On double cyclic codes over Z4Z4

Let R=Z4R=Z4 be the integer ring mod 4. A double cyclic code of length (r,s)(r,s) over R   is a set that can be partitioned into two parts that any cyclic shift of the coordinates of both parts leaves invariant the code. These codes can be viewed as R[x]R[x]-submodules of R[x]/(xr−1)×R[x]/(xs−1)R[x]/(xr−1)×R[x]/(xs−1). In this paper, we determine the generator polynomials of this family of codes as R[x]R[x]-submodules of R[x]/(xr−1)×R[x]/(xs−1)R[x]/(xr−1)×R[x]/(xs−1). Further, we also give the minimal generating sets of this family of codes as R  -submodules of R[x]/(xr−1)×R[x]/(xs−1)R[x]/(xr−1)×R[x]/(xs−1). Some optimal or suboptimal nonlinear binary codes are obtained from this family of codes. Finally, we determine the relationship of generators between the double cyclic code and its dual.