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

Abstract

We construct a class of ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes generated by pairs of polynomials, study their algebraic structures, and obtain the generator matrix of any code in the class. Using a probabilistic method, we prove 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 minimal distance 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 also almost $1$. As an obvious consequence, the ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive cyclic codes are asymptotically good.

References in corpus (2)

• Quasi-cyclic Codes of Index 1.5
• Z2Z4-Additive Cyclic Codes: Kernel and Rank