List decoding of a class of affine variety codes

TL;DR

This paper improves bounds on the zeros of multivariate polynomials over finite sets and applies these results to develop an enhanced list decoding algorithm for affine variety codes.

Contribution

It introduces new bounds on polynomial zeros and leverages them to create a more effective list decoding method for affine variety codes.

Findings

Derived improved bounds on the number of zeros of multivariate polynomials

Designed a list decoding algorithm with better performance for affine variety codes

Enhanced decoding capability for a broad class of affine variety codes

Abstract

Consider a polynomial $F$ in $m$ variables and a finite point ensemble $S=S_1 \times ... \times S_m$. When given the leading monomial of $F$ with respect to a lexicographic ordering we derive improved information on the possible number of zeros of $F$ of multiplicity at least $r$ from $S$. We then use this information to design a list decoding algorithm for a large class of affine variety codes.