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.

令p为任何奇数质数，令m，  k为任意正整数。在Galois环\（\ mathrm {GR}（p ^ 2，m）\）上构造长度为\（p ^ k \）的自对偶循环码是构造长度为\的自对偶循环码的关键（p ^ KN \）在整数剩余类环\（{\ mathbb {Z}} _ {p ^ 2} \）为任意正整数ñ满足\（\ mathrm {GCD}（p，N）= 1 \ ）。到目前为止，现有文献仅确定了这些自对偶循环码的数量（Des Codes Cryptogr 63：105-112，2012）。在本文中，我们为长度不同的所有自对偶循环码给出了一种有效的构造通过使用具有特定类型的矩阵的Kronecker乘积的列向量，在\（\ mathrm {GR}（p ^ 2，m）\）上的\（p ^ k \）上。在此基础上，我们还使用二项式系数进一步获得了所有这些自对偶循环码的显式表达式。