On the Hermitian curve, its intersections with some conics and their applications to affine-variety codes and Hermitian codes

TL;DR

This paper explores the intersections of the Hermitian curve with conics, providing geometric characterizations of small-weight codewords in Hermitian codes, and offers explicit formulas for counting minimum and second-weight codewords.

Contribution

It introduces a method to construct ideals for affine-variety codes, classifies intersections with lines and parabolas, and derives explicit formulas for codeword counts in Hermitian codes.

Findings

Complete classification of Hermitian curve intersections with lines and parabolas

Explicit formulas for counting minimum-weight codewords

Geometric characterizations of small-weight codewords

Abstract

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We classify completely the intersections of the Hermitian curve with lines and parabolas (in the $\mathbb{F}_{q^2}$ affine plane). Starting from both results, we are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any $\mathbb{F}_{q^2}$. From the geometric characterization, we obtain explicit formulae. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with $d\leq q$ and all second-weight codewords for distance-$3,4$ codes.