A zero-dimensional approach to Hermitian codes

TL;DR

This paper investigates Hermitian codes using algebraic geometry, providing a cohomological framework to understand their dual minimum distance and describing the geometry of their minimum-weight codewords.

Contribution

It introduces a cohomological characterization of dual minimum distance for Hermitian codes and analyzes the geometry of minimum-weight codewords.

Findings

Cohomological description of dual minimum distance

Geometric characterization of minimum-weight codewords

Applicable to Hermitian s-point codes

Abstract

We study the algebraic geometry of a family of evaluation codes from plane smooth curves defined over any field. In particular, we provide a cohomological characterization of their dual minimum distance. After having discussed some general results on zero-dimensional subschemes of the plane, we focus on the interesting case of Hermitian $s$-point codes, describing the geometry of their dual minimum-weight codewords.