We construct a class of ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic Gilbert-Varshamov bound at $\frac{1+p^{s-r}}{2}\delta$ is greater than $\frac{1}{2}$ and the relative distance of the code is convergent to δ, while the rate is convergent to $\frac{1}{1+p^{s-r}}$ for $0<\delta <\frac{1}{1+p^{s-r}}$ and $1\le r<s$. As a consequence, we prove that there exist numerous asymptotically good ${\mathbb{Z}}_{p^r}{\mathbb{Z}}_{p^s}$-additive cyclic codes.

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渐近好 ${\mathbb{ž}}_{p^{\mathrm{[R}}}{\mathbb{ž}}_{p^s}$附加循环码

我们构造一类 ${\mathbb{ž}}_{p^{\mathrm{[R}}}{\mathbb{ž}}_{p^s}$-由多项式对生成的加性循环码，其中p是质数。基于概率论，我们确定此类代码的渐近率和相对距离：渐近的Gilbert-Varshamov界在$\frac{\mathrm{1个}+p^{s-\mathrm{[R}}}{2}\delta$ 大于 $\frac{\mathrm{1个}}{2}$代码的相对距离收敛到δ，速率收敛到$\frac{\mathrm{1个}}{\mathrm{1个}+p^{s-\mathrm{[R}}}$ 对于 $0<\delta <\frac{\mathrm{1个}}{\mathrm{1个}+p^{s-\mathrm{[R}}}$ 和 $\mathrm{1个}\le \mathrm{[R}<s$。结果，我们证明了存在许多渐近良好的${\mathbb{ž}}_{p^{\mathrm{[R}}}{\mathbb{ž}}_{p^s}$-加性循环码。