We introduce a new product on the ambient space $F_q^n$ as a generalization of Euclidean, Hermitian and $δ$ 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 $λ$-constacyclic codes with respect to this product. Further, we extend and analyse this notion of the product for codes over semisimple rings.

翻译：我们引入了一个在环境空间$F_q^n$上的新乘积，作为欧几里得乘积、埃尔米特乘积与$δ$乘积的推广。我们给出了该对偶码的一些一般性质、与欧几里得对偶的关系，以及自正交码、自对偶码、包含对偶码和LCD码的定义与刻画，同时给出了若干存在性条件。此外，我们还针对重复码、二元码和$λ$-常循环码等几类码，计算了它们在此乘积意义下的对偶码。进一步，我们将该乘积概念扩展并分析至半单环上的码。