We construct a class of \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes generated by pairs of polynomials, where p is a prime number. The generator matrix of this class of codes is obtained. By establishing the relationship between the random \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic code and random quasi-cyclic code of index 2 over \(\mathbb {Z}_{p}\), the asymptotic properties of the rates and relative distances of this class of codes are studied. As a consequence, we prove that \(\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}\)-additive cyclic codes are asymptotically good since the asymptotic GV-bound at \(\frac {1+p^{s-1}}{2}\delta \) is greater than \(\frac {1}{2}\), the relative distance of the code is convergent to δ, while the rate is convergent to \(\frac {1}{1+p^{s-1}}\) for \(0< \delta < \frac {1}{1+p^{s-1}}\).

中文翻译：

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ℤpℤps$ \ mathbb {Z} _ {p} \ mathbb {Z} _ {p ^ {s}} $-渐近性好的循环代码

我们构造了一类\（\ mathbb {Z} _ {p} \ mathbb {Z} _ {p ^ {s}} \）-由多项式对生成的可加循环码，其中p是质数。获得此类代码的生成器矩阵。通过建立随机\（\ mathbb {Z} _ {p} \ mathbb {Z} _ {p ^ {s}} \）之间的关系，在\（\ mathbb {Z} _ {p} \），研究此类代码的速率和相对距离的渐近性质。结果，我们证明\（\ mathbb {Z} _ {p} \ mathbb {Z} _ {p ^ {s}} \）的加法循环代码在\（\ frac {1 + p ^ {s-1}} {2} \ delta \）大于\（\ frac {1} {2} \），代码的相对距离收敛于δ，而速率收敛于\（\ frac {1} {1 + p ^ {s-1}} \）为\（0 <\ delta <\ frac {1} {1 + p ^ {s-1}} \）。