The minimum distance of parameterized codes on projective tori (Q429761)

Let \(K\) be a finite field with \(q\) elements and \(X\) be an \textit{algebraic toric set} on \(\mathbb{P}^{s-1}\), that is, \[ X := \left\{ \left[ x_1^{v_{11}}\dots x_n^{v_{1n}}, \dots, x_1^{v_{s1}}\dots x_n^{v_{sn}} \right] \in \mathbb{P}^{s-1} \mid x_i \in K^{*} \text{ for all } i\right\}, \] where \(v_1 := (v_{11},\dots, v_{1n}),\dots, v_s := (v_{s1},\dots, v_{sn})\) is a given subset of \(\mathbb{N}^n\). 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 \textit{evaluation map} \(ev_d\) is the map \[ ev_d: S_d \to K^{|X|}, \qquad f \mapsto \left( \frac{f(P_1)}{f_0(P_1)},\dots, \frac{f(P_m)}{f_0(P_m)} \right), \] where \(X = \left\{[P_1],\dots, [P_m]\right\}\). The image of \(ev_d\), denoted \(C_X(d)\) is a linear code called an \textit{evaluation code} associated to \(X\) or a \textit{projective Reed-Muller code} over the set \(X\). The \textit{minimum distance} of the linear code \(C_X(d)\) is a basic parameter given by \[ \delta_d = \min\{\|v\| \mid 0 \neq v \in C_X(d) \}, \] where \(\|v\|\) denotes the number of non-zero entries of \(v\). In this paper, it is shown that \(I(X)\) is a \textit{complete intersection} if and only if \(X\) is a projective torus. In this case, the authors determine the minimum distance of any parameterized linear code arising from \(X\).