GMD functions for scheme-based linear codes and algebraic invariants of Geramita ideals

12/16/2018

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by   Susan M. Cooper, et al.

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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 Tohăneanu--Van Tuyl and Eisenbud-Green-Harris.