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 \(\mathbb {Z}_{p^r}\mathbb {Z}_{p^s}\mathbb {Z}_{p^t}\) ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}{\mathbb{Z}}_{p^t}$ -additive cyclic codes generated by 3-tuples of polynomials, where p is a prime number and \(1 \le r \le s \le t\) $1\le r\le s\le t$ . 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 \(\delta \) $\delta$ satisfying \(0< \delta < 1\) $0<\delta <1$ such that the asymptotic Gilbert-Varshamov bound at \(\left( \frac{k+l+n}{3p^{r-1}}\delta \right) \) $\left(\frac{k+l+n}{3p^{r-1}},\delta \right)$ is greater than \(\frac{1}{2}\) $\frac{1}{2}$ , we demonstrate that the relative distance of the random code converges to \(\delta \) $\delta$ , while the rate of the random code converges to \(\frac{1}{k+l+n}\) $\frac{1}{k+l+n}$ . Finally, we conclude that the \(\mathbb {Z}_{p^r}\mathbb {Z}_{p^s}\mathbb {Z}_{p^t}\) ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}{\mathbb{Z}}_{p^t}$ -additive cyclic codes exhibit asymptotically good properties.