Notes on LCD Codes over Frobenius Rings

Let \mathcal {R} be a commutative Frobenius local ring. A result that the injective hull of an LCD code \mathcal {C} over \mathcal {R} is free of dimension \ell (\mathcal {C}) , where \ell (\mathcal {C}) is the minimum over the cardinalities of the generating sets of \mathcal {C} , is proved in this correspondence. Applying this result, a concise proof for the main result in a recent paper by Sanjit Bhowmick et al. is derived. Furthermore, the LCD \lambda constacyclic codes with \lambda being a unit, \pi (\lambda {2})=1 and \lambda {2} \neq 1 , where \pi is the natural projection of \mathcal {R} to its residue field, are characterized, as another application of our result.