On double cyclic codes over Z4
Let $R=\mathbb{Z}_4$ be the integer ring mod $4$ A double cyclic code of length (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] submodules of $R[x](x^r 1)\times R[x](x^s 1) In this paper, we determine the generator polynomials of this family of codes as $R[x] submodules of $R[x](x^r 1)\times R[x](x^s 1) Further, we also give the minimal generating sets of this family of codes as $R$ submodules of $R[x](x^r 1)\times R[x](x^s 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.
Finite Fields Their Appl.