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 Cα,YQ. We share procedures implementing them in Macaulay2. Finally, we give a lower bound for the minimum distance of Cα,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.