Additive conjucyclic codes over $\F_{q^2}$ are closed under the conjugated cyclic shift and play an important role in constructing quantum error-correcting codes (QECCs). However, a systematic algebraic theory for such codes over general finite fields has been lacking. In this paper, we develop a unified framework by establishing a trace-based $\F_q$-linear isomorphism between $\F_{q^2}^n$ and $\F_q^{2n}$. This correspondence shows that additive conjucyclic codes of length $n$ correspond bijectively to $q$-ary linear cyclic codes of length $2n$, translating their structural analysis to the well-understood setting of cyclic codes. Using this isomorphism, we determine the enumeration of such codes and give explicit forms of their generator matrices. We then introduce an alternating inner product on $\F_{q^2}^n$, which is shown to be compatible with the symplectic inner product on $\F_q^{2n}$ under the trace isomorphism. Based on this inner product, we characterize the dual-containing condition for additive conjucyclic codes and derive explicit parity-check matrices. Finally, we construct $q$-ary QECCs from dual-containing additive conjucyclic codes. Our results unify and generalize previous studies on quaternary additive conjucyclic codes and present a construction method for $q$-ary QECCs from additive conjucyclic codes, together with an illustrative example.

翻译：加法共轭循环码在$\F_{q^2}$上对共轭循环移位封闭，并在构造量子纠错码（QECCs）中发挥重要作用。然而，目前尚缺乏针对一般有限域上此类码的系统代数理论。本文通过建立$\F_{q^2}^n$与$\F_q^{2n}$之间基于迹的$\F_q$-线性同构，构建了一个统一框架。该对应表明，长度为$n$的加法共轭循环码与长度为$2n$的$q$元线性循环码一一对应，从而将此类码的结构分析转化为已充分研究的循环码理论。利用该同构，我们确定了此类码的计数问题，并给出了其生成矩阵的显式形式。随后，我们引入了$\F_{q^2}^n$上的交错内积，并证明在迹同构下该内积与$\F_q^{2n}$上的辛内积相容。基于此内积，我们刻画了加法共轭循环码的对偶包含条件，并推导了显式校验矩阵。最后，我们从对偶包含加法共轭循环码构造了$q$元量子纠错码。本文结果统一并推广了此前关于四元加法共轭循环码的研究，提出了一种基于加法共轭循环码构造$q$元QECCs的方法，并给出了示例说明。