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We construct a class of $\mathbb{Z}_2\mathbb{Z}_4$-additive cyclic codesgenerated by pairs of polynomials, study their algebraic structures, and obtainthe generator matrix of any code in the class. Using a probabilistic method, weprove that, for any positive real number $\delta<1/3$ such that the entropy at$3\delta/2$ is less than $1/2$, the probability that the relative minimaldistance of a random code in the class is greater than $\delta$ is almost $1$;and the probability that the rate of the random code equals to $1/3$ is alsoalmost $1$. As an obvious consequence, the $\mathbb{Z}_2\mathbb{Z}_4$-additivecyclic codes are asymptotically good.

$\mathbb{Z}_2\mathbb{Z}_4$-Additive Cyclic Codes Are Asymptotically Good