Algebra Engrg. Comm. Comput., 18, 379-389.
  4. Boucher, D., Sole, P., Ulmer, F., (2008). “Skew constacylic codes over Galois rings”, Adv. Math. Commun., 2, 273-292.
  5. 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.
  6. Boucher, D., Ulmer, F., (2009). “Coding with skew polynomial rings”, J. of Symbolic Comput., 44, 1644–1656.
  7. Boucher, D., Ulmer, F., (2014). “Linear codes using skew polynomials with automorphisms and derivations”, Des. Codes Cryptogr., 70, 405–431.
  8. 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.
  9. 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.
  10. Ore, O., (1933). “Theory of Non-Commutative Polynomials”, Ann. Math., 2nd Ser, 34(3), 480–508.
  11. Prange, E., (1957). “Cyclic error-correcting codes in two symbols”, Air Force Cambridge Research Center, Cambridge, MA, Tech. Rep. AFCRC-TN, 57-103.
  12. 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.
  13. Spiegel, E., (1977). “Codes over Z_m”, Information and Control, 35, 48-51.
  14. Spiegel, E., (1978). “Codes over Z_m (revisited)”, Information and Control, 37, 100-104.