Hermitian codes from higher degree places

TL;DR

This paper improves bounds on the minimum distances of Hermitian algebraic-geometry codes from higher degree places by analyzing Weierstrass gap sequences and geometric methods, leading to more accurate estimates.

Contribution

It determines the Weierstrass gap sequence at degree 3 places and enhances minimum distance bounds for Hermitian codes using geometric approaches.

Findings

Improved minimum distance bounds for Hermitian codes from degree 3 places.

Explicit determination of Weierstrass gap sequence at degree 3 places.

Comparison of bounds with actual minimum distances and existing estimates.

Abstract

Matthews and Michel investigated the minimum distances in certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree 3 place, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian 1-point codes, as well as with estimates due Xing and Chen.