dual codes and LCD codes
arXiv:2606.13467
The paper defines a new product on vector spaces over finite fields that generalizes Euclidean, Hermitian, and δ products, and uses it to study dual, self‑orthogonal, self‑dual, dual‑containing, and LCD codes, including explicit duals for several code families and extensions to semisimple rings.
Abstract
We introduce a new product on the ambient space as a generalization of Euclidean, Hermitian and 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 -constacyclic codes with respect to this product. Further, we extend and analyse this notion of the product for codes over semisimple rings.