F₂F₂[u²]F₂[u³]-additive cyclic codes are asymptotically good

In this paper, we construct a class of F₂F₂[u²]F₂[u³]-additive cyclic codes generated by 3-tuples of polynomials, where F₂ is the binary field, F₂[u²]=F₂+uF₂ (u²=0) and F₂[u³]=F₂+uF₂+u²F₂ (u³=0). We provide their algebraic structure and show that generator matrices can be obtained for all codes of this class. Using a random Bernoulli variable, we investigate the asymptotic properties in this class of codes. Furthermore, let 0<δ<1 be a real number and k,l and t be pairwise co-prime positive odd integers such that the entropy at [Formula presented] is less than [Formula presented], we prove that the relative minimum homogeneous distances converge to δ, and the rates of the random codes converge to [Formula presented]. Consequently, F₂F₂[u²]F₂[u³]-additive cyclic codes are asymptotically good.