부산대학교 도서관

A ${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -additive code ${\mathcal{ C}}\subseteq {\mathbb {Z}}_{2}^\alpha \times {\mathbb {Z}}_{4}^\beta $ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb {Z}}_{2}$ coordinates and the set of ${\mathbb {Z}}_{4}$ coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let $\Phi ({\mathcal{ C}})$ be the binary Gray map image of ${\mathcal{ C}}$ . We study the rank and the dimension of the kernel of a ${\mathbb {Z}}_{2} {\mathbb {Z}}_{4} $ -additive cyclic code ${\mathcal{ C}}$ , that is, the dimensions of the binary linear codes $\langle \Phi ({\mathcal{ C}}) \rangle $ and $\ker (\Phi ({\mathcal{ C}}))$ . We give upper and lower bounds for these parameters. It is known that the codes $\langle \Phi ({\mathcal{ C}}) \rangle $ and $\ker (\Phi ({\mathcal{ C}}))$ are binary images of ${\mathbb {Z}}_{2} {\mathbb {Z}}_{4}$ -additive codes that we denote by ${\mathcal{ R}}({\mathcal{ C}})$ and ${\mathcal{ K}}({\mathcal{ C}})$ , respectively. Moreover, we show that ${\mathcal{ R}}({\mathcal{ C}})$ and ${\mathcal{ K}}({\mathcal{ C}})$ are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code ${\mathcal{ C}}$ .