$${\pmb {{\mathbb {Z}}}}_p{\pmb {{\mathbb {Z}}}}_p[v]$$ZpZp[v] -additive cyclic codes are asymptotically good

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.