$\mathbb {Z}_{p}\mathbb {Z}_{p^{s}}$-additive cyclic codes are asymptotically good

Abstract

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 $0< \delta < \frac {1}{1+p^{s-1}}$.