On the vanishing ideal of an algebraic toric set and its parametrized linear codes (Q2909802)

Let \(K\) be a finite field and \(X\) an algebraic toric set in a toric variety \({\mathbb{P}}^{s-1}\). The vanishing ideal \(I(X)\) is the ideal of \(S=K[t_1,\dots,t_s]\) generated by the homogeneous polynomials which vanish on \(X\). Let \(d\geq 0\) be fixed and define \(f_0(t_1,\dots,t_s)=t_1^d\). The evaluation map \(\mathrm{ev}_d\) is the map \(\mathrm{ev}_d :S_d\to K^m\) given by \(f\mapsto (f(P_1)/f_0(P_1),\dots , f(P_m)/f_0(P_m))\), where \(X=\{[P_1],\dots, [P_m]\}\). The image of \(ev_d\), denoted \(C_X(d)\), is a linear code. It is this code which is studied here. In this paper, an upper bound is proven for the minimum distance of these codes is given, along with some general facts about \(I(X)\).