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.

令X是一个有限的域上具有圆环\（T_ {X} \）的完全简单复曲面变体。对于任何矩阵Q，我们感兴趣的是子组\（Y_ {Q} \）的\（T_ {X} \）通过的列参数Q。我们给出了一种算法，用于获得唯一格L的基础，该唯一格L的格理想\（I_ {L} \）为\（I（Y_ {Q}）\）。我们还提供了两种直接的算法方法来计算\（Y_ {Q} \）的顺序，即相应代码\（{{\ mathcal {C}}} _ {\ mathbf {\ alpha}，Y_Q的长度} \）。我们共享实施这些程序的程序\（{\ texttt {Macaulay2}} \）。最后，我们得到下界的最小距离\（{{\ mathcal {C}}} _ {\ mathbf {\阿尔法}，Y_Q} \） ，取子组的参数化描述的优点Y_Q \（\ ）。作为应用，我们计算了Hirzebruch曲面\（{\ mathcal {H}} _ {\ ell} \）上复曲面代码的主要参数，从而概括了汉森给出的相应结果。