${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -Additive Cyclic Codes, Generator Polynomials, and Dual Codes

A ℤ
₂
ℤ
₄
-additive code C ⊆ ℤ
₂
^α
x 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. These codes can be identified as submodules of the ℤ
₄
[x]-module ℤ
₂
[x]/(x
^α
- 1) x ℤ
₄
[x]/(x
^β
- 1). The parameters of a ℤ
₂
ℤ
₄
-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a ℤ
₂
ℤ
₄
-additive cyclic code are determined in terms of the generator polynomials of the code C.