We construct a class of \({\mathbb {Z}}_p{\mathbb {Z}}_p[v]\)-additive cyclic codes, where p is a prime number and \(v^2=v\). We determine the asymptotic properties of the relative minimum distance and rate of this class of codes. We prove that, for any positive real number \(0<\delta <1\) such that the p-ary entropy at \(\frac{k+l}{2}\delta \) is less than \(\frac{1}{2}\), the relative minimum distance of the random code is convergent to \(\delta \) and the rate of the random code is convergent to \(\frac{1}{k+l}\), where p, k, l are pairwise coprime positive integers.

我们构造一类\（{\ mathbb {Z}} _ p {\ mathbb {Z}} _ p [v] \）可加循环码，其中p是素数，\（v ^ 2 = v \）。我们确定此类代码的相对最小距离和速率的渐近性质。我们证明，对于任何正实数\（0 <\ delta <1 \），使得\（\ frac {k + 1} {2} \ delta \）的p-熵小于\（\ frac {1} {2} \），随机码的相对最小距离收敛于\（\ delta \），随机码的速率收敛于\（\ frac {1} {k + 1} \）其中p，  k，  l 是成对的互质数正整数。