The dual geometry of Hermitian two-point codes

TL;DR

This paper explores the algebraic geometry underlying Hermitian two-point codes, revealing geometric structures of their dual codes and characterizing minimum-weight codewords as collinear point sets.

Contribution

It provides a geometric interpretation of the dual minimum distance and explicitly characterizes dual codewords for Hermitian two-point codes.

Findings

Dual minimum distance has a geometric nature.

Minimum-weight dual codewords often form collinear point sets.

Explicit geometric descriptions of dual code supports are provided.

Abstract

In this paper we study the algebraic geometry of any two-point code on the Hermitian curve and reveal the purely geometric nature of their dual minimum distance. We describe the minimum-weight codewords of many of their dual codes through an explicit geometric characterization of their supports. In particular, we show that they appear as sets of collinear points in many cases.