Geometric representation of binary codes and computation of weight enumerators

TL;DR

This paper introduces a geometric triangular configuration for linear binary codes that captures their weight enumerator, extending the theory of Pfaffian orientations from graphs to codes, and establishes a bijection with perfect matchings.

Contribution

It constructs a geometric representation of binary codes via triangular configurations and links their cycle spaces to perfect matchings, advancing code theory and combinatorial geometry.

Findings

Provides a geometric construction for weight enumerators of binary codes.

Establishes a bijection between cycle spaces and perfect matchings.

Extends Pfaffian orientation theory from graphs to binary codes.

Abstract

For every linear binary code $C$, we construct a geometric triangular configuration $\Delta$ so that the weight enumerator of $C$ is obtained by a simple formula from the weight enumerator of the cycle space of $\Delta$. The triangular configuration $\Delta$ thus provides a geometric representation of $C$ which carries its weight enumerator. This is the initial step in the suggestion by M. Loebl, to extend the theory of Pfaffian orientations from graphs to general linear binary codes. Then we carry out also the second step by constructing, for every triangular configuration $\Delta$, a triangular configuration $\Delta'$ and a bijection between the cycle space of $\Delta$ and the set of the perfect matchings of $\Delta'$.