On parameterised toric codes

TL;DR

This paper introduces algorithms for analyzing parameterized subgroups of toric varieties over finite fields, enabling the computation of their algebraic properties and code parameters, with applications to toric codes on Hirzebruch surfaces.

Contribution

It provides new algorithms to determine the lattice basis, order, and minimum distance bounds of parameterized toric subgroups, extending previous work on toric codes.

Findings

Algorithms for lattice basis and subgroup order computation

Procedures implemented in Macaulay2

Lower bounds for minimum distance of toric codes

Abstract

Let $X$ be a complete simplicial toric variety over a finite field with a split torus $T_X$. For any matrix $Q$, we are interested in the subgroup $Y_Q$ of $T_X$ parameterized by the columns of $Q$. We give an algorithm for obtaining a basis for the unique lattice $L$ whose lattice ideal $I_L$ is $I(Y_Q)$. We also give two direct algorithmic methods to compute the order of $Y_Q$, which is the length of the corresponding code ${\cC}_{\aa,Y_Q}$. We share procedures implementing them in \verb|Macaulay2|. Finally, we give a lower bound for the minimum distance of ${\cC}_{\aa,Y_Q}$, taking advantage of the parametric description of the subgroup $Y_Q$. As an application, we compute the main parameters of the toric codes on Hirzebruch surfaces $\cl H_{\ell}$ generalizing the corresponding result given by Hansen.