$W-\delta -\mu$ dual codes and LCD codes

We introduce a new product on the ambient space $F_q^n$ as a generalization of Euclidean, Hermitian and $\delta$ products. We give some general properties of the dual codes, relation with Euclidean duals, definition and characterization of self orthogonal, self dual, dual containing and LCD codes along with certain existence conditions. Also, we calculate the dual codes of some classes of codes like repetition, binary and $\lambda$-constacyclic codes with respect to this product. Further, we extend and analyse this notion of the product for codes over semisimple rings.