  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 $\mathbb{F}_{q}$.
