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

In this paper, we construct a class of $F_2{F}_2\left[u^2\right]F_2\left[u^3\right]$-additive cyclic codes generated by 3-tuples of polynomials, where $F_2$ is the binary field, $F_2\left[u^2\right]=F_2+u{F}_2$ ($u^2=0$) and $F_2\left[u^3\right]=F_2+u{F}_2+u^2{F}_2$ ($u^3=0$). 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 $0<\delta <1$ be a real number and $k,l$ and t be pairwise co-prime positive odd integers such that the entropy at $\frac{(k+l+t)\delta }{3}$ is less than $\frac{2}{3}$, we prove that the relative minimum homogeneous distances converge to δ, and the rates of the random codes converge to $\frac{1}{k+l+t}$. Consequently, $F_2{F}_2\left[u^2\right]F_2\left[u^3\right]$-additive cyclic codes are asymptotically good.