On the minimum distance and the minimum weight of Goppa codes from a quotient of the Hermitian curve

TL;DR

This paper investigates evaluation codes from quotients of the Hermitian curve, focusing on their dual minimum distance and minimum weight, providing geometric characterizations and applications to classical Goppa codes.

Contribution

It offers a geometric analysis of the dual minimum distance and minimum weight of codes from Hermitian curve quotients, including explicit descriptions of minimum-weight codewords.

Findings

Complete description of minimum-weight codewords in many cases

Geometric characterization of code supports

Application to classical Goppa codes

Abstract

In this paper we study evaluation codes arising from plane quotients of the Hermitian curve, defined by affine equations of the form $y^q+y=x^m$, $q$ being a prime power and $m$ a positive integer which divides $q+1$. The dual minimum distance and minimum weight of such codes are studied from a geometric point of view. In many cases we completely describe the minimum-weight codewords of their dual codes through a geometric characterization of the supports, and provide their number. Finally, we apply our results to describe Goppa codes of classical interest on such curves.