Galois lcd codes over rings

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.