On parameterized toric codes

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 C-alpha,C-YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of C-alpha,C-YQ, 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 H-l generalizing the corresponding result given by Hansen.