Toric Codes and Lattice Ideals
share
Let X be a complete simplicial toric variety over a finite field F_q with homogeneous coordinate ring S=F_q[x_1,...,x_r] and split torus T_X (F^*_q)^n. We prove that vanishing ideal of a subset Y of the torus T_X is a lattice ideal if and only if Y is a subgroup. We show that these subgroups are exactly those subsets that are parameterized by Laurents monomials. We give an algorithm for determining this parametrization if the subgroup is the zero locus of a lattice ideal in the torus. We also show that vanishing ideals of subgroups of T_X are radical homogeneous lattice ideals of dimension r-n. We identify the lattice corresponding to a degenerate torus in X and completely characterize when its lattice ideal is a complete intersection. We compute dimension and length of some generalized toric codes defined on these degenerate tori.
READ FULL TEXT