Z2Z4-Additive Cyclic Codes: Kernel and Rank

A Z2Z4-additive code C subset of Z_2^alpha x Z_4^beta is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z_2 and the set of Z_4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Phi© be the binary Gray image of C. We study the rank and the dimension of the kernel of a Z2Z4-additive cyclic code C, that is, the dimensions of the binary linear codes <Phi©> and ker(Phi©). We give upper and lower bounds for these parameters. It is known that the codes <Phi©> and ker(Phi©) are binary images of Z2Z4-additive codes R© and K©, respectively. Moreover, we show that R© and K© are also cyclic and we determine the generator polynomials of these codes in terms of the generator polynomials of the code C.