On $$\mathbb {Z}_{p^r} \mathbb {Z}_{p^s} \mathbb {Z}_{p^t}$$ -additive cyclic codes exhibit asymptotically good properties

In this paper, we construct a class of \(\mathbb {Z}_{p^r}\mathbb {Z}_{p^s}\mathbb {Z}_{p^t}\)-additive cyclic codes generated by 3-tuples of polynomials, where p is a prime number and \(1 \le r \le s \le t\). We investigate the algebraic structure of these codes and establish that it is possible to determine generator matrices for a subfamily of codes within this class. We employ a probabilistic approach to analyze the asymptotic properties of these codes. For any positive real number \(\delta \) satisfying \(0< \delta < 1\) such that the asymptotic Gilbert-Varshamov bound at \(\left( \frac{k+l+n}{3p^{r-1}}\delta \right) \) is greater than \(\frac{1}{2}\), we demonstrate that the relative distance of the random code converges to \(\delta \), while the rate of the random code converges to \(\frac{1}{k+l+n}\). Finally, we conclude that the \(\mathbb {Z}_{p^r}\mathbb {Z}_{p^s}\mathbb {Z}_{p^t}\)-additive cyclic codes exhibit asymptotically good properties.