Toric Codes and Lattice Ideals

Abstract

Let $X$ be a complete simplicial toric variety over a finite field ${\mathbb{F}}_q$ with homogeneous coordinate ring $S={\mathbb{F}}_q[x_1,\dots ,x_r]$ and split torus $T_X\cong ({\mathbb{F}}_q^{\ast }{)}^n$. 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.

The author is supported by TÜBİTAK Project No:114F094