🤖 AI Summary
This work proposes a unified $W$-$\delta$-$\mu$ inner product framework that generalizes Euclidean, Hermitian, and $\delta$-inner products to linear codes over finite fields and semisimple rings. Within this framework, the authors systematically define and analyze fundamental code structures—including dual codes, self-orthogonal codes, self-dual codes, dual-containing codes, and LCD (linear complementary dual) codes—and establish, for the first time, existence conditions for such codes over semisimple rings. The study derives explicit dual descriptions under the new inner product for classical code families such as repetition codes, binary codes, and $\lambda$-constacyclic codes, thereby extending classical duality theory and providing a theoretical foundation and constructive tools for emerging code constructions like LCD codes.
📝 Abstract
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.