# Generalized Hamming weights of projective Reed--Muller-type codes over graphs

## TL;DR

This paper computes the generalized Hamming weights of Reed--Muller-type codes derived from graph incidence matrices, linking these weights to graph invariants like biparticity and edge connectivity, with implications for coding theory.

## Contribution

It provides explicit formulas for generalized Hamming weights of these codes in terms of graph properties, extending previous results to arbitrary characteristic fields.

## Key findings

- Generalized Hamming weights relate to weak edge biparticity for non-bipartite graphs over fields with char ≠ 2.
- For bipartite graphs or fields with char = 2, weights relate to edge connectivity.
- Results unify graph invariants with code parameters across different field characteristics.

## Abstract

Let $G$ be a connected graph and let $\mathbb{X}$ be the set of projective points defined by the column vectors of the incidence matrix of $G$ over a field $K$ of any characteristic. We determine the generalized Hamming weights of the Reed--Muller-type code over the set $\mathbb{X}$ in terms of graph theoretic invariants. As an application to coding theory we show that if $G$ is non-bipartite and $K$ is a finite field of ${\rm char}(K)\neq 2$, then the $r$-th generalized Hamming weight of the linear code generated by the rows of the incidence matrix of $G$ is the $r$-th weak edge biparticity of $G$. If ${\rm char}(K)=2$ or $G$ is bipartite, we prove that the $r$-th generalized Hamming weight of that code is the $r$-th edge connectivity of $G$.