An explicit expression for all distinct self-dual cyclic codes of length $$p^k$$ p k over Galois ring $$\mathrm{GR}(p^2,m)$$ GR ( p 2 , m )

Let p be any odd prime number and let m, k be arbitrary positive integers. The construction for self-dual cyclic codes of length $$p^k$$ over the Galois ring $$\mathrm{GR}(p^2,m)$$ is the key to construct self-dual cyclic codes of length $$p^kn$$ over the integer residue class ring $${\mathbb {Z}}_{p^2}$$ for any positive integer n satisfying $$\mathrm{gcd}(p,n)=1$$ . So far, existing literature has only determined the number of these self-dual cyclic codes (Des Codes Cryptogr 63:105–112, 2012). In this paper, we give an efficient construction for all distinct self-dual cyclic codes of length $$p^k$$ over $$\mathrm{GR}(p^2,m)$$ by using column vectors of Kronecker products of matrices with specific types. On this basis, we further obtain an explicit expression for all these self-dual cyclic codes by using binomial coefficients.