In this paper, we are interested in finding an algebraic structure of conjucyclic codes of length n over the finite field ${\mathbb{F}}_4$. We show that conjucyclic codes of length n over ${\mathbb{F}}_4$ are related to binary cyclic codes of length 2n and show that there is a canonical bijective correspondence between the two sets. We illustrate how the factorization of the polynomial $x^{2n}+1$ plays a critical role in each setting. Moreover, we construct the generator and parity check matrices of conjucyclic codes of length n over ${\mathbb{F}}_4$.

中文翻译：

---

上加性共循环码的代数结构 ${\mathbb{F}}_4$

在本文中，我们有兴趣寻找有限域上长度为n的共轭码的代数结构${\mathbb{F}}_4$。我们证明长度为n的共轭码超过${\mathbb{F}}_4$它们与长度为2 n的二进制循环码有关，并且表明两组之间存在规范的双射对应。我们说明了多项式的因式分解$X^{2ñ}+\mathrm{1个}$在每种情况下都起着至关重要的作用。此外，我们构建的长度的conjucyclic码发生器和奇偶校验矩阵Ñ过${\mathbb{F}}_4$。