On parameterised 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 \({\cC}_{\aa,Y_Q}\). We share procedures implementing them in \verb|Macaulay2|. Finally, we give a lower bound for the minimum distance of \({\cC}_{\aa,Y_Q}\), 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 \(\cl H_{\ell}\) generalizing the corresponding result given by Hansen.