${\mathbb{Z}_{2}\mathbb{Z}_{4}}$ -Additive Cyclic Codes: Kernel and Rank

A ℤ
₂
ℤ
₄
-additive code C ⊆ ℤ
₂
^α
× ℤ
₄
^β
is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ℤ
₂
coordinates and the set of ℤ
₄
coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a ℤ
₂
ℤ
₄
-additive cyclic code C, that is, the dimensions of the binary linear codes (Φ(C)) and ker (Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes (Φ(C)) and ker (Φ(C)) are binary images of ℤ
₂
ℤ
₄
-additive codes that we denote by R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and determine the generator polynomials of these codes in terms of the generator polynomials of the code C.