Notes on LCD Codes Over Frobenius Rings

Let R be a commutative Frobenius local ring. A result that the injective hull of an LCD code C over R. is free of dimension 1(C), where 1(C) is the minimum over the cardinalities of the generating sets of 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 λ-constacyclic codes with λ being a unit, π(λ ² ) = 1 and λ ² ≠ 1, where π is the natural projection of R. to its residue field, are characterized, as another application of our result.