A note on cyclic codes over ℤ4 + uℤ4

In this paper, we have studied cyclic codes over the ring R = ℤ4 + uℤ4, u2 = 0. We have provided the general form of the generators of a cyclic code over R and obtained a minimal spanning set for such codes and determined their ranks. We have determined a necessary condition and a sufficient condition for cyclic codes over R to be R-free. For n = 2k, we have shown that R[x] 〈xn−1〉 is a local ring, and the complete ideal structure of R[x] 〈xn−1〉 is determined. Some examples are presented.

[1]  Alexis Bonnecaze,et al.  Cyclic Codes and Self-Dual Codes Over , 1999 .

[2]  Taher Abualrub,et al.  On the generators of Z4 cyclic codes of length 2e , 2003, IEEE Trans. Inf. Theory.

[3]  Maheshanand Bhaintwal,et al.  On quasi-cyclic codes over $${\mathbb{Z}_q}$$ , 2009, Applicable Algebra in Engineering, Communication and Computing.

[4]  Taher Abualrub,et al.  Cyclic Codes of Length 2e over Z4 , 2001, Electron. Notes Discret. Math..

[5]  N. J. A. Sloane,et al.  The Z4-linearity of Kerdock, Preparata, Goethals, and related codes , 1994, IEEE Trans. Inf. Theory.

[6]  Minjia Shi,et al.  Some results on cyclic codes over F2 + UpsilonF2. , 2010 .

[8]  Taher Abualrub,et al.  Cyclic codes over the rings Z2 + uZ2 and Z2 + uZ2 + u2Z2 , 2007, Des. Codes Cryptogr..

[9]  H. Q. Dinh.  Constacyclic Codes of Length BBF2+uBBF2 , 2009, IEEE Trans. Inf. Theory.

[10]  Sergio R. López-Permouth,et al.  Cyclic Codes over the Integers Modulopm , 1997 .

[11]  Parampalli Udaya,et al.  Cyclic Codes and Self-Dual Codes Over F2 + uF2 , 1999, IEEE Trans. Inf. Theory.

[12]  YildizBahattin,et al.  Linear codes over Z4+uZ4 , 2014 .

[13]  B. R. McDonald.  Finite Rings With Identity , 1974 .

[14]  Suat Karadeniz,et al.  Linear Codes over Z_4+uZ_4: MacWilliams identities, projections, and formally self-dual codes , 2014, Finite Fields Their Appl..

[15]  Suat Karadeniz,et al.  Cyclic codes over F2+uF2+vF2+uvF2 , 2011, Des. Codes Cryptogr..

[16]  Thomas Blackford.  Cyclic Codes Over Z4 of Oddly Even Length , 2003, Discret. Appl. Math..

[17]  Vera Pless,et al.  Cyclic codes and quadratic residue codes over Z4 , 1996, IEEE Trans. Inf. Theory.

[18]  Thomas Blackford,et al.  Cyclic Codes Over Z4 of Oddly Even Length , 2001, Discret. Appl. Math..

[19]  Steven T. Dougherty,et al.  Cyclic Codes Over$$\mathbb{Z}_{4}$$ of Even Length , 2006, Des. Codes Cryptogr..

[20]  Bahattin Yildiz,et al.  On cyclic codes over ℤ4 + uℤ4 and their ℤ4-images , 2014, Int. J. Inf. Coding Theory.

[21]  Sergio R. López-Permouth,et al.  Cyclic and negacyclic codes over finite chain rings , 2004, IEEE Transactions on Information Theory.