Generalized minimum distance functions and algebraic invariants of Geramita ideals

Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function δI(d,r) of a graded ideal I in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that δI is non-decreasing as a function of r and non-increasing as a function of...

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Bibliographic Details
Published inAdvances in applied mathematics Vol. 112; p. 101940
Main Authors Cooper, Susan M., Seceleanu, Alexandra, Tohăneanu, Ştefan O., Pinto, Maria Vaz, Villarreal, Rafael H.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.01.2020
Subjects
Basic parameters
Complete intersections
Degree
Fat points
Generalized minimum distance
Hilbert function
Linear codes
Reed–Muller-type codes
Unmixed ideal
secondary
Basic parameters
Generalized minimum distance
Fat points
Degree
Unmixed ideal
Linear codes
Hilbert function
Reed–Muller-type codes
Complete intersections
primary
Online AccessGet full text
ISSN0196-8858
1090-2074
DOI10.1016/j.aam.2019.101940

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Summary:Motivated by notions from coding theory, we study the generalized minimum distance (GMD) function δI(d,r) of a graded ideal I in a polynomial ring over an arbitrary field using commutative algebraic methods. It is shown that δI 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 δI 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 Tohăneanu–Van Tuyl and Eisenbud-Green-Harris.
ISSN:0196-8858
1090-2074
DOI:10.1016/j.aam.2019.101940