Toric Codes and Lattice Ideals

TL;DR

This paper characterizes when the vanishing ideal of a subset of a torus is a lattice ideal, relates it to subgroup structures, and explores implications for toric codes and their algebraic properties.

Contribution

It provides a complete characterization of lattice ideals associated with subgroups of tori in toric varieties and analyzes their algebraic and coding-theoretic properties.

Findings

Vanishing ideal of a subgroup is a radical homogeneous lattice ideal.

Subgroups parameterized by Laurent monomials correspond to lattice ideals.

Dimension and length of generalized toric codes on degenerate tori are computed.

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 $T_X\cong (\mathbb{F}^*_q)^n$. 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.