\(\mathbb {F}_{q}\mathbb {F}_{q}[u]\)-additive constacyclic codes are asymptotically good

We construct a class of \(\mathbb {F}_{q}\mathbb {F}_{q}[u]\) ${\mathbb{F}}_q{\mathbb{F}}_q[u]$ -additive constacyclic codes, where \(q=p^{m}\) $q=p^m$ (p is an odd prime and m is a positive integer) and \(u^{3}=u\) $u^3=u$ . We study their algebraic structure, and determine the asymptotic rates and relative distances of this class of codes. For the rates R of this class of codes, let \(0<\delta <\frac{1}{3}\) $0<\delta <\frac{1}{3}$ and the q-ary entropy \(h_{q}(2\delta )<\frac{1}{2}\) $h_q(2\delta )<\frac{1}{2}$ , there is a sequence of random \(\mathbb {F}_{q}\mathbb {F}_{q}[u]\) ${\mathbb{F}}_q{\mathbb{F}}_q[u]$ -additive constacyclic code \(\mathcal{C}\) $\mathcal{C}$ with \(R(\mathcal{C})\) $R(\mathcal{C})$ converges to \(\frac{1}{4}\) $\frac{1}{4}$ with probability. Under such conditions, we further determine that the relative distance is convergent to \(\delta \) $\delta$ with probability. We prove that \(\mathbb {F}_{q}\mathbb {F}_{q}[u]\) ${\mathbb{F}}_q{\mathbb{F}}_q[u]$ -additive constacyclic codes over finite non-chain ring \(\mathbb {F}_{q}[u]\) ${\mathbb{F}}_q[u]$ are asymptotically good