Asymptotically good $ \mathbb{Z}_{p}\mathbb{Z}_{p}[u]/\langle u^{t}\rangle $-additive cyclic codes

Abstract

$ \mathbb{Z}_{2}(\mathbb{Z}_{2}+u\mathbb{Z}_{2}) $-additive cyclic codes were proved to be asymptotically good in 2020 by Yao et al., where $ u^{2} = 0 $. We extend the study to the double cyclic codes over two finite commutative chain rings. Let $ R_{t} = \mathbb{Z}_{p}[u]/\langle u^{t}\rangle = \mathbb{Z}_{p}+u\mathbb{Z}_{p}+u^{2}\mathbb{Z}_{p}+\ldots+u^{t-1}\mathbb{Z}_{p} $ be a chain ring, where $ u^{t} = 0 $. We construct a class of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes generated by pairs of polynomials, where $ p $ is a prime number. By using probabilistic methods, we study the asymptotic behaviour of the rates and relative minimum distances of a certain class of the codes. We show that there exists an asymptotically good infinite sequence of $ \mathbb{Z}_{p}R_{t} $-additive cyclic codes with the relative minimum distance of the code is convergent to $ \delta $, and the rate is convergent to $ \frac{1}{1+p^{t-1}} $ for $ 0< \delta< \frac{1}{1+p^{t-1}} $, and $ t\geq1 $.