On the structure of cyclic codes over the ring Z2s [u]/⟨u k ⟩

In this paper, we consider cyclic codes of odd length n over the local, non-chain ring R = Z_2^s [u] ∕〈u^k〉 = Z_2^s + uZ_2^s +…+ u^k−1Z_2^s (uk=0), for any integers s ≥ 1 and k ≥ 2. An explicit algebraic representation of such codes is obtained. This algebraic structure is then used to establish the duals of all cyclic codes. Among others, all self-dual cyclic codes of odd length n over the ring R are determined. Moreover, some examples are provided which produce several optimal codes.