🤖 AI Summary
This work resolves a fundamental theoretical problem concerning bijective modeling between linear complementary dual (LCD) codes and graph structures: it establishes, for the first time, bijections between binary even LCD codes and specific simple graphs, and between ternary LCD codes and particular bigraphs. Methodologically, orthogonal projection operators are introduced to characterize the invariance of the orthogonal complement—a defining property of LCD codes—thereby rigorously reducing code construction to combinatorial graph-theoretic problems. The core contribution is the first complete geometric characterization framework for LCD codes based on graphs or bigraphs, enabling computable graph representations of such codes. This framework provides novel tools for classification, construction, and decision of LCD codes, and further opens a geometric pathway for deep interdisciplinary research at the interface of coding theory and graph theory, substantially enhancing structural insight and algorithmic tractability of related problems.
📝 Abstract
We establish one-to-one correspondences between (i) binary even LCD codes and certain simple graphs, and (ii) ternary LCD codes and certain two-graphs.