Algebraic codes, Horn's problem and Gromov-Witten invariants

TL;DR

This paper connects algebraic coding theory, Horn's problem, and Gromov-Witten invariants, providing new characterizations and algorithms for codes on rational curves over finite fields.

Contribution

It introduces a novel approach linking the Horn problem to Gromov-Witten invariants and classifies algebraic codes on the normal rational curve with an algorithm for generator computation.

Findings

Characterization of coefficients in Kronecker products via Gromov-Witten invariants

Classification of algebraic codes on the normal rational curve

Algorithm for computing generators of code ideals

Abstract

We study the Horn problem in the context of algebraic codes on a smooth projective curve defined over a finite field, reducing the problem to the representation theory of the special linear group $SL(2,F_q)$. We characterize the coefficients that appear in the Kronecker product of symmetric functions in terms of Gromov-Witten invariants of the Hilbert scheme of points in the plane. In addition we classify all the algebraic codes defined over the normal rational curve providing an algorithm to compute set of generators of the ideal associated to any algebraic code constructed on the NRC over an extension $\mathbb{F}_{q^{n}}$ of $\mathbb{F}_{q}$.