Linear codes arising from geometrical operations

We construct linear codes over the finite field ${\mathbb{F}}_q$ 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 analyze 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 ${\mathbb{F}}_2$ using these geometric methods. Thanks to the previously established geometric properties, we can precisely determine the parameters of these families.