Z2-Triple cyclic codes and their duals

A Z2-triple cyclic code of block length (r,s, t) is a binary code of length r + s+ t such that the code is partitioned into three parts of lengths r, s and t such that each of the three parts is invariant under the cyclic shifts of the coordinates. Such a code can be viewed as Z2[x]-submodules of Z2[x] 〈xr−1〉 × Z2[x] 〈xs−1〉 × Z2[x] 〈xt−1〉 , in polynomial representation. In this paper, we determine the structure of these codes. We have obtained the form of the generators for such codes. Further, a minimal generating set for such a code is obtained. Also, we study the structure of the duals of these codes via the generators of the codes.