On parameterized toric codes

Abstract

Let X be a complete simplicial toric variety over a finite field with a split torus $T_{X}$. For any matrix Q, we are interested in the subgroup $Y_{Q}$ of $T_{X}$ parameterized by the columns of Q. We give an algorithm for obtaining a basis for the unique lattice L whose lattice ideal $I_{L}$ is $I(Y_{Q})$. We also give two direct algorithmic methods to compute the order of $Y_{Q}$, which is the length of the corresponding code ${{\mathcal {C}}}_{\mathbf{\alpha },Y_Q}$. We share procedures implementing them in ${\texttt {Macaulay2}}$. Finally, we give a lower bound for the minimum distance of ${{\mathcal {C}}}_{\mathbf{\alpha },Y_Q}$, taking advantage of the parametric description of the subgroup $Y_Q$. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces ${\mathcal {H}}_{\ell }$ generalizing the corresponding result given by Hansen.