Classification of self-dual cyclic codes over the chain ring Z_p[ u] / ⟨ u³⟩

We classify all the cyclic self-dual codes of length p^k over the finite chain ring R: = Z_p[ u] / ⟨ u³⟩ , which is not a Galois ring, where p is a prime number and k is a positive integer. First, we find all the dual codes of cyclic codes over R of length p^k for every prime p. We then prove that if a cyclic code over R of length p^k is self-dual, then p should be equal to 2. Furthermore, we completely determine the generators of all the cyclic self-dual codes over Z₂[ u] / ⟨ u³⟩ of length 2 ^k. Finally, we obtain a mass formula for counting cyclic self-dual codes over Z₂[ u] / ⟨ u³⟩ of length 2 ^k.