We define the generalized Galois inner product for codes over Frobenius rings, and then present the Galois linear complementary dual (LCD) codes over such rings which generalize the Galois LCD codes over finite fields defined in a recent reference. We describe the judging criterions for a generalized Galois LCD code over a Frobenius ring by introducing the rank of a matrix over such a ring. We also present necessary and sufficient conditions for the generalized Galois LCD codes over chain rings. The structure of the generalized Galois LCD codes over chain rings can be used to construct Galois LCD codes over finite fields. By using the structure of the generalized Galois LCD codes, we also give the description of constacyclic LCD codes over a class of chain rings.

中文翻译：

---

环上的伽罗瓦 LCD 代码

我们为 Frobenius 环上的代码定义了广义 Galois 内积，然后在这些环上展示了 Galois 线性互补对偶 (LCD) 代码，该代码在最近的参考文献中定义的有限域上推广了 Galois LCD 代码。我们通过在 Frobenius 环上引入矩阵的秩来描述在 Frobenius 环上的广义 Galois LCD 代码的判断标准。我们还提出了链环上广义伽罗瓦 LCD 代码的充要条件。链环上的广义 Galois LCD 码的结构可用于构造有限域上的 Galois LCD 码。利用广义伽罗瓦LCD码的结构，我们还给出了一类链环上的恒环LCD码的描述。