Algebraic methods for parameterized codes and invariants of vanishing ideals over finite fields

TL;DR

This paper investigates the algebraic structure of vanishing ideals over finite fields, introduces parameterized codes from these structures, and provides methods to compute their parameters, with specific results for codes from graph-based parameterizations.

Contribution

It demonstrates that the vanishing ideal I(X) is a lattice ideal and develops algebraic techniques to analyze the parameters of associated parameterized codes, especially those from graph structures.

Findings

I(X) is a lattice ideal.

Methods to compute code dimension, length, and minimum distance.

Upper bounds for minimum distance in non-bipartite graph cases.

Abstract

Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length and to make our results more precise. If the graph is non-bipartite, we show an upper bound for the minimum distance. We also study the underlying geometric structure of X.