Rational points on cubic surfaces and AG codes from the Norm-Trace curve

TL;DR

This paper characterizes intersections between the Norm-Trace curve and cubic polynomials over finite fields, and analyzes rational points on cubic surfaces to inform algebraic geometry codes.

Contribution

It generalizes previous intersection results and provides explicit bounds for rational points on cubic surfaces, enhancing understanding of AG codes from these curves.

Findings

Complete characterization of intersections between Norm-Trace and cubic polynomial curves

Explicit bounds for rational points on cubic surfaces over finite fields

Detailed information on the weight spectrum of certain AG codes

Abstract

In this paper we give a complete characterization of the intersections between the Norm-Trace curve over $\mathbb{F}_{q^3}$ and the curves of the form $y=ax^3+bx^2+cx+d$, generalizing a previous result by Bonini and Sala, providing more detailed information about the weight spectrum of one-point AG codes arising from such curve. We also derive, with explicit computations, some general bounds for the number of rational points on a cubic surface defined over $\mathbb{F}_{q}$.