Conjucyclic codes are an important and special family of classical error-correcting codes, which have been used to construct binary quantum error-correcting codes (QECCs). However, at present, the research on the conjucyclic codes is extremely insufficient. This paper will explore the algebraic structure of additive conjucyclic codes over $\mathbb{F}_{q^{2}}$ for the first time. Mainly via the trace function from $\mathbb{F}_{q^{2}}$ down $\mathbb{F}_{q}$, we will firstly build an isomorphic mapping between $q^2$-ary additive conjucyclic codes and $q$-ary linear cyclic codes. Since the mapping preserves the weight and orthogonality, then the dual structure of these codes with respect to the alternating inner product will be described. Then a new construction of QECCs from conjucyclic codes can be obtained. Finally, the enumeration of $q^2$-ary additive conjucyclic codes of length $n$ and the explicit forms of their generator and parity-check matrices will be determined.

翻译：共循环码是一类重要且特殊的经典纠错码，已被用于构建二进制量子纠错码（QECCs）。然而，目前对共循环码的研究仍然极其不足。本文首次探索了$\mathbb{F}_{q^{2}}$上加性共循环码的代数结构。主要通过从$\mathbb{F}_{q^{2}}$到$\mathbb{F}_{q}$的迹函数，我们首先建立$q^2$元加性共循环码与$q$元线性循环码之间的同构映射。由于该映射保持重量和正交性，因此将描述这些码关于交错内积的对偶结构。进而可以基于共循环码得到量子纠错码的一种新构造。最后，将确定长度为$n$的$q^2$元加性共循环码的计数及其生成矩阵和校验矩阵的显式形式。