# Generalized minimum distance functions and algebraic invariants of Geramita ideals

## TL;DR

This paper explores the generalized minimum distance function of graded ideals, especially Geramita ideals, revealing monotonic properties and their relation to algebraic invariants, with implications for coding theory and conjectures.

## Contribution

It introduces new properties of the GMD function for Geramita ideals and connects algebraic invariants to coding theory concepts, addressing open conjectures.

## Key findings

- GMD function is non-decreasing in r and non-increasing in d.
- For finite field ideals, GMD decreases with d until stabilization.
- Geramita ideals are graded, unmixed, 1-dimensional with linear associated primes.

## Abstract

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $\delta_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $\delta_I$ is non-decreasing as a function of $r$ and non-increasing as a function of $d$. For vanishing ideals over finite fields, we show that $\delta_I$ is strictly decreasing as a function of $d$ until it stabilizes. We also study algebraic invariants of Geramita ideals. Those ideals are graded, unmixed, $1$-dimensional and their associated primes are generated by linear forms. We also examine GMD functions of complete intersections and show some special cases of two conjectures of Toh\u{a}neanu--Van Tuyl and Eisenbud-Green-Harris.