GMD functions for scheme-based linear codes and algebraic invariants of Geramita ideals.
Susan Cooper, Alexandra Seceleanu, Ştefan Ovidiu I. Tohăneanu, Maria Vaz Pinto, Rafael H. Villarreal · arXiv (Cornell University) · 2018
Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function $δ_I(d,r)$ of a graded ideal $I$ in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that $δ_I$ 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 $δ_I$ 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 Tohaneanu--Van Tuyl and Eisenbud-Green-Harris.