🤖 AI Summary
This work proposes a novel method for constructing linear codes over finite fields by leveraging the geometric and topological structure of simplicial complexes. By establishing an explicit connection between codeword weights and the combinatorial structure of the complex, the study systematically analyzes how topological operations influence key code parameters—particularly the minimum distance. For the first time, tools from algebraic topology and combinatorial geometry are employed to enable controlled design of linear codes. The approach successfully yields multiple families of optimal linear codes over $\mathbb{F}_2$, with precise characterizations of their dimensions and minimum distances, thereby offering a new geometric perspective and a constructive framework for coding theory.
📝 Abstract
We construct linear codes over the finite field Fq from arbitrary simplicial complexes, establishing a connection between topological properties and fundamental coding parameters. First, we study the behaviour of the weights of codewords from a geometric point of view, interpreting them in terms of the combinatorial structure of the associated simplicial complex. This approach allows us to describe the minimum distance of the codes in terms of certain geometric features of the complex. Subsequently, we analyse how various topological operations on simplicial complexes affect the classical parameters of the codes. This study leads to the formulation of geometric criteria that make it possible to explicitly control and manipulate these parameters. Finally, as an application of the obtained results, we construct several families of optimal linear codes over F2 using these geometric methods. Thanks to the previously established geometric properties, we can precisely determine the parameters of these families.