Classification of self-dual cyclic codes over the chain ring $$\mathbb Z_p[u]/\langle u^3 \rangle $$

Abstract

We classify all the cyclic self-dual codes of length $$p^k$$ over the finite chain ring $$\mathcal R:=\mathbb Z_p[u]/\langle u^3 \rangle $$ , 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 $${\mathcal R}$$ of length $$p^k$$ for every prime p. We then prove that if a cyclic code over $${\mathcal R}$$ 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 $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ . Finally, we obtain a mass formula for counting cyclic self-dual codes over $$\mathbb Z_2[u]/\langle u^3 \rangle $$ of length $$2^k$$ .