In this paper, we construct a class of \(\mathbb Z_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes generated by 3-tuples of polynomials. We discuss their algebraic structure and show that generator matrices can be constructed for all codes in this class. We study asymptotic properties of this class of codes by using a Bernoulli random variable. Moreover, let \(0< \delta < 1\) be a real number such that the entropy \(h_{4}(\frac{(k+l+t)\delta }{6})<\frac{1}{4},\) we show that the relative minimum distance converges to \(\delta\) and the rate of the random codes converges to \(\frac{1}{k+l+t},\) where k, l, and t are pairwise co-prime positive odd integers. Finally, we conclude that the \({\mathbb {Z}}_{4}{\mathbb {Z}}_{4}{\mathbb {Z}}_{4}\)-additive cyclic codes are asymptotically good.