Classification of self-dual cyclic codes over the chain ring Z_p[u]/〈u~3〉

摘要

We classify all the cyclic self-dual codes of length p(k) over the finite chain ring R := Z(p)[u]/ u(3), 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 Z(p)[u]/ u(3) of length 2(k). Finally, we obtain a mass formula for counting cyclic self-dual codes over Z(p)[u]/ u(3) of length 2(k).