Binary Images of ${\mathbb{Z}_2\mathbb{Z}_4}$ -Additive Cyclic Codes

A 762764-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
and the set of ℤ
₄
coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. We study the binary images of ℤ
₂
ℤ
₄
-additive cyclic codes. We determine all ℤ
₂
ℤ
₄
-additive cyclic codes with odd β whose Gray images are linear binary codes. In this case, it is shown that such binary codes are permutation equivalent (by the Nechaev permutation) to ℤ
₂
-double cyclic codes. Finally, the generator polynomials of these binary codes are given.