On Linear Codes over$${\mathbb{Z}}_{2^{s}}$$
Designs, Codes and Cryptography
On the Quasi-cyclicity of the Gray Map Image of a Class of Codes over Galois Rings
ICMCTA '08 Proceedings of the 2nd international Castle meeting on Coding Theory and Applications
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For any integer $k \geq 1$, an isometry between codes over $\mathbb{Z}_{2^{k+1}}$ and codes over $\mathbb{Z}_4$ is defined and used to give an equivalent generalization of the Gray map to the one introduced in [C. Carlet, IEEE Trans. Inform. Theory, 44 (1998), pp. 1543-1547]. Several results related to the linearity or nonlinearity of codes over $\mathbb{Z}_{2^{k+1}}$, as well as its corresponding images under this map, are given. These results are similar to those presented in Theorems 4, 5, and 6 of [A. R. Hammons, Jr., P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, IEEE Trans. Inform. Theory, 40 (1994), pp. 301-319] for codes over $\mathbb{Z}_4$.