On the regularity index of the minimum distance function in projective nested Cartesian codes

TL;DR

This paper derives a formula for the regularity index of the minimum distance function in projective nested Cartesian codes, linking it to the v-number of the vanishing ideal and characterizing Cayley–Bacharach properties.

Contribution

It provides a new explicit formula for the regularity index and an arithmetical criterion for Cayley–Bacharach property in this coding context.

Findings

Derived a formula for the regularity index of the minimum distance function.

Connected the regularity index to the v-number of the vanishing ideal.

Provided an arithmetical criterion for Cayley–Bacharach property.

Abstract

Let $X$ be a projective nested product of fields and let $\delta_X(d)$ be the minimum distance in degree $d\geq 1$ of the projective nested Cartesian code $C_X(d)$. The regularity index ${\rm reg}(\delta_X)$ of the minimum distance function $\delta_X$ is the minimum integer $d_0\geq 0$ such that $\delta_X(d)=1$ for $d\geq d_0$. We give a formula for ${\rm reg}(\delta_X)$ by determining an indicator function of least degree for each point of $X$ and using the fact that ${\rm reg}(\delta_X)$ is the ${\rm v}$-number of the vanishing ideal $I_X$ of $X$. Then we give an arithmetical criterion that characterizes when $X$ is Cayley--Bacharach.