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 α , Y Q . We share procedures implementing them in Macaulay 2 . Finally, we give a lower bound for the minimum distance of C α , 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 H ℓ generalizing the corresponding result given by Hansen.