ℤ4-Double Cyclic Codes Are Asymptotically Good

We construct a class of <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula>-double cyclic codes generated by pairs of polynomials. Based on the probabilistic method, we prove the asymptotic properties of this class of codes: for any positive real number <inline-formula> <tex-math notation="LaTeX">$0< \delta < 1$ </tex-math></inline-formula> such that the 4-ary entropy at <inline-formula> <tex-math notation="LaTeX">$\frac {k+l}{2}\delta $ </tex-math></inline-formula> is less than <inline-formula> <tex-math notation="LaTeX">$\frac {1}{4}$ </tex-math></inline-formula>, the rate of the random code is convergent to <inline-formula> <tex-math notation="LaTeX">$\frac {1}{k+l}$ </tex-math></inline-formula> and the relative distance of the code is convergent to <inline-formula> <tex-math notation="LaTeX">$\delta $ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$l$ </tex-math></inline-formula> are pairwise coprime positive odd integers. As a result, the <inline-formula> <tex-math notation="LaTeX">$\mathbb {Z}_{4}$ </tex-math></inline-formula>-double cyclic codes are asymptotically good.