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arXiv · 1911.09350

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

Yun Fan·Hualu Liu
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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.

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Yun Fan, Hualu Liu. 2019-11-21. $\mathbb{Z}_2\mathbb{Z}_4$-Additive Cyclic Codes Are Asymptotically Good. https://arxiv.org/abs/1911.09350

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