Linear codes arising from geometrical operation

TL;DR

This paper introduces a geometric approach to constructing linear codes from simplicial complexes, linking topological features to code parameters and enabling the design of optimal codes over F2.

Contribution

It establishes a novel connection between topological properties of simplicial complexes and the parameters of linear codes, allowing explicit control and construction of optimal codes.

Findings

Describes minimum distance in terms of geometric features

Analyzes effects of topological operations on code parameters

Constructs families of optimal codes over F2

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.